Contributions to Algebraic Geometry: Impanga Lecture Notes

EMS Series of Congress Reports

EMS Congress Reports publishes volumes originating from conferences or seminars focusing on any field of pure or applied mathematics. The individual volumes include an introduction into their subject and review of the contributions in this context. Articles are required to undergo a refereeing process and are accepted only if they contain a survey or significant results not published elsewhere in the literature. Previously published: Trends in Representation Theory of Algebras and Related Topics, Andrzej Skowro´nski (ed.) K-Theory and Noncommutative Geometry, Guillermo Cortiñas et al. (eds.) Classification of Algebraic Varieties, Carel Faber, Gerard van der Geer and Eduard Looijenga (eds.) Surveys in Stochastic Processes, Jochen Blath, Peter Imkeller and Sylvie Rœlly (eds.) Representations of Algebras and Related Topics, Andrzej Skowro´nski and Kunio Yamagata (eds.)

Contributions to Algebraic Geometry Impanga Lecture Notes Piotr Pragacz Editor

Editor: Piotr Pragacz Institute of Mathematics Polish Academy of Sciences ul. S´ niadeckich 8 00-956 Warszawa Poland E-mail: [email protected]

2010 Mathematics Subject Classification: 11S15, 13D10, 14-02, 14B05, 14B12, 14C17, 14C20, 14C35, 14D06, 14D15, 14D20, 14D23, 14E15, 14E30, 14F43, 14H10, 14H40, 14H42, 14J17, 14J28, 14J30, 14J32, 14J50, 14J70, 14K10, 14K25, 14L30, 14M15, 14M17, 14M25, 14N10, 14N15, 32G10, 32Q45, 34A30, 53D05, 55N91; 01-02, 01A70, 05E05, 11S85, 13A35, 13D10, 14C30, 14F18, 14J26, 14J32, 14N20, 19D55, 32S25, 53D20, 57R45 Key words: K3 surface, Enriques surface, Calabi–Yau threefold, linear system, Seshadri constant, multiplier ideal, differential form, log canonical treshold, Mori theory, canonical ring, deformation of morphism, Hodge numbers, logarithmic differential form, Prym variety, moduli space, syzygy, Wro´n ski determinant, linear ODE, ramification locus, Schubert calculus, Grassmannian, Schubert variety, Schur function, singularity, Thom polynomial, P-ideal, equivariant localization, toric variety, symplectic manifold, toric stack, symplectic quotient, equivariant cohomology, Bloch group, field extension, norm

ISBN 978-3-03719-114-9 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2012 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the authors’ TEX files: I. Zimmermann, Freiburg Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

A tribute to Oscar Zariski

Preface This book is an outgrowth of the Impanga Conference on Algebraic Geometry, held at the Banach Center in B¸edlewo on July 4–10, 2010. For detailed information about this event, see http://www.impan.pl/~impanga/school The articles in this volume cover a broad range of topics in algebraic geometry: classical varieties, linear systems, birational geometry, Minimal Model Program, moduli spaces, toric varieties, enumerative theory of singularities, equivariant cohomology and arithmetic questions. The book is based on contributions by conference speakers and participants, including two articles by mathematicians who were unable to attend the meeting. Here is a brief summary of the content of the volume. The article by Klaus Altmann et al. describes the language of polyhedral divisors, which is used when working with T -varieties. This language is explained in parallel to the well established theory of toric varieties. In addition to basic constructions, subjects discussed include: singularities, separatedness and properness, divisors and intersection theory, cohomology, Cox rings, polarizations and equivariant deformations. Dave Anderson gives an extensive introduction to equivariant cohomology in algebraic geometry. The first part is an overview, including basic definitions and examples. In the second part, the author discusses one of the most useful aspects of the theory: the possibility of localizing at fixed points without losing information. The third part focuses on Grassmannians, and describes some recent positivity results for their equivariant cohomology rings. The article by Thomas Bauer et al. discusses open problems in the theory of linear systems. The discussion revolves around the speciality and the postulation problems as well as the containment problems for various powers of ideals. The main motivations come from the Harbourne–Hirschowitz, Nagata and the Bounded Negativity Conjectures. Gergely Bérczi investigates the moduli space of holomorphic map germs from the complex line into complex compact manifolds. Two major applications are discussed: to Thom polynomials for Morin singularities in global singularity theory, and to the Green–Griffiths conjecture in the theory of hyperbolic algebraic varieties. The article by Paolo Cascini and Vladmir Lazi´c gives an introduction to some recent developments in Mori theory aimed for a less experienced reader. In particular, the authors’ recent direct proof of the finite generation of the canonical ring is discussed. Sławomir Cynk and Sławomir Rams present various results on invariants of a resolution of a singular projective hypersurface. In particular, they prove some defect-type formulas. The proofs use, in an essential way, the sheaves of logarithmic differential forms.

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Gavril Farkas studies the geometry of the moduli space of Prym varieties. Several applications of Prym varieties in algebraic geometry are presented. The exposition begins with a historical discussion of the life and achievements of Friedrich Prym. Topics treated in subsequent sections include singularities and Kodaira dimension of the moduli space, syzygies of Prym-canonical embedding and the geometry of the moduli space Rg for small genus. The article by Letterio Gatto and Inna Scherbak concerns Wro´nskians in algebraic geometry. Wro´nskians, usually introduced in standard courses in Ordinary Differential Equations (ODE), are a useful tool in algebraic geometry to detect ramification loci of linear systems. The authors describe some “materializations” of the Wro´nskian, and of its close relatives, the generalized Wro´nskians (labelled by partitions), in algebraic geometry. Emphasis is put on the relationship between Schubert calculus and ODE. The article by Kevin Hitchinson and Masha Vlasenko supports Burt Totaro’s expectation that Bloch’s higher Chow groups of a field k can be computed using a very small class of affine algebraic varieties (linear spaces in the right coordinates), whereas the current definition uses essentially all algebraic cycles in affine space. The authors consider a simple modification of CH2 .Spec.k/; 3/ using only linear subvarieties in affine spaces, and show that it maps surjectively to the Bloch group B.k/ for any infinite field k. They also describe the kernel of this map. Andreas Hochenegger and Frederik Witt describe some constructions in symplectic toric geometry. The starting point is the Delzant construction of a symplectic toric manifold from a smooth polytope. The article then discusses the refinements for rational but not necessarily smooth polytopes due to Lerman and Tolman, leading to symplectic toric orbifolds or, more generally, symplectic toric DM stacks. The authors show that the latter stacks are isomorphic to the stacks obtained by Borisov et al. whenever the stacky fan is induced by a polytope. Clemens Jörder and Stefan Kebekus address the following question. Let f W Y ! X be the inclusion map of a compact reduced subspace of a complex manifold, and let F  TX be a subsheaf of the tangent bundle which is closed under the Lie bracket, but not necessarily a sheaf of OX -algebras. When do infinitesimal deformations of f which are induced by F lift to positive-dimensional deformations of f , where f is deformed along the sheaf F ? In the case where X is complex-symplectic and F is the sheaf of locally Hamiltonian vector fields, this partially reproduces known results on unobstructedness of deformations of Lagrangian submanifolds. Michał Kapustka studies the variety of isotropic five-spaces of a degenerate fourform in a seven-dimensional vector space. It is a degeneration of the adjoint variety of the simple Lie group G2 . It is also the image of P5 under the map induced by the system of quadrics containing a twisted cubic. Using degenerations of this twisted cubic to three lines, the author constructs geometric transitions between Calabi–Yau threefolds. The article by Mateusz Michałek contains the notes from the lectures of Stefan Kebekus on hyperbolicity properties of moduli stacks and generalizations of Shafarevich hyperbolicity to higher dimensions. In the first part, various sheaves of differential forms on singular algebraic varieties are discussed. The second part collects some

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facts on the Minimal Model Program and logarithmic sheaves. The last part is devoted mainly to hyperbolicity of moduli. It also discusses: pull-backs of differerentials, a generalization of the Bogomolov–Sommese vanishing, and a special case of the Lipman–Zariski conjecture. Mircea Musta¸ta˘ investigates an invariant of singularities, which plays an important role in birational geometry: the log canonical threshold. After its definition, properties and examples, the author describes an analogous invariant: the F -pure threshold, that comes up in positive characteristic in commutative algebra. Then a sketch of the proof of Shokurov’s ACC conjecture for ambient smooth varieties is presented, and a connection of this conjecture with Termination of Flips is described. The last part discusses an asymptotic version of the log canonical threshold in the context of graded sequences of ideals. The article by Shigeru Mukai concerns K3 surfaces and Enriques surfaces. The author studies the connection between symplectic symmetries of K3 surfaces and the Mathieu group M24 . He also investigates its Enriques analogy, that is, a conjectural connection between semi-symplectic symmetries of Enriques surfaces and another Mathieu group M12 . Özer Öztürk and Piotr Pragacz study Thom polynomials of singularities of maps, by expanding them in the basis of Schur functions (labelled by partitions). The article contains some necessary conditions on a partition to appear in the set of indices of the Schur function expansion of a Thom polynomial. Moreover, the authors describe several recursions for the coefficients of the Schur functions in these expansions. The article also shows old and new computations of the Thom polynomials of some singularities. Marek Szyjewski investigates the kernel of the norm map on power classes for cyclic field extensions. In particular, he gives several results on the exactness of the “generalized Gross–Fischer exact sequence”. Halszka Tutaj-Gasi´nska discusses a certain connection between the (local) positivity of line bundles on smooth projective varieties and the symplectic packing of balls into symplectic manifolds. The article collects some results on Seshadri constants and packing constants – two kinds of numerical invariants appearing in this connection. A special emphasis is put on the toric situation. We dedicate the whole book to the memory of Oscar Zariski – the father of modern algebraic geometry. The opening article by Piotr Blass discusses his life and achievements. In particular, the influence of Zariski’s work and the work of his students on contemporary algebraic geometry is emphasized. This volume completes the Impanga trilogy whose first two volumes are “Topics in cohomological studies of algebraic varieties” and “Algebraic cycles, sheaves, shtukas, and moduli”, published by Birkhäuser in 2005 and 2008, respectively. Acknowledgments. The Editor wishes to thank the authors for their contributions. He is grateful to Halszka Tutaj-Gasi´nska for her help with the Impanga conference in B¸edlewo. Thanks also go to Manfred Karbe and Irene Zimmermann from European

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Mathematical Society Publishing House for a pleasant cooperation during the preparation of this book. The Editor is grateful for partial support by MNiSW grant N N201 608040 during the preparation of this volume. Warszawa, June 2012

The Editor

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii The influence of Oscar Zariski on algebraic geometry by Piotr Blass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 The geometry of T -varieties by Klaus Altmann, Nathan Owen Ilten, Lars Petersen, Hendrik Süß, and Robert Vollmert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Introduction to equivariant cohomology in algebraic geometry by Dave Anderson. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71 Recent developments and open problems in linear series by Thomas Bauer, Cristiano Bocci, Susan Cooper, Sandra Di Rocco, Marcin Dumnicki, Brian Harbourne, Kelly Jabbusch, Andreas L. Knutsen, Alex Küronya, Rick Miranda, Joaquim Roé, Hal Schenck, Tomasz Szemberg, and Zach Teitler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Moduli of map germs, Thom polynomials and the Green–Griffiths conjecture by Gergely Bérczi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 The Minimal Model Program revisited by Paolo Cascini and Vladimir Lazi´c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Invariants of hypersurfaces and logarithmic differential forms by Sławomir Cynk and Sławomir Rams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Prym varieties and their moduli by Gavril Farkas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 On generalized Wro´nskians by Letterio Gatto and Inna Scherbak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Lines crossing a tetrahedron and the Bloch group by Kevin Hutchinson and Masha Vlasenko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

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On complex and symplectic toric stacks by Andreas Hochenegger and Frederik Witt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Deformation along subsheaves, II by Clemens Jörder and Stefan Kebekus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Some degenerations of G2 and Calabi–Yau varieties by Michał Kapustka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Notes on Kebekus’ lectures on differential forms on singular spaces by Mateusz Michałek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 Lecture notes on K3 and Enriques surfaces by Shigeru Mukai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 IMPANGA lecture notes on log canonical thresholds by Mircea Musta¸ta˘ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 On Schur function expansions of Thom polynomials by Özer Öztürk and Piotr Pragacz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 A note on the kernel of the norm map by Marek Szyjewski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 Seshadri and packing constants by Halszka Tutaj-Gasi´nska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

The influence of Oscar Zariski on algebraic geometry Piotr Blass

Introduction Oscar Zariski was born in 1899 in Kobryn in the Ukraine (today Kobryn lies in Belarus) in a Jewish family. His first name, as he once told me, was sometimes pronounced Aszer in his childhood. He loved mathematics and original creative thinking about mathematics from a very early age. He recalled the exhilaration of doing mathematics as a boy. His father died when Oscar was very young; his mother was a business lady, as he used to say. She would sell various things in the Jewish Nalewki district in Warsaw.

Oscar Zariski (1899–1986) Photo by George M. Bergman (Archives of the Mathematisches Forschungsinstitut Oberwolfach)

In 1921, Oscar Zariski went to study in Rome. He had previously studied in Kiev and recalled that he was strongly interested in algebra and also in number theory. The latter subject was by tradition strongly cultivated in Russia. This paper was originally published in the Australian Mathematical Society Gazette vol. 16, No. 6 (December 1989). We thank the Australian Mathematical Publishing Association Inc. for permission to reprint an updated version of this paper. The updated version of the paper has been prepared by Piotr Pragacz; the typing and pictures are due to Maria Donten-Bury.

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He went to Rome on a Polish passport. His stay in Rome must have been very exciting. It lasted from 1921 until 1927. He became a student at the University of Rome and he married a wonderful woman who was his unfailing companion and soulmate for the next 65 years, Yole. They were inseparable and she was a tower of strength to him in good and bad times. In fact, during the latter years of his life when his hearing became impaired, she took over a lot of his communication tasks with other people. He continued to be creative almost to the last year of his life.

1 Zariski in Rome But let us return to Rome, 1921. There were three mathematicians at the University of Rome who became synonymous with the beauty, excitement and perhaps slightly cavalier approach to proofs in the Italian school of algebraic geometry. Guido Castelnuovo, Federigo Enriques and Francesco Severi. The first two were of Jewish origin and were related to each other. Yole Zariski came from the family that belonged to the same highly sophisticated social group as they did (in my impression at least). Zariski always spoke very warmly about Castelnuovo and Enriques. I don’t feel that they liked Severi very much, although they respected his work. This may be a good moment to trace Zariski’s mathematical ancestry. Italian algebraic geometry started with Luigi Cremona who was a fighter in Garibaldi’s army, became a senator, and was contemporary with the Romanic Revival in Italy and perhaps a part of it. (Yes, mathematicians do fit into general culture!) Cremona, who had apparently studied with Chasles, influenced Corrado Segre who, in turn, taught Castelnuovo. Castelnuovo influenced Enriques – this really was a partnership – and finally Guido Castelnuovo became Oscar Zariski’s thesis adviser. I remember Zariski telling me about an important and dramatic conversation with Castelnuovo during his early days in Rome. Castelnuovo was so impressed with young Zariski that he helped to cut a lot of “red tape” to speed up Oscar’s studies and became Zariski’s dissertation adviser. The Italians considered Zariski to be an “unpolished diamond”. They sensed that his view of geometry would eventually be different from their own. Castelnuovo once told him “You are here with us but you are not one of us”. This was not said in reproach but good-naturedly for Castelnuovo himself told Zariski time and time again that the methods of the Italian geometric school had done all they could do, had reached a dead end and were inadequate for further progress in the field of algebraic geometry. (This is reported by Zariski in the introduction to his collected papers.) Castelnuovo perhaps suspected that the way out of this predicament would lie in increasing use of algebra and topology in algebraic geometry and, well aware of Oscar’s algebraic inclinations, he suggested to him a thesis problem that was closely related to Galois theory and to topology.

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2 Zariski’s thesis problem (over C) He proved the following based on the results of his thesis: Given an algebraic equation f .x; y/ D 0 of genus > 6 and generic (of general moduli) it is not possible to introduce a parameter t , a rational function of x, y, so that x and y can be expressed through t by radicals. Another formulation is contained in the following theorem. Let X be a curve. We call a map of curves X ! P 1 solvable if and only if the corresponding extension k.x/  k.t/ is a field extension solvable by radicals. Theorem 1. A general curve of genus  7 admits no solvable map into P 1 . During his stay in Rome Zariski was constantly involved in and exposed to research on algebraic surfaces (over C). This was the favourite topic of his teachers. He says, in the introduction to [6], In my student days in Rome algebraic geometry was almost synonymous with the theory of algebraic surfaces. This was the topic about which my Italian teachers lectured most frequently and in which arguments and controversy were also most frequent. Old proofs were questioned, corrections were offered and these corrections were – rightly so – questioned in their turn. At any rate the theory of algebraic surfaces was very much on my mind... Nevertheless, most of his publications from this period still deal with algebraic curves and also certain foundational philosophical questions (e.g. Dedekind’s theory of real numbers and Cantor’s and Zermelo’s recently created set theory). To that interest, he was influenced by Enriques, who was himself a philosopher and a historian of mathematics. Zariski must have considered the theory of algebraic curves to be a necessary training ground for an algebraic geometer. Indeed, when I asked him to teach me algebraic geometry in Harvard in 1970 he made sure that I had some knowledge on algebraic curves. When I asked him later about the best introduction to this subject, he said: the book by Enriques-Chisini [3].

3 Illinois and Johns Hopkins In 1927, Oscar and Yole left Italy for the United States. The reasons for this were, I think, the emergence of fascism in Italy and difficulty of finding a suitable academic position in Italy. Upon arrival in the US, the Zariskis spent some time in Illinois at a rather mediocre university but soon his brilliance was recognized and he was offered a position at Johns Hopkins University in Baltimore. This was the school that brought Sylvester to the USA before. Little did they realize that they had another Sylvester on their hands. Or did they?

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Zariski spent considerable time preparing his monograph Algebraic surfaces. In it he presented the current status of the theory of algebraic surfaces as of 1933. He examined every argument carefully and found a number of significant gaps in the classical proofs of the Italian school. As he says: “The geometric paradise was lost once and for all” – new tools, and a new framework and language were needed. This was a crisis. But to a mind like Zariski’s in 1937 a crisis was just an exciting opportunity. He found his new tools in the commutative algebra and valuation theory that were being developed by Krull and van der Waerden who actually was trying his hand quite impressively in applying modern algebra to algebraic geometry even slightly before Zariski. But we are getting ahead of the story. During 1927–1937, Zariski made frequent trips to talk with Solomon Lefschetz in Princeton. Castelnuovo had the highest respect for Lefschetz and told Zariski about his work. Solomon Lefschetz, another Jewish immigrant from Russia, “stuck the harpoon of topology into the whale of algebraic geometry”. Lefschetz was another great genius with an unusual, even romantic, life story. Originally trained as an engineer he lost both hands in a terrible industrial accident. He had to give up his career and entered Clark University in Worcester, Massachusetts, to get his PhD in mathematics. I was on the Faculty of Clark for a year, and I have had the pleasure of examining Lefschetz’s thesis written under the supervision of Storey. It was very concrete, quite “Italian”, geometry. Lefschetz spent many years in Nebraska and thirteen years in Lawrence, Kansas, in complete isolation, which he later considered a blessing. He read papers of Picard and Poincaré about integrals and their periods on algebraic varieties. He was deeply impressed by Picard’s method of fibering an algebraic surface by a suitable pencil of curves (now called a Lefschetz pencil). Using such pencils and monodromy, Lefschetz obtained very deep and subtle results on the topology and algebraic geometry of algebraic varieties over C and later over fields of characteristics zero. Lefschetz’s genius was recognized and he was called to be a professor in Princeton (miracles still happen in America – at least they did in 1924). While he was talking to Lefschetz, Zariski was also doing work on topology of algebraic varieties. Under Lefschetz’s influence a young American, Walker, rigorously proved a difficult and important theorem: resolution of singularities of algebraic surfaces over algebraically closed fields of characteristic zero. Zariski examined Walker’s proof and declared it to be correct in his Algebraic surfaces monograph mentioned above. Let me quote from Zariski’s 1934 introduction to Algebraic surfaces: It is especially true in algebraic geometry that in this domain the methods employed are at least as important as the results. The author has therefore avoided as much as possible purely formal accounts of the theory... and then due to exigencies of simplicity and rigor the proofs given in the text differ to a greater or lesser extent from the proofs given in the original papers.

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Thus Zariski essentially proves anew and clarifies a great deal of material. This book was the work of a master geometer. During the period 1935–50, Zariski continued his work on topology of algebraic varieties, their fundamental group, purity of branch locus, cyclic multiple planes, etc. For lack of space, I must refer the reader for the details to the very descriptive and insightful summaries of the editors of Zariski’s collected papers [7].

4 Zariski applies modern algebra to resolution of singularities Between 1935–37, Zariski studied modern algebra, saying “I had to start somewhere”. He took valuation theory and the notion of integral dependence from Krull’s Idealtheorie [5] and applied them to algebraic varieties – more specifically – to two problems: I. Local uniformisation, and II. reduction of singularities or resolution of singularities. Much later, in 1958, he wrote about: III. Purity of branch locus. I will restrict myself to describing problem II, i.e. resolution of singularities and Zariski’s contribution to it. Definition 1. Let V  Pkn be an irreducible projective algebraic variety (i.e. a set of common zeros of a set of homogeneous forms in the homogeneous variables fx0 ; x1 ; : : : ; xn g). We call a projective variety W  Pkm a desingularisation of V if there exists a regular algebraic morphism  W W ! V such that 1. W is a smooth variety; 2.  is proper, i.e. universally closed for any base change (which is automatically satisfied in our context); 3.  induces an isomorphism of  1 .V  Sing V / with V  Sing V . The simplest example is a plane curve with a node. For example, V W x 3 Cy 3 xy D 0. The essential features of this curve can be pictured as follows:

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N , the origin, is a node, a very well known type of singularity. The desingularisation W is a smooth (space) curve where the two branches at N are pulled apart.

An equally famous and simple example is the cusp V W y 2 x 3 . This can be pictured as follows:

The origin C is a cusp, the desingularisation W is a straight line S with a mapping  W W ! V that can be pictured as follows:

Here is an example of a surface singularity:

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O is a biplanar double point of V . The desingularisation  W W ! V can be pictured as follows:

Here L1 and L2 are (projective) lines that are contracted by  to the point O. The map  restricted to W  .L1 [ L2 / establishes an isomorphism with V  fOg. In other words, after blowing up or quadratic transformation the singular point is replaced by  a pair of projective lines intersecting at one point. (The intersection matrix  1 .) is 2 1 2 Singularities do not have to be isolated. For example, x 3 D y 2 z has a whole curve of cusps x D z D 0. This can be pictured (roughly) as follows

(This singularity gets “worse” when y D 0.) Resolution can be a very complicated process and general existence theorems tend to be very hard in higher dimensions. Zariski, first of all, showed that the algebraic notion of integral closure, which he called normalisation, gives resolution of singularities for curves. Thus: dim X D 1, any characteristic, resolution is possible, dim X D 2, the case of algebraic surfaces; he proved that resolution could be obtained by alternating use of normalisation and quadratic transformations with point centers: Normalize ! Blow up ! Normalize ! : : :

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His proof worked in characteristic zero and, i.e., he proved the same type of result as the one by Walker, mentioned before. However, he masterfully wrote his paper in such a way that made it very transparent what still had to be done in char p > 0. This was then completed by some brilliant work of Abhyankar in the early 1950s. Abhyankar was Zariski’s first student at Harvard. So resolution for surfaces is now well known in all characteristics. A beautiful and conceptual proof was supplied by another student of Zariski, Joseph Lipman, around 1980. Zariski then turned to the very difficult case of a 3-dimensional variety in characteristic zero. Again he succeeded but the proof was now very long (70 pages in the Annals of Mathematics; reprinted in [7], volume I, where all of Zariski’s resolution papers may be found). He says in the introduction (about the n-dimensional problem): How much more difficult is the general problem is of course impossible to say with certainty and precision at the present moment. We are inclined to conjecture that the difficulties in the general case and in the three-dimensional case are of comparable order of magnitude... the threedimensional case offers an excellent testing ground... Again the paper was written in such a way that Abhyankar was able to extend this proof brilliantly to the fields of characteristics p > 5. (This has been only recently extended to all positive characteristics by V. Cossart and O. Piltant [2].) Heisuke Hironaka, advised and prompted by Zariski, then solved the general case and proved that resolution of singularities exists in all dimensions in characteristic zero. This was in 1964 – the Annals paper [4] is one of the best ever written (it is 217 pages long). As Abyankar points out (Kyoto 2008), Hironaka proved this result at first in dimension 4. In Abyankar’s opinion, people working on resolution of singularities in positive characteristic, should follow this strategy. Hironaka received a Field’s Medal for this achievement. I remember talking with Zariski about that achievement of Hironaka. I could still feel in 1971 the excitement over this result; somehow Zariski made it clear to me that he was very helpful in Hironaka’s work without taking away any of the credit, rightly earned by his brilliant former student. Zariski not only proved general theorems about resolution, he also knew how to resolve singularities in practice when they are given by explicit equations. He taught that to his students and to me. He also knew how to use resolution to study differentials and numerical invariants of varieties. He could thus make precise a great deal of Italian geometry. In fact, being able to resolve singularities is a trademark of the Zariski school. Abhyankar once said of Zariski: “Without his blessing, who can resolve singularities?” In the near future – I feel – the knowledge of resolution of singularities and its possible computer implementations should become useful to engineers and other scientists who work with systems of algebraic equations. In connection with the resolution problem, I mentioned the Indian Mathematician Abhyankar several times. He is a professor in Purdue and in Poona, India, he has had a large number of students who consider Zariski paramguru (guru of your guru). Thus

The influence of Oscar Zariski on algebraic geometry

9

Zariski’s influence is being felt in the new generation of geometers in Japan (Hironaka’s influence) and in India (Abhyankar’s). In the US, David Mumford, Michael Artin, Joseph Lipman and Steve Kleiman have given great impetus to algebraic geometry and have had numerous students. Daniel Gorenstein was also an early Zariski student. His thesis was about curves – hence Gorenstein rings. He moved out of algebraic geometry but we will forgive him since he led the magnificent effort to classify finite simple groups.

5 Linear systems, simple points, Zariski’s Main Theorem During the period 1937–45, Zariski, in addition to his work in resolution and local uniformisation, took up in a rigorous way such topics as linear systems, simple points, Bertini theorems, applying modern algebra to all of these topics that were studied less rigorously by the Italian geometers. Around 1945–46, he started to develop his theory of holomorphic functions and continuation in abstract, algebraic geometry. His teaching load at Johns Hopkins was 18 hours a week; it was wartime. He was however invited to spend at least one year in São Paulo in January 1945. There he developed his theory of holomorphic functions in relative peace and quiet. He had a superlative audience consisting of one person, André Weil, with whom he took frequent walks and talks together. An important paper appeared in 1946 in Brazil and was completed in a 1951 AMS Memoir. There are several noteworthy effects that came out of Zariski’s theory of holomorphic functions: (i) Zariski’s Main Theorem, and (ii) the connectedness principle. Also, (iii) it inspired Grothendieck’s theory of formal schemes and several deep theorems in cohomology of schemes and thus is now the mainstream and lifeblood of modern algebraic geometry. The easiest one to explain is the connectedness principle. Enriques stated it as follows: If an irreducible variety V varies in a continuous system and degenerates into a reducible variety V0 , then V0 is connected.

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(Over C it is obvious because V0 is a continuous image of V but over an abstract field it is much harder.) As for (iii), first of all Grothendieck reformulated and generalized the connectedness theorem as follows: If f W V 1 ! V is a proper morphism and f OV 1 D OV , then the geometric fibers of f are connected. See Artin’s introduction to Zariski’s collected papers, ([7]) volume II.

6 Back to surfaces (with vengeance). Main course After much foundational work, Zariski returned to his old Italian love: algebraic surfaces. Now he had powerful algebraic tools, resolution of singularities, Bertini Theorems, and adequate notions of smoothness. He could proceed much more confidently. Also, in many cases he was able to deal with varieties of any dimension. Mumford remarked that this must have seemed like a dessert after the foundational work. Zariski good-naturedly corrected him and said to him this was the main course. (Mumford was perhaps Oscar’s favorite and most trusted student.) Zariski moved to Harvard in 1949. He was at the pinnacle of his career and became world famous. (It is remarkable that his greatest achievements started when he was almost 40 years old, thus dispelling once and for all the myth that mathematics is a young person’s game. There is hope for all of us!) During 1946–1955, he reigned supreme in algebraic geometry. Then came Serre and Grothendieck, but we will come to this later. Zariski published several important papers about linear systems, algebraic surfaces and algebraic varieties of higher dimension, studying a number of global questions this time. In the period 1948–1962 roughly, he dealt with such topics as invariance of arithmetic genus under birational transformations, the so-called lemma of Enriques– Severi–Zariski, Riemann–Roch for surfaces, and minimal models of surfaces. In a paper written in 1958, he generalized a famous theorem of his teacher Castelnuovo to surfaces in characteristic p > 0. Let us try to explain Castelnuovo’s theorem (and criterion). A surface S is called rational if it can be parametrized “almost everywhere” by two independent parameters. In purely algebraic terms, k.S/ D the function field of S Š k.T1 ; T2 /; with T1 , T2 algebraically independent over k. A surface S is called unirational if there is an extension k.S/ ! k.T1 ; T2 / : When k D C, Castelnuovo proved that S unirational ) S rational

(1)

The influence of Oscar Zariski on algebraic geometry

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(about 1895). This work was a jewel of Italian geometry, Castelnuovo’s argument being long and subtle. Castelnuovo used his criterion ³ arithmetic genus of S D 0 ) S rational. (2) bigenus of S D 0 In modern terms (a simply connected surface with no regular 2-forms of weight two is rational), ³ h2 .S /  h1 .S / D 0 ) S rational. h0 .2K/ D 0 Zariski generalized (2) to all fields of char p > 0, p D 2 being the hardest case in his approach. However (1) is false in char p > 0; there exist unirational surfaces that are not rational! (Unless you assume k.S/ ! k.T1 ; T2 / is a separable extension, in which case (1) is true, as Zariski showed in 1958.) Zariski wrote down an explicit example: p  3 prime F : z p D x pC1 C y pC1 

x2 2



y2 2

Fz WD closure of F in projective space,  W Fx ! Fz desingularisation The desingularisation is:

Clearly, k.Fz / D k.x; y; z/  .inseparable ext/k.x 1=p ; y 1=p /. Thus Fz is unirational but Zariski checked that, for example, dxdz=.y p  y/ defines a regular differential 2-form on Fz . Thus Fz cannot be rational. This example led to my own thesis topic, suggested to me by Hironaka in 1970. Zariski’s example (half a page) has blossomed into a substantial theory of Zariski surfaces (a 450 page monograph [1] that uses all the tools of modern algebraic geometry and is tied up with computer science and, it seems, coding theory. Thus in this case, as in many others, Zariski’s idea has blossomed into a large theory. How typical!

7 Zariski at Harvard (1949–1986) Zariski officially retired at Harvard around 1970. Upon Hassler Whitney’s suggestion (and that of a lot of other people, or so they say now), Zariski was called to Harvard in approximately 1944. At a recent algebraic geometry meeting, someone (perhaps

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P. Blass

Abhyankar) asked for Zariski’s students and students of his students to raise their hands. Almost everybody in the room did. Zariski can be considered the father of the American algebraic geometry school. He did not stand in the path of progress, on the contrary he welcomed it. While at Harvard from 1949 Zariski quickly established himself as the undisputed leader in algebra and algebraic geometry. In fact, Garrett Birkhoff stopped teaching algebra after a couple of years and moved to computer science. Harvard had great graduate students. Zariski’s students included Abhyankar, Gorenstein, Mike Artin (Emil Artin’s son), David Mumford, Steven Kleiman, Joseph Lipman, Heisuke Hironaka and Alberto Azevedo (from Brazil). Mumford and Hironaka went on to receive Fields Medals. In 1955 Serre and in 1958 Grothendieck suddenly revolutionized algebraic geometry by introducing the notion of sheaves, schemes and cohomology. They were inspired by Zariski but in some ways their theories could go much further. Zariski, who was approaching 60 at that time, organized a summer school in algebraic sheaf theory. He wrote an account of Serre’s work. Grothendieck was welcomed at Harvard and taught a remarkable class with Mumford, Artin, Hironaka, Tate (who was on the faculty), Shatz and others in the audience. Zariski’s students became Grothendieck followers, but they never forgot what they learned from Zariski. Thus the Zariski school adopted the scheme theoretic and cohomological techniques of Grothendieck. Grothendieck dedicated his EGA Elements of Algebraic Geometry treatise to Oscar Zariski and André Weil. Zariski spent the last fifteen or so years of his life working on the question of equisingularity. Once again he created an important and impressive theory which, roughly speaking, attempts to compare singularities at different points on varieties and decide when they are in some sense the same (or similar). It was touching and inspiring to see him work in his 80s. He agonized when he felt that his mathematical powers could be leaving him. I overheard him once talking about it to Mumford. “Maybe I should quit,” said the 85 year old Oscar. “Take a little vacation,” said Mumford. And so he did. His wife, Yole, was incredibly important to him during this time, as always. His hearing became disturbed with constant ringing. It was hard to converse with him. You had to write everything. He was depressed because results were coming slowly.

8 Conclusion – personal memories Zariski was serious and professional about mathematics and that was picked up by all of his students. He claimed to be “slow”, which forced people to really explain things. His approach was to do something almost every day. A lemma a day... (he used to say). Let me add a personal note. I met Oscar at Harvard when I was about 21 years old. He was retired but I worked with him as much as I could. He was very helpful with my thesis problem; I would meet with him once a week to talk about geometry and my

The influence of Oscar Zariski on algebraic geometry

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progress, although Hironaka was my official advisor. Sometimes we would all go to lunch together at the Harvard Faculty Club. Zariski told me quite a lot about his youth; I came from a nearby part of the world. (He visited his country of birth, Russia, in 1937.) I visited him at Purdue where he would spend his summer (to get away from Harvard “that was like a madhouse sometimes”). I named my own son Oscar. I guess we all loved him like a father. A lot of us came to his memorial meeting in September 1986. His was a wonderful life. He is survived by his wife, children, several grandchildren and most importantly by numerous geometers all over the world. As a teacher he was quite strict and made you want to learn and do all that you are capable of doing just to keep up with him. A word of praise from him was something you really treasured. But somehow he made you feel like part of the family. In 1973, I was on leave from the Israeli army and from the war. I decided that I had to see Oscar. He was warm, his house enlivened by a couple of grandchildren. His presence and Yole’s and their conversation made me stronger to face the hardships and come back to mathematics. Probably the greatest praise I ever heard from him was that he called my thesis interesting and wrote me a nice letter about it. I saved all of his letters and mathematical notes. There was no conflict between research and teaching for Zariski. Teaching extended his research and increased its impact a hundredfold. He was a truly wise and happy man. There are not many like him. Oscar Zariski died in 1986, so in 2011 we celebrate the 25-th anniversary of his death.

In summary Zariski transformed algebraic geometry from its semi-art, semi-science status into both art and science. He made it mathematically precise without sacrificing any of its beauty. The most fundamental topology on an algebraic variety or a scheme is called the Zariski topology. Thus he is remembered whenever modern algebraic geometry is done. Also the terms “Zariski tangent space”, “Zariski decomposition” for surfaces, are very common in algebraic geometry. He was flexible enough to welcome and encourage the most modern trends as long as they contributed to solving hard classical problems. The total impact of the Zariski school may well be historically comparable to Riemann’s or Hilbert’s, especially when combined with its logical allies and successors, Grothendieck’s school and Šafareviˇc’s in Russia. Great triumphs of mathematics are the solution of the Mordell conjecture by Faltings and the proof of the Fermat’s Last Theorem by Wiles, which are built on Grothendieck’s machinery. Zariski provided a powerful bridge spanning several decades between nineteenth century mathematics and early twentieth century mathematics to the most modern developments.

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Quote from Zariski The Italian geometers have erected, on somewhat shaky foundations, a stupendous edifices: the theory of algebraic surfaces. It is the main object of modern algebraic geometers to strengthen, preserve and further embellish this edifice, while at the same time building up also the theory of varieties of higher dimensions. The bitter complaint that Poincaré had directed, in his time, against the modern theory of functions of real variable cannot be deservedly directed against modern algebraic geometry. We are not intent on proving that our fathers were wrong. On the contrary our whole purpose is to prove that our fathers were right. The arithmetic trend in algebraic geometry is in itself a radical departure from the past. This trend goes back to Dedekind and Weber who have developed, in their classical memoir, an arithmetic theory of fields of algebraic functions of one variable. Abstract algebraic geometry is a direct continuation of the work of Dedekind and Weber except that our chief object is the study of fields of algebraic functions of more then one variable. The work of Dedekind and Weber has been greatly facilitated by previous developments of classical ideal theory. Similarly, modern algebraic geometry has become a reality partly because of the previous development of the great theory of ideals. But here the similarity ends. Classical ideal theory strikes at the very core of the theory of functions of one variable and there is in fact a striking parallelism between this theory and the theory of algebraic numbers. On the other hand the general theory of ideals strikes almost at the foundations of algebraic geometry and falls short of the deeper problems which we face at the post foundational stage. Furthermore there is nothing in modern commutative algebra that can be regarded even remotely as a development parallel to the theory of algebraic function fields of more than one variable. This theory is after all itself a chapter of algebra, but it is a chapter about which modern algebraists know very little. All our knowledge here comes from geometry. For all these reasons it is undeniably true that the arithmetisation of algebraic geometry represents a substantial advance in algebra itself. In helping geometry, modern algebra is helping itself above all. We maintain that abstract algebraic geometry is one of the best things that has happened to commutative algebra in a long time. [7]

Remark. For a more detailed account of Oscar Zariski’s life and work, see the book by Carol Parikh, The unreal life of Oscar Zariski, Academic Press (1991).

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References [1]

P. Blass and J. Lang, Zariski surfaces and differential equations in characteristic p > 0. Monogr. Textbooks Pure Appl. Math. 106, Marcel Dekker, New York 1987.

[2] V. Cossart and O. Piltant, Resolution of singularities of threefolds in positive characteristic II. J. Algebra 321 (2009), 1836–1976. [3]

F. Enriques and O. Chisini, Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche. In three volumes, Zanichelli, Bologna 1915–24.

[4]

Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. of Math. 79 (1964), 109–326.

[5]

W. Krull, Idealtheorie. Ergeb. Math. Grenzgeb. 46, Springer-Verlag, Berlin 1968.

[6]

O. Zariski, Algebraic surfaces. Chelsea Publishing, New York 1948.

[7]

O. Zariski, Collected papers. In four volumes, MIT Press, Cambridge, Mass., London 1972.

Piotr Blass, Ulam University, 113 West Tara Lakes Drive, Boynton Beach, Florida, 33436-6763, U.S.A. E-mail: [email protected]

The geometry of T -varieties Klaus Altmann, Nathan Owen Ilten, Lars Petersen, Hendrik Süß, and Robert Vollmert

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 2 Affine T -varieties . . . . . . . . . . . . . . . . . . . . 3 The functor TV . . . . . . . . . . . . . . . . . . . . . 4 How to construct p-divisors . . . . . . . . . . . . . . . 5 Globalization in complexity 1 . . . . . . . . . . . . . . 6 The T -orbit decomposition . . . . . . . . . . . . . . . 7 Cartier divisors . . . . . . . . . . . . . . . . . . . . . 8 Canonical divisors, positivity, and divisor ideals . . . . 9 Cohomology groups of line bundles in complexity 1 . . 10 Cox rings as affine T -varieties . . . . . . . . . . . . . 11 Invariant valuations and proper equivariant morphisms 12 Resolution of singularities . . . . . . . . . . . . . . . 13 (Log-)terminality and rationality of singularities . . . . 14 Polarizations . . . . . . . . . . . . . . . . . . . . . . . 15 Toric intersection theory . . . . . . . . . . . . . . . . 16 Deformations . . . . . . . . . . . . . . . . . . . . . . 17 Related constructions . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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17 18 22 25 27 30 32 35 37 39 42 44 48 50 56 60 64 66

1 Introduction 1.1 C  -actions and toric varieties. The present paper is a survey about complex T varieties, i.e. about normal (n-dimensional) varieties X over C with the effective action of a torus T WD .C  /k . The case k D 1 is a classical one – especially singularities with a so-called “good” C  -action have been studied intensively by Pinkham [Pin74], [Pin77], [Pin78]. The case n D k is classical as well – first studied by Demazure [Dem70], such varieties are called “toric varieties”. The basic theory encodes the category of toric varieties and equivariant morphisms in purely combinatorial terms. Many famous theories in algebraic geometry have their combinatorial counterpart and thus illustrate the geometry from an alternative point of view. The dictionary between algebraic geometry and combinatorics sometimes even helps to establish new theories. The most prominent example of this phenomenon might be the development of mirror symmetry in the 1990s. Building on the notion of reflexive polytopes, Batyrev gave a first systematic mathematical treatment of this subject [Bat94].

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1.2 Higher complexity. For some features, however, it is useful to consider lower dimensional torus actions as well. For instance, when deforming a toric variety, the embedded torus T still acts naturally on the total- and base-spaces Xz ! S, but it is usually too small to provide a toric structure on them. The adjacent fibers Xs are even worse: depending on the isotropy group of T at the point s 2 S , only a subtorus of T still acts. Thus, it is worth to study the general case as well. The difference n  k is then called the complexity of a T -variety X . While complexity 0 means toric, the next case of complexity 1 was systematically studied by Timashev (even for more general algebraic groups) in [Tim97]. On the other hand, Flenner and Zaidenberg [FZ03] gave a very useful description of C  -surfaces even for “non-good” actions. Starting in 2003, there is a series of papers dealing with a general treatment of T -varieties in terms of so-called polyhedral divisors. The idea is to catch the non-combinatorial part of a T -variety X in an .n  k/-dimensional variety Y which is a sort of quotient Y D X=T . Now, X can be described by presenting a “polyhedral” divisor D on Y with coefficients being not numbers but instead convex polyhedra in the vector space NQ WD N ˝Z Q where N is the lattice of one-parameter subgroups of T . 1.3 What this paper is about. The idea of the present paper is to give an introduction to this subject and to serve as a survey for the many recent papers on T -varieties. Moreover, since the notion of polyhedral divisors and the theory of T -varieties closely follows the concept of toric varieties, we will treat both cases in parallel. This means that the present paper also serves as a quick introduction to the fascinating field of toric varieties. For a broader discussion and detailed proofs of facts related to toric geometry which are merely mentioned here, the reader is asked to consult any of the standard textbooks like [KKMSD73], [Dan78], [Ful93], and [Oda88]. The subjects of non-toric T -varieties are covered in [AH03], [AH06], [AHS08], [AH08], [IS10], [Süß], [PS], [IS], [IVa], [HI], [Vol10], [HS10], [AW], and [AP]. There are also applications of the theory of T -varieties and polyhedral divisors in affine geometry [Lie10b], [Lie10a], [Lie11] and coding theory [IS10] which are not covered by this survey. Very recently, SL2 -actions on affine T-varieties have also been studied [AL].

2 Affine T -varieties 2.1 Affine toric varieties. Let M and N be two mutually dual, free abelian groups of rank k. In other words, both are isomorphic to Zk , and we have a natural perfect pairing M  N ! Z. Then T WD Spec CŒM  D N ˝Z C  is the coordinate free version of the torus .C  /k mentioned in (1.1). On the other hand, N D HomalgGr .C  ; T / is the set of one-parameter subgroups of T , and M D HomalgGr .T; C  / equals the character group of T . We denote by MQ WD M ˝Z Q and NQ WD N ˝Z Q the associated Q-vector spaces.

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m i 1 PmWe alwaysi assume that the generators a for a polyhedral cone  D ha ; : : : ; a i WD iD1 Q0 a  NQ are primitive elements of N , i.e. they are not proper multiples of other elements of N . Dropping this assumption is the first step towards the theory of toric stacks which we will not pursue here, cf. [BCS05]. Moreover, we will often identify a primitive element a 2 N with the ray Q0  a it generates. Given  as above, we define its dual cone as  _ WD fu 2 MQ j h; ui  0g. If  does not contain a non-trivial linear subspace (i.e. 0 2  is a vertex), then dim  _ D k, and we use the semigroup algebra CŒ _ \ M  WD ˚u2 _ \M C u to define the k-dimensional toric variety TV. / WD TV.; N / WD Spec CŒ _ \ M :

This is a normal variety. If dim  D k, the (finite) set E   _ \ M of indecomposable elements is called the Hilbert basis of  _ . Furthermore, TV. /  C E is defined by a binomial ideal which arises from the relations between the elements of the Hilbert basis E, cf. (3.2). Example 1. Let  D h.1; 2/; .1; 2/i  Q2 . This gives rise to  _ D hŒ2; 1; Œ2; 1i and E D fŒe; 1 j 2  e  2g, illustrated in Figure 1. The resulting TV. /  C 5 is the cone over the rational normal curve of degree 4, and its defining ideal is generated  by the six minors expressing the inequality rank yy01 yy12 yy23 yy34  1:

y0 y1 y2 y3 y4

(a) σ

(b) σ∨

Figure 1. Cones for Example 1.

2.2 Toric bouquets. The previous notion allows for a generalization. If   NQ is a polyhedron, then we denote by tail./ WD fa 2 NQ j a C   g its so-called tailcone. Assume that this cone is pointed, i.e. contains no non-trivial linear subspace; then  has a non-empty set of vertices V ./. Denoting their compact convex hull by c , the polyhedron  splits into the “Minkowski” sum c C tail./. Moreover,  gives rise to its inner normal fan N ./ consisting of the linearity regions of the function minh; i W tail./_ ! Q. Note that the cones of N ./ are in a one-to-one correspondence to the faces F   via the map F 7! N .F; / WD fu 2 MQ j hF; ui D minh; uig. We can now define TV./ WD Spec CŒN ./ \ M 

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where CŒN ./ \ M  WD CŒ _ \ M  as a C-vector space. Note, however, that the multiplication is given by ´ 0 uCu if u, u0 belong to a common cone of N ./; u u0    WD 0 otherwise: A scheme of the form TV./ is called a toric bouquet. We obtain the decomposition [ TV./ D TV.Q0  .  v// v2V./

into irreducible (toric) components, compare with (6.1). Note that there is a oneto-one correspondence between polyhedral cones and affine toric varieties, whereas the construction of a toric bouquet only depends on the normal fan N ./. Hence, dilations of the compact edges of  and translations do not affect the structure of the corresponding bouquet. Example 2. Consider the cone  WD h.1; 0/; .1; 1/i  Q2 together with the  -polyhedron  D .0; 0/.0; 1/C D c Ctail./ and its inner normal fan N ./ as depicted in Figure 2. Observe that Spec CŒN ./ \ M  is equidimensional and consists of two irreducible components isomorphic to A2 which are glued along an affine line. N (F1 , Δ) F2 N (F2 , Δ)

F1 (a) Δ

(b) N (Δ)

Figure 2. An affine toric bouquet, cf. Example 2.

2.3 Polyhedral and p-divisors. Let M , N ,   NQ be as in (2.1). In particular, we assume that  contains no non-trivial linear subspace. These data then give rise to a semigroup (with respect to Minkowski addition) PolC Q .N;  / WD f  NQ j  polyhedron with tail./ D  g: Note that PolC Q .N;  / satisfies the cancellation law and contains  as its neutral element. If, in addition, Y is a normal, projective variety over C, then we denote by CaDiv0 .Y / the semigroup of effective Cartier divisors and call elements X DD DZ ˝ Z 2 PolC Q .N;  / ˝Z0 CaDiv0 .Y / Z

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polyhedral divisors on Y in N (or simply on .Y; N /) with tail.D/ WD  . We will also allow ; as an element of PolC Q .N;  / which satisfies ; C  WD ;. This somewhat bizarre coefficient is needed to deal with non-compact, open loci defined as Loc.D/ WD S Y n DZ D; Z. For each u 2 .tail.D//_ , we may then consider the evaluation D.u/ WD

X

minhDZ ; ui  ZjLoc D 2 CaDivQ .Loc D/:

DZ ¤;

This is an ordinary Q-divisor with supp D.u/  supp D WD Loc D \

S DZ ¤;

Z.

Definition 3. D is called a p-divisor if all evaluations D.u/ are semiample (i.e. have positive, base point free multiples) and, additionally, are big for u 2 int.tail.D//_ . Note that this condition is void if Loc.D/ is affine. Polyhedral divisors have the propertyL that D.u/ C D.u0 /  D.u C u0 /. Hence, they lead to a sheaf of rings OLoc .D/ WD u2 _ \M OLoc .D.u// u giving rise to the schemes

e

T V .D/ WD SpecLoc.D/ O.D/ and the affine TV.D/ WD Spec .Loc.D/; O.D//: The latter space does not change if D is pulled back via a birational modification Y 0 ! Y or if D is altered by a principal polyhedral divisor on Y , that is, an element in the image of the natural map N ˝Z C.Y / ! PolQ .N;  / ˝Z CaDiv.Y /. Two p-divisors which differ by chains of those operations are called equivalent. Note that this implies that Y can always be replaced by a log-resolution. Theorem 4 ([AH06], Theorems (3.1), (3.4); Corollary (8.12)). The map D 7! TV.D/ yields a bijection between equivalence classes of p-divisors and normal, affine Cvarieties with an effective T -action. 2.4 Products of T -varieties. Consider some T -variety X and a T 0 -variety X 0 . Then the product X  X 0 carries the natural structure of a T  T 0 -variety. As we shall see, the combinatorial data describing X  X 0 arise as a kind of product of the combinatorial data describing X and X 0 . For simplicity, we will only consider the affine case, although the construction easily globalizes. Thus, consider p-divisors D; D 0 on respectively Y , Y 0 , with tailcones  ,  0 , respectively. We define the product p-divisor D  D 0 on Y  Y 0 as follows: X X 0 0 .DZ   0 / ˝ .Z  Y 0 / C .  DZ D  D0 D 0 / ˝ .Y  Z /: ZY

Z 0 Y 0

Proposition 5. For any p-divisors D and D 0 , TV.D  D 0 / Š TV.D/  TV.D 0 /.

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Proof. For any uQ D .u; u0 / 2 M ˚ M 0 ,   Q  Loc.D  D 0 /; O.D  D 0 .u//   D  Loc.D  D 0 /; O.D.u/  Y 0 / ˝ O.Y  D 0 .u0 //     D  Loc.D/; O.D.u// ˝  Loc.D 0 /; O.D 0 .u0 // where the last equality is due to the Künneth formula for coherent sheaves, see [Gro63, Theorem 6.7.8]. Remark 6. A special case of the above is when X and X 0 are toric varieties. Here, the proposition simplifies to TV.   0 / Š TV. /  TV. 0 /.

3 The functor TV 3.1 Maps between toric varieties. We will now see that all constructions from the previous section are functorial. We start with the setting given in (2.1). A Z-linear map F W N 0 ! N satisfying FQ . 0 /   gives rise to a morphism TV.F / W TV. 0 ; N 0 / ! TV.; N / of affine toric varieties via F _ . _ \ M /  . 0 /_ \ M 0 . For example, if E   _ \ M is a Hilbert basis, then the embedding TV. / ,! C E from (2.1) is induced by the map E W N ! ZE . 3.2 Maps between T -varieties. A generalization of the functoriality to the setting of (2.3) also has to take care of the underlying variety Y . Let .Y 0 ; D 0 ; N 0 / and .Y; D; N / be p-divisors. Consider now any tuple .F; ; f/ 0 consisting  W Y 0 ! Y , and an element P of a map F W N !0 N , a dominant morphism 0  f D vi ˝ fi 2 N ˝ C.Y / , satisfying F D  ' D C div.f/ (to be checked for all coefficients separately). Here, X 0 0 F D 0 WD F .DZ 0/ ˝ Z Z0

is the push forward of D 0 by F , '  D WD

X

DZ ˝ '  .Z/

Z

is the pull back of D by , and div.f/ D

X .vi / ˝ div.fi / i

is the principal polyhedral divisor associated to f. Such a tuple provides us with naturally defined morphisms

e

e

e

T V .F; '; f/ W T V .D 0 ; N 0 / ! T V .D; N /;

The geometry of T -varieties

23

and TV.F; '; f/ W TV.D 0 ; N 0 / ! TV.D; N /: This construction yields an equivalence of categories between polyhedral cones and affine toric varieties in the toric case (where only the map F plays a role). To obtain a similar result for the relation between p-divisors and T -varieties one first needs to define the category of the former by turning both types of equivalences mentioned in (2.3) into isomorphisms. This can be done by the common technique of localization which is known from the construction of derived categories. In the category of T varieties we restrict to morphisms W X 0 ! X, with T: .X 0 /  X being dense. We will call such morphisms orbit dominating. Theorem 7 ([AH06], Corollary 8.14). The functor TV induces an equivalence from the category of p-divisors to the category of normal affine varieties with effective torus action and orbit dominating equivariant morphisms. 3.3 Open embeddings. Let us fix a torus T . To glue affine T -varieties together one has to understand T -equivariant open embeddings. In the toric case, an inclusion FQ W  0 ,!  of cones in NQ (i.e. F D idN with  0   ) provides an open embedding TV.F / if and only if  0 is a face of . Indeed, if  0 D face.; u/ WD  \ u? is cut out by the supporting hyperplane u 2  _ , then CŒ 0 _ \ M  D CŒ _ \ M u equals the localization by u . Thus, TV. 0 / D Œu ¤ 0  TV. /. Example 8. The origin f0g is a common face of all cones  . Hence, the torus T D TV.0; N /  TV.; N / appears as an open subset in all toric varieties. Similarly, if D is a p-divisor on Y and f 2 OLoc D .D.u//, then f .y/u 2 C.Y /ŒM   C.TV.D// is an M -homogeneous rational function on TV.D/. The localization procedure of the toric setting now generalizes to to the present setting: the open subset Œf u ¤ 0  TV.D/ is again a T -variety. Its associated p-divisor Df u is still supported on Y and its tailcone is tail.Df u / D face.tail.D/; u/. Moreover, Df u D face.D; u/ C ; ˝ .div f C D.u// where the face operator is supposed to be applied to the polyhedral coefficients only. More specifically, face.; u/ WD fa 2  j ha; ui D minh; uig. Note that Zu .f / WD div f C D.u/ is an effective divisor which depends “continuously” upon f . In contrast to the toric case, not all T -invariant open subsets are of the form Œf u ¤ 0. We will now present a precise characterization of open embeddings, which is quite technical, but apparently does not easily simplify in the general case. For the much nicer case of complexity one, however, P we refer to (5.2). We need the following notation: for a polyhedral divisor D D Z DZ ˝ Z and a not necessarily P closed point y 2 Y we define Dy WD y2Z DZ 2 PolC Q .N;  /. For any two p-divisors D, D 0 on a common .Y; N /, we say that D 0  D if the 0  DZ . Two such p-divisors induce polyhedral coefficients respectively satisfy DZ

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K. Altmann, N. O. Ilten, L. Petersen, H. Süß, and R. Vollmert

e

e

maps Q W T V .D 0 / ! T V .D/ and  W TV.D 0 / ! TV.D/. The former is an open 0 are faces of embedding if and only if D 0  D, i.e. if and only if all coefficients DZ 0 the corresponding DZ . In particular, this implies that tail.D /  tail.D/. Proposition 9 ([AHS08], Proposition (3.4)). The morphism  is an open embedding if and only if D 0  D and, additionally, for each y 2 Loc.D 0 / there are uy 2 tail.D/_ \M and a divisor Zy 2 jD.uy /j such that y … supp Zy , Dy0 D face.Dy ; uy /, and face.Dz0 ; uy / D face.Dz ; uy / for all z 2 Loc.D/ n supp.Zy /. Example 10. Let U  X D Spec A be an open embedding of normal affine varieties. By normality, D D X n U is a Weil divisor. Then the open embedding U  X of trivial T -varieties corresponds to the divisors D 0 D ; ˝ D  D D 0 on X . For y 2 Loc.D 0 / D U , choose fy in the ideal of D  X that doesn’t vanish at y. Then uy WD 0 and Zy WD div.fy / satisfy the conditions of Proposition 9. Note that U is not necessarily of the form Œf ¤ 0. Example 11. Consider the divisors D 0 D ; ˝ f1g  D D Œ1; 1/ ˝ f1g

e

on P 1 with tailcone Œ0; 1/. Then T V .D/ ! TV.D/ is the blowup of A2 at the origin, and the open subset T V .D 0 /  T V .D/ intersects the exceptional divisor, so TV.D 0 / ! TV.D/ is not injective.

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3.4 General toric and T -varieties. The characterization of open embeddings for affine toric varieties via the face relation between polyhedral cones leads directly to the notion of a (polyhedral) fan which is a special instance of a polyhedral subdivision: Definition 12. A finite set † of polyhedral cones in NQ is called a fan if the intersection of two cones from † is a common face of both and, moreover, if † contains all faces of its elements. (The normal fan N ./ from (2.2) is an example.) Given a fan † we can glue the affine toric varieties associated to its elements, namely [ TV.†/ WD TV. / with TV. / \ TV. / D TV. \ /;  2†

and fTV./ j  2 † maximalg is an open, affine, T -invariant covering of TV.†/. This S variety is automatically separated, and it is compact if and only if j†j WD 2†  D NQ . Moreover, the functoriality globalizes; here the right notion for a morphism .N 0 ; †0 / ! .N; †/ is a linear map F W N 0 ! N such that for all  0 2 †0 there is a  2 † with F . 0 /  . The above construction can be generalized in a straight-forward, if somewhat technical, manner to describe general T -varieties. Let D; D 0 be p-divisors on .Y; N /.

The geometry of T -varieties

25

0 Their intersection D \ D 0 is the polyhedral divisor with Z-coefficient DZ \ DZ 0 0 for any divisor Z on Y . If D  D, we say D is a face of D if the induced map  W TV.D 0 / ! TV.D/ is an open embedding, i.e. the conditions of Proposition 9 hold.

Definition 13. A finite set  of p-divisors on .Y; N / is called a divisorial fan if the intersection of two p-divisors from  is a common face of both and, moreover,  is closed under taking intersections. Similar to the toric case, we can construct a scheme from a divisorial fan  via [ TV.D/ with TV.D/ \ TV.D 0 / D TV.D \ D 0 /: TV./ WD D2

In general, the resulting scheme may not be separated, but there is a combinatorial criterion for checking this, see [AHS08, Theorem 7.5]. On the other hand, by covering any T -variety X with invariant affine open subsets, it is possible to construct a divisorial fan  with X D TV./, see [AHS08, Theorem 5.6]. In general, the notion of divisorial fan can be cumbersome to work with, in large part due to the technical nature of Proposition 9. However, a manageable substitute can be obtained in complexity 1, see (5.3).

4 How to construct p-divisors 4.1 Toric varieties. The affine toric setting presented in (2.1) fits into the language of (2.3) when setting Y D Spec C. Since divisors on points are void, the only information carried by D is its tailcone . Thus, T V .D/ D TV.D/ D TV. / in this case.

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4.2 Toric downgrades. The toric case, however, can also provide us with more interesting examples. To this end we consider a toric variety TV.ı; Nz / and fix a subtorus action T ,! Tz . Now, what is the description of TV.ı; Nz / as a T -variety? Assuming that the embedding T ,! Tz is induced from a surjection of the corresponding character z ! groups p W M ! M , we denote the kernel by MY and obtain two mutually dual exact sequences (1) and (2). !  _ . On the dual Setting  WD NQ \ ı, the map p gives us a surjection ı _ ! z side, denote the surjection N ! ! NY by q. However, instead of just considering the cone q.ı/, we denote by † the coarsest fan refining the images of all faces of ı under the map q. Then, j†j D q.ı/, and the set †.1/ of rays in †, i.e. its one-dimensional cones, contains the set q.ı.1//. 0

/ MY

r.u/ WD p 1 .u/ \ ı

 _ 

/M z / ı_

p

/M / / _ 3 u

/0

(1)

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K. Altmann, N. O. Ilten, L. Petersen, H. Süß, and R. Vollmert

0o

NY o

q

a 2 †.1/  j†j o o

Nz o ıo

N o ?_

q 1 .a/ \ ı DW Da

0

(2)

Next, we perform the following two mutually dual constructions. For u 2  _ and a 2 j†j, let r.u/  MY;Q and Da  NQ be as indicated in the diagram above. Note z of p, i.e. a splitting of the two that we have to make use of some section s W M ,! M 1 sequences, to shift both polyhedra from p .u/ and q 1 .a/ into the respective fibers over 0. Both polyhedra are linked to each other by the equality minha; r.u/i D minhDa ; ui which is an easy consequence (via a D q.x/ and u D p.y/) of the following lemma appearing in the proof of [AH03, Proposition 8.5]. Lemma 14. x 2 ı, y 2 ı _ ) minhq 1 q.x/\ı; yiCminhx; p 1 p.y/\ı _ i D hx; yi. Thus, defining Y WD TV.†/, we can refer to the upcoming discussion in (6.1) for the construction of T -invariant P Weil divisors orb.a/  TV.†/ for a 2 †.1/ to define the p-divisor D ı WD a2†.1/ Da ˝ orb.a/ on .Y; N /. Although we have indexed the coefficients of D by a ray a instead of a prime divisor, this is harmless since we may always identify a with the divisor orb.a/  TV.†/. It follows from the characterization of toric nef Cartier divisors and the above equality involving r.u/ and Da that TV.ı/ D TV.D ı /. Remark 15. Note that q does not yield a map .Nz ; ı/ ! .NY ; †/ in general. To achieve this, one has to further subdivide ı into the fan ıQ consisting of the preimages Q D T V .D/, which illustrates both of the cones of †. Moreover, observe that TV.ı/ the contraction T V .D/ ! TV.D/ as well as the morphism T V .D/ ! Y .

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4.3 General affine T -varieties. Let X  C n be an equivariantly embedded, normal, affine T -variety which intersects the torus non-trivially, and denote by f1 ; : : : ; fm M -homogeneous generating equations for X . If T acts diagonally and effectively on C n , then each variable xi has a degree in M which gives rise to a surjection z WD Zn ! p W M ! M , e 7! deg x as in the previous section. Setting ı WD Qn0 z as mentioned in (4.2) to shift and ui WD deg fi 2 M , we use the section s W M ,! M 0 s.ui / the equations fi to the Laurent polynomials fi WD  fi 2 CŒMY . They define a closed subvariety of the torus TY WD Spec CŒMY , and we use this to modify the definition of Y from (4.2) to the normalization of V .f10 ; : : : ; fm0 /  TV.†/: If orb.a/ are Cartier divisors, then we can consider their pull backs Za WD orb.a/ \ Y P. Thus, we arrive at the description X D TV.D/ as a T -variety, where D D a2†.1/ Da ˝ Za .

The geometry of T -varieties

27

Remark 16. It is sometimes easier to consider T -equivariant embeddings X ,! TV.ı/ instead of X ,! C n . In this case, everything also works out as discussed in the paragraph above. 4.4 Slices of divisorial fans. At the end of (3.4) we introduced the notion of divisorial fans, which encode invariant open affine coverings of general T -varieties. The face conditions guarantees that for any divisorial fan  D fD i g and any prime divisor Z  Y , the set i g Z D fDZ is a polyhedral subdivision in NQ called a slice. Indeed, since open embeddings between affine T -varieties TV.D 0 / ,! TV.D/ imply that D 0  D, the polyhedra i 0 DZ  NQ are supposed to be glued along the common faces DZ  DZ . These slices can be viewed another way as well. Similarly to (4.3), one can try to embed a general T -variety X into a toric variety TV.F / with some fan F of polyhedral cones ı in NzQ . Now, the method of (4.2) can be copied to yield an  on TV.†/ describing TV.F / as a T -variety denoted by TV./ and, afterwards, one follows (4.3) to find the right Y  TV.†/ to which  should be restricted in order to describe X . In this setting, the slices Z appear as intersections of the fan F with certain affine subspaces, explaining the choice of terminology. It is tempting to think that the slices Z capture the entire information of . However, the important point is that one also must keep track of which of the polyhedral cells inside the different DZ belong to a common p-divisor. This is related to the question of how much is contracting along T V ./ ! TV./. Thus, one is supposed to introduce labels for the cells in Z . Again, we refer to (5.3) for a nice way of overcoming this problem in complexity one.

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5 Globalization in complexity 1 P 5.1 The degree polyhedron. Let D D P 2Y DP ˝ P denote a p-divisor of complexity one on the curve Y with tailcone  . Since prime divisors Z coincide with closed points P 2 Y we will from now on replace the letter Z by P in this setting. As in the definition of DP in (3.3) we can now introduce the degree X deg D WD DP 2 PolC Q .N;  /: P 2Y

Note that it satisfies the equation minhdeg D; ui D deg D.u/. In contrast, considering the sum of the polyhedral coefficients of a p-divisor in higher complexity does not make much sense. Nonetheless, one may define the degree of a p-divisor on a polarized Y via the following construction: let C  Y be a curve or, more generally, a numerical class of curves in Y . Then there is a well defined generalized polyhedron .C  D/ 2 PolQ .N; / where the term “generalized” refers to an element in the Grothendieck

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group associated to PolC Q .N;  /. Hence, it is representable as the formal difference of two polyhedra. 5.2 Using deg D to characterize open embeddings. For a p-divisor D on a curve Y it is as obvious as important to observe that deg D ¤ ; , Loc D D Y: Moreover, if D is a polyhedral divisor on Y then it is not hard to see that D is a p-divisor , deg D ¨ tail.D/ and D.w/ has a principal multiple for all w 2 .tail.D//_ with w ? \ .deg D/ ¤ 0. Note that the latter condition is automatically fulfilled if Y D P 1 . Nonetheless, degree polyhedra play their most important roles in the characterization of open embeddings since they transform the technical condition from (3.3) into a very natural and geometric one. Theorem 17 ([IS], Lemma 1.4). Let Y be a curve and D 0 , D be a polyhedral and a p-divisor, respectively, such that D 0  D. Then, D 0 is a p-divisor and TV.D 0 / ,! TV.D/ is a T -equivariant open embedding if and only if deg D 0 D deg D \ tail.D 0 /. In particular, if  D fD i g is a divisorial fan as in (3.4) then the set ftail.D i / j D i 2 g forms the so-called tailfan tail./, and the subsets deg D i ¨ tail.D i / glue together to a proper subset deg  ¨ j tail./j. The single degree polyhedra can be recovered as deg D i D deg  \ tail.D i /. 5.3 Non-affine T -varieties in complexity 1. Another important feature of curves as base spaces Y is that Loc D  Y is either projective (if deg D ¤ ;) or affine (if deg D D ;). In the latter case, T V .D/ D TV.D/, and gluing those cells does not cause problems. For the other case, an easy observation is

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Lemma 18. Let D be a p-divisor on a curve Y with deg D ¤ ;. Then, for all P 2 Y , we have that dim DP D dim.tail.D//. Indeed, we have the following chain of inequalities dim.tail.D//  dim DP  dim.deg D/  dim.tail.D// each of which results from a combination of translations and inclusions. Consider a divisorial fan  D fD i g on a curve. For the following, we will make the reasonable assumption that each (lower-dimensional) cone of † D tail./ is the face of a full-dimensional cone of tail./. For example, this is satisfied if the support of tail./ is a polyhedral cone or the entire space NQ . Now we have P seen in (4.4) and (5.2) that a divisorial fan  D fD i g on a curve determines a pair . P P ˝ P; deg/. This motivates the following definition:

The geometry of T -varieties

29

P Definition 19. Consider a pair . P P ˝ P; deg/ where P are all polyhedral subdivisions with some common tailfan † and deg  j†j. This pair is called P anf-divisor if for any full-dimensional  2 tail./ with deg \  ¤ ;, D  WD Z is a pdivisor and deg \  D deg D  . Here Z denotes the unique polyhedron in Z with tail.Z / D  . P Proposition 20 ([Süß, IS, Proposition 1.6]). Consider an f-divisor . P P ˝P; deg/. Under the above assumptions, there is a divisorial fan  with slices P and degree deg. For any other such divisorial fan  0 , we have TV./ D TV. 0 /. Indeed, we can take the divisorial fan  to consist of p-divisors D  for full-dimensional  2 tail./ with non-empty degree, for  with empty degree any finite set of p-divisors providing an open affine covering of T V .D  /, and intersections thereof.

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Remark 21. P Since the subset deg  j tail./j is completely determined by the equality deg \ D D P DP , it is just necessary to know if deg D is empty or not. This leads to the notion of markings (of tailcones with deg D ¤ ;) in [Süß], [IS]. P If  is an f-divisor, i.e. a pair . P P ˝ P; deg/, we write TV./ for the T -variety it determines as in proposition 20. Note that we use the same symbol for divisorial fans and f-divisors, since they both determine general T -varieties (although the former also contains the information of a specific affine cover). As has already been observed in the affine case of (2.3), different f-divisors ;  0 might yield the same (or equivariantly isomorphic) T -varieties TV./ D TV. 0 /. ThisP is the case if there are isomorphisms ' W Y !PY 0 and F W N ! N 0 and an element i vi ˝ fi 2 N ˝ C.Y 0 / such that 0 '.P C i ordP .fi /vi D F .P / holds for every P 2 Y and we have deg0 D F .deg/. / Example 22. We consider the projectivized cotangent bundle on the first Hirzebruch surface F1 . The non-trivial slices of its divisorial fan  over P 1 are illustrated in Figure 3. The associated tailfan † and degree deg  are given in Figure 4.

(a)  0

(b)  1

(c)  ∞

Figure 3. Non-trivial slices of .P . F1 //, cf. Example 22.

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K. Altmann, N. O. Ilten, L. Petersen, H. Süß, and R. Vollmert

(a) Σ = tail( )

(b) deg 

  Figure 4. Tailfan and degree of  P . F1 / , cf. Example 22.

6 The T -orbit decomposition 6.1 T -orbits in toric varieties. Let   NQ be a not necessarily full-dimensional, pointed polyhedral cone. Then, denoting N WD N= spanZ . \ N /, the affine toric variety TV.; N / contains a unique closed T -orbit orb. / WD Spec CŒ ? \ M  D N ˝Z C  D TV.0; N / with dim orb. / D rank N  dim  . If   is a face, then we obtain the diagram TV.0; N / _ open

 TV.; N N /

Spec CΠ ? \ M  _ open  Spec CΠ_ \ ? \ M 

 orb. _ /

closed

open

  orb. /

/ TV. / _ open

closed

 / TV. /

with N WD im. ! N;Q /. This construction does Falso work in the global setting: if † is a fan, then there is a stratification TV.†/ D 2† orb. / with orb. /  orb. / ,  . In particular, we obtain the T -invariant Weil divisors of TV.†/ as orb.a/ for a 2 †.1/. Moreover, the closure of the T -orbits are again toric varieties with x N / and † x WD fN j   2 †g being a fan in N . orb. / D TV.†; Note that we can, alternatively, generalize the T -orbit decomposition of the affine TV./ to the situation of toric bouquets TV./ from (2.2). Its T -orbits correspond to the faces of . In particular, as already mentioned in (2.2), the irreducible components of TV./ are counted by the vertices v 2 V ./. 6.2 T -orbits for p-divisors. Let D be an integral p-divisor on a projective variety Y , i.e. assume that the coefficients DZ of the Cartier divisors Z are lattice polyhedra. Equivalently, we ask for D.u/ to be an ordinary Cartier divisor for all u 2 tail.D/_ \M . Then,  W T V .D/ ! Y is a flat map fitting into the following diagram:

e

e

T V .D/ 

Y o

?_

 Loc D

p

/ TV.D/  / Spec .Loc D; OLoc /

The geometry of T -varieties

31

For y P 2 Loc D  Y , the fiber  1 .y/ equals the toric bouquet TV.Dy / with Dy WD y2Z DZ 2 PolC Q .N;  / as it was already used in (3.3). Note that the neutral element tail.D/ serves as the sum of the empty set of summands. Hence, there is a decomposition of T V .D/ into T -orbits orb.y; F / where y 2 Loc D is a closed point and F  Dy is a non-empty face. Their dimension is dim orb.y; F / D codimN F . While every T -orbit orb.y; F /  T V .D/ maps, via the contraction p, isomorphically into T V .D/, some of them might be identified with each other:

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Proposition 23 ([AH06], Theorem 10.1). Let .y; F / and .y 0 ; F 0 / correspond to the T -orbits orb.y; F / and orb.y 0 ; F 0 /. Then they become identified along p if and only if N .F; / D N .F 0 ; 0 / DW (cf. (2.2)) and ˆu .y/ D ˆu .y 0 / for u 2 int , where ˆu denotes the stabilized morphism associated to the semi-ample divisor D.u/. If D is a general, i.e. not integral p-divisor, then a similar result holds true. However, one has to deal with the sublattices Sy WD fu 2 M j D.u/ is principal in yg as was done in [AH06, §7]. 6.3 Invariant Weil divisors on T -varieties. In contrast to (6.2), we are now going to use the notation orb.y; F / also for non-closed points y 2 Y , e.g. for generic points

of subvarieties of Y . Note that the dimension of their closures is then given as dim orb.y; F / D dim yN C codimN F . We conclude from (6.2) that there are two kinds of T -invariant prime divisors in T V .D/: on the one hand, we have the so-called vertical divisors D.Z;v/ WD orb. .Z/; v/

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for prime divisors Z  Y and vertices v 2 DZ . On the other hand, there are the so-called horizontal divisors D% WD orb. .Y /; %/ with % being a ray of the cone tail.D/ D D.Y / . The T -invariant prime divisors on TV.D/ correspond exactly to those on T V .D/ which are not contracted via p W T V .D/ ! TV.D/. In the special case of complexity one, i.e. if Y is a curve, the vertical divisors D.P;v/ (with P 2 Y.C/ and v 2 P ) survive completely in TV.D/. In contrast, some of the horizontal divisors D% may be contracted.

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Definition 24. Let  be an f-divisor on Y . The set of rays % 2 .tail.//.1/ for which f the prime divisor D% is not contracted via the map p W TV./ ! TV./ is denoted by R WD R./. Remark 25. The set R./ can be determined from .; deg /. Indeed, R./ D f 2 .tail.//.1/ j  \ deg  D ;g, cf. for example [Pet10, Proposition 2.1].

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7 Cartier divisors 7.1 Invariant principal divisors on toric varieties. Let †  NQ be a polyhedral fan. The monomials u in CŒM   C.TV.†// are the T -invariant, rational functions on the toric variety TV.†/. Hence, the principal divisor associated to such a function must be a linear combination of the T -invariant Weil divisors orb.a/ from (6.1). The coefficients can be obtained by restricting the situation to the open subsets C  .C  /k1 Š TV.a; N /  TV.†/ for each a 2 †.1/ separately. This gives us X div.u / D ha; ui  orb.a/; a2†.1/

P i.e. the orbits of codimension one satisfy the relation a2†.1/ a ˝ orb.a/ 0. Note that every divisor is linearly equivalent to an invariant one. Assuming that † contains a cone whose dimension equals the rank of N , this leads to the famous exact sequence  div  0 ! M ! DivT TV.†/ D Z†.1/ ! Cl.TV.†// ! 0: 7.2 Invariant principal divisors on T -varieties. Let D be a p-divisor on Y . Similarly to (3.3), the place of the monomials u in (7.1) is now taken by the functions f .y/u 2 C.Y /ŒM   C.TV.D//. Identifying a ray % 2 tail.D/.1/ with its primitive lattice vector and denoting by .v/ the smallest integer k  1 for v 2 NQ such that k  v is a lattice point, one has the following characterization of T -invariant principal divisors: Theorem 26 ([PS], Proposition 3.14). The principal divisor which is associated to f .y/ u 2 C.Y /ŒM  on T V .D/ or TV.D/ is given by X     X div f .y/ u D h%; uiD% C .v/ hv; ui C ordZ f  D.Z;v/

e %

.Z;v/

where, if focused on TV.D/, one is supposed to omit all prime divisors being contracted via p W T V .D/ ! TV.D/.

e

A proof which (formally) locally inverts the toric downgrade construction from (4.2) is given in [AP, (2.4)]. Hence, the claim becomes a direct consequence of (7.1). 7.3 Invariant Cartier divisors on toric varieties. We consider an invariant Cartier divisor D on TV.†/. Restricting to an open affine chart TV. /  TV.†/ we have that DjTV./ D div.u / with u 2 M WD M=.M \  ? /. The compatibility condition on intersections gives us that  .u / D u where denotes a face of  and  W M ! M is the natural restriction. Hence, we can regard fu j  2 †g as a piecewise continuous linear function D W j†j ! Q which is locally defined as W j†j ! Q which is linear on the D jjj D u . Vice versa, any continuous function

The geometry of T -varieties

33

cones  2 † and integral on j†j \ N can be evaluated along the primitive generators of the rays  2 †.1/ and thus defines an invariant Cartier divisor D on TV.†/. A special instance of the above correspondence shows that every lattice polyhedron r  MQ such that † is a subdivision of the inner normal fan N .r/, cf. (2.2) and (14.1), gives rise to a Cartier divisor div.r/ whose sheaf OTV.†/ .div.r// is globally generated by the monomials corresponding to r \ M . Locally, on the chart TV. /  TV.†/ for a maximal cone  2 †, one has OTV.†/ .div.r// D u  OTV.†/ with u 2 r \ M being the vertex corresponding to the unique cone ı 2 N .r/ containing  . The associated Weil divisor is then given by X minha; ri  orb.a/: div.r/ D  a2†.1/

7.4 Invariant Cartier divisors in complexity 1. Let  be an f-divisor on the curve Y . A divisorial support function on  is a collection .hP /P 2Y of continuous piecewise affine linear functions hP W jP j ! Q such that (1) hP has integral slope and integral translation on every polyhedron in the polyhedral complex P  NQ ; (2) all hP have the same linear part DW h ; (3) the set of points P 2 Y for which hP differs from h is finite. Observe that we may restrict hP to a subcomplex of P . Similarly, we may restrict a divisorial support function h to a p-divisor D 2  which we will denote by hjD . In addition, we can associate a divisorial support function sf.D/ to any Cartier divisor D 2 CaDiv Y by setting sf.D/P coeff P .D/. Moreover, we can consider any element u 2 M as a divisorial support function by setting sf.u/P u. Definition 27. A divisorial support function h on  is called principal if h D sf.u/ C sf.D/ for some u 2 M and some principal divisor D on Y . It is called Cartier if its restriction hjD is principal for every D 2  with Loc D D Y . The set of Cartier support functions is a free abelian group which we denote by CaSF./. The following result not only plays an important role for the proof of Proposition 29 but it is also interesting by itself, since it generalizes the fact that the Picard group of an affine toric variety is trivial. Lemma 28 ([PS], Proposition 3.1). Assume that D has complete locus. Then every invariant Cartier divisor on TV.D/ is already principal. Let TV./ be a complexity-one T -variety and denote by T-CaDiv.TV.// the free abelian group of T -invariant Cartier divisors on TV./. Choosing affine invariant charts on which the Cartier divisor in question trivializes, one can use its local representations f .y/u 2 C.Y /ŒM  to define piecewise affine functions hP WD ordP f Ch; ui. It is then not hard to see that these functions glue and yield an element h 2 CaSF./ which leads to the following correspondence:

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Proposition 29 ([PS], Proposition 3.10). T-CaDiv.TV.// Š CaSF./ as free abelian groups. We will sometimes identify an element h 2 CaSF./ with its induced T -invariant Cartier divisor Dh via this correspondence. As a consequence of Theorem 26 we obtain that the Weil divisor associated to a given Cartier divisor h D .hP /P on TV./ is equal to X X  h.%/D%  .v/hP .v/D.P;v/ : %

.P;v/

Remark 30. Let D denote a Cartier divisor on the toric variety TV.†/ which is associated to the integral piecewise linear function W j†j ! Q (see (7.3)). Performing a downgrade to complexity one (cf. (4.2)), we denote the induced f-divisor by . The associated Cartier support function h or, more specifically, its only non-trivial parts h0 and h1 then result from the restriction of to the slices 0 and 1 , respectively. 7.5 The divisor class group. Using the results from (6.3) and (7.2) one obtains that (cf. [PS, Section 3]) L L % Z  D% ˚ D.Z;v/ Z  D.Z;v/ ˛; Cl.TV.// D ˝ P P u.%/D% C D.Z;v/ .v/.hv; ui C aZ /D.Z;v/ P where u runs over all elements of M and Z aZ Z over all principal divisors on Y . However, assuming that TV./ is a complete rational T -variety of complexity one, it is possible to find a representation of Cl.TV.// which is analogous to the exact sequence from P(7.1) as follows. Let  D P 2P 1 P ˝ ŒP  be a complete f-divisor on P 1 . In particular, deg  ¨ j tail./j D NQ . Choose a minimal finite subset P  P 1 containing at least two points and such that P is trivial (i.e. P D tail./) for P 2 P 1 n P . If v is a vertex of the slice P , we denote P .v/ D P . Let us define V WD fv 2 P .0/ j P 2 P g together with the maps Q W ZV [R ! ZP =Z and

 W ZV [R ! N

with e.v/ 7! .v/ e.p.v// and e.%/ 7! 0; with e.v/ 7! .v/v and e.%/ 7! %

with e.v/ and e.%/ denoting the natural basis vectors. Proposition 31 ([AP], Section 2). For a complete rational complexity-one T -variety TV./, one has the following exact sequence:  _ 0 ! .ZP =Z/_ ˚ M ! ZV [R ! Cl.TV.// ! 0 where the first map is induced from .Q; /. Example 32. We return to the projectivized cotangent bundle of F1 , cf. Example 22. From the discussion above we see that its Picard rank is equal to 2  2 C 7 D 3 which confirms a classical result.

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8 Canonical divisors, positivity, and divisor ideals 8.1 The canonical divisor. Given an n-dimensional toric variety TV.†/ the invariant e1 en ^    ^ d , where fe1 ; : : : ; en g denotes a basis of logarithmic n-form ! WD d en e1 M , defines a rational differential form on TV.†/. Moreover, the latter turns out to P be independent of the chosen basis and thus leads to the description KTV.†/ D  %2†.1/ D% . Considering a torus action of complexity k > 0, let KY denote a representation of the canonical divisor on the base Y , i.e. KY D div !Y for some rational differential form !Y 2 1 .Y /. Then one can construct a rational differential form de1 dek

1 .TV.// 3 !TV./ WD !Y ^ e ^    ^ e 1 k on the T -variety TV./ where, as above, fe1 ; : : : ; ek g denotes a Z-basis of the lattice M . By (formally) locally inverting the toric downgrade construction of (4.2) the following equality is an immediate consequence of (7.1) and the above representation of KY : X X KTV./ D ..v/ coef P KY C .v/  1/ D.P;v/  D% : .P;v/

%

Example 33. Let us revisit P . F1 / from Example 22. Recall that R./ D ; and note that the anticanonical divisor KP . F1 / is Cartier. As a Weil divisor it may be represented as   KP . F1 / D 2 D.Œ0 ;.0;0// C D.Œ0 ;.0;1// C D.Œ0 ;.0;1// : Our aim now is to provide the relevant data of the associated Cartier support function h WD sf.KP . F1 / /. To this end we denote the maximal cones of the tailfan by 1 ; : : : ; 8 (starting with the cone h.1; 0/; .1; 1/i and counting counter-clockwise). To illustrate the associated piecewise linear functions hP over the relevant slices P one may use the following list which displays the local representations sf.ui / C sf.Ei / of KP . F1 / jTV.D i / where D i is the p-divisor which is associated to the maximal cone i : 1 2 3 4 ui Œ2; 0 Œ0; 2 Œ2; 2 Œ2; 0 Ei 2Œ0 C 2Œ1 0 0 2Œ0 C 2Œ1 ui Ei

5 6 7 8 Œ2; 0 Œ2; 2 Œ0; 2 Œ2; 0 0 0 2Œ0 C 2Œ1 2Œ0 C 2Œ1

More specifically, this means that for every point P 2 P 1 we have that hP jD i D P

ui C coeff P Ei which is to be considered as an affine linear map NQ  jDPi j ! Q.

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8.2 Positivity criteria. We consider a complete complexity-one T -variety TV./. For a cone  2 tail./ of maximal dimension and a point P 2 Y we denote by P 2 P the unique polyhedron whose tailcone is equal to  . Given a Cartier support function h 2 CaSF./ we may write hP jP D u. / C aP . / and define hj .0/ WD P P aP ./P . Definition 34. A function f W C ! Q defined over some convex subset C  Qk is called concave if f .tx1 C .1  t /x2 /  tf .x1 / C .1  t /f .x2 / for all x1 ; x2 2 C and 0  t  1. Given a polyhedral subdivision of C , the function f is called strictly concave if the inequality from above becomes strict for all pairs of points .x1 ; x2 / 2 C 2 which lie in different maximal cells. Proposition 35 ([PS], Section 3.4). (1) Dh is nef if and only if hP is concave for all P 2 Y and deg hj .0/  0 for every maximal cone  2 tail./. (2) Dh is semiample if and only if hP is concave for all P 2 Y and hj .0/ is semiample for every maximal cone  2 tail./. (3) Dh is ample if and only if hP is strictly concave for all P 2 Y and hjtail.D/ .0/ is ample for all maximal D 2  with affine locus. Similar results for the case of varieties with general reductive group actions of complexity one can be found in [Tim00]. It is well known for toric varieties that the anti-canonical divisor is always big. This is not true in general for higher complexity T -varieties. Nevertheless, for rational T -varieties of complexity one, it is sufficient to impose the condition of connected isotropy groups. In our language, this is equivalent to the condition that all the vertices of P are lattice points. Proposition 36. For a rational T -variety X of complexity one with connected isotropy groups, the anti-canonical divisor is big. P Proof. By (8.1) we get an effective anti-canonical divisor KX D  D C 2  P D for any point Q 2 Y . Moreover, the effective cone is generated by inQ;v v variant divisor classes. Now, for every D D D or D D DP;v we see that the class of .KX /  D is effective. This implies that KX sits in the interior of the effective cone, and is thus big. 8.3 The ideal of a Weil divisor. Previous considerations allow us to express the ideal I  .TV.D/; OTV.D/ / D .Loc D; OLoc .D// of a T -invariant prime divisor: I.D.Z;v/ / D D

L

u2tail.D/_ \M ff

2 .Loc; D.u// j div.f u /  D.Z;v/ g  u

u2tail.D/_ \M ff

2 .Loc; D.u// j hv; ui C ordZ f > 0g  u :

L

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Since, if f belongs to .Loc D; D.u//, the expression minhDZ ; ui C ordZ f is nonnegative, we obtain for D.Z;v/ the affine coordinate ring L .Loc D; O/=I D u2N .v;DZ /\M I hv;ui2Z .Loc; D.u//= .Loc; D.u/  Z/ L  u2N .v;DZ /\M I hv;ui2Z .Z; D.u/jZ /: z ! Z denote the normalization map. Furthermore, we define the cone Let ' W Z .Z;v/ WD Q0  .DZ  v/ and denote by  W Z ,! Y the inclusion. Then we have that X .DW C .Z;v/ / ˝ .' ı / W D.Z;v/ WD W Y

z with lattice N.Z;v/ D .N C Zv/  NQ such that defines a p-divisor over Z

B

TV.D.Z;v/ / D D.Z;v/

equals the normalization of the prime divisor D.Z;v/  TV.D/ (see 6.1). Note that .' ı / Z is defined only up to linear equivalence, and different choices give rise to isomorphic T -varieties. For a prime divisor D  TV.D/ of horizontal type we obtain the ideal M % WD O.D.u// _ n%? u2DY

in an analogous way. The corresponding p-divisor X D% WD pr % .DZ /  Z; Z

lives on Y and has tailcone pr % .tail.D// with lattice pr % .N /, where pr % is the projection NQ ! NQ =Q  %. As above, we have that D% D TV.D% /.

9 Cohomology groups of line bundles in complexity 1 9.1 Toric varieties. Consider an equivariant line bundle L on the toric variety TV.†/. i The induced torus action on the cohomology spaces  L/ yields  a weight LH .TV.†/; i decomposition of the latter, i.e. H .TV.†/; L/ D u2M H i TV.†/; L .u/. As in (7.3), we denote by W j†j ! Q a continuous piecewise linear function representing the line bundle L. Introducing the closed subsets Zu WD fv 2 NR j h u; v i  .v/g, the complex with entries H 0 .; Zu I C/ associated to the covering of j†j by the cones ˇ  2 † can be identified with the usual Cech complex for TV.†/ given by the covering via the affine open subsets TV. /. This gives us that     H i TV.†/; L .u/ D H i j†j; Zu I C ;

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cf. [Dem70]. It follows that if † is complete and h is concave, e.g. L is globally generated, then H i .TV.†/; L/ D 0 for i > 0; see [Dan78, Corollary 7.3]. One also has an analogous result for toric bouquets which is proved by induction on the number of irreducible components and the fact that the restriction homomorphism on the level of global sections is surjective (cf. [Dan78, Lemma 6.8.1]): Proposition 37 ([Pet10], Section 2.4). Let X be a complete equidimensional toric bouquet and L a nef line bundle on X . Then H i .X; L/ D 0 for i > 0. 9.2 Global sections. Recall from (7.3) that a globally generated equivariant line bundle L on a complete L toric variety TV.†/ corresponds to a polytope r  MQ such that .TV.†/; L/ D u2r\M Cu . In the following we will see how to generalize this formula to complexity-one T -varieties. Consider a T -invariant Weil divisor D on a complexity-one T -variety TV./ over the curve Y and define the polyhedron D WD convHull fu 2 MQ j h%; ui   coef % D for all % 2 Rg; where coef % D denotes the coefficient of D% inside D. Moreover, there is a map D  W D ! DivQ Y with coef P D  .u/ WD minfhv; ui C coef .P;v/ D=.v/ j v 2 P .0/g. As a direct consequence of Theorem 26 one obtains that      X; OX .D/ .u/ D  Y; OY .D  .u// ; cf. [PS, Section 3.3]. In the case that D D Dh is Cartier the function D  is usually denoted by h and we write h instead of D . Example 38. Let us return to Example 33 and compute dim .P . F1 /; KP . F1 / /. The weight polytope of the anticanonical divisor is pictured in Figure 5. Together with all weights .u1 ; u2 / D u 2 h the following list displays the degree of the induced divisor h .u/ on P 1 : 0 1 2 1 0 1 2 2 1 0 1 2 1 0 u1 2 2 2 1 1 1 1 0 0 0 0 0 1 1 u2 deg h .u/ 0 0 0 0 1 1 0 0 1 2 1 0 0 0 Summing up over all degrees yields dim .X; KX / D 20. 9.3 Higher cohomology groups for complexity-one torus actions. We suspect that higher cohomology group computations for equivariant line bundles on complexityone T -varieties are rather hard in general, in particular if one cannot directly make use

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3 2 1 0 -3

-2

-1

0

1

2

3

-1 -2

Figure 5. Weight polytope associated to KP . F1 /, cf. Example 38.

of a quotient map to the base curve Y . Hence, we restrict to the case where X D Xz . Invoking Proposition 37, one can directly generalize the toric vanishing result from (9.1) with a “cohomology and base change” argument for the flat projective morphism  W Xz ! Y to the complexity-one case: Proposition 39 ([Pet10], Section 2.4). Let X D Xz be a complete complexity-one T variety over the base curve Y . For any nef T -invariant Cartier divisor Dh on X we have that Ri  OX .Dh / D 0 for i > 0. Hence, M   H i .X; OX .Dh // D H i Y; OY .h .u// : u2h \M

In particular, H i .X; OX .Dh // D 0 for i  2. A precursor of this result for smooth projective surfaces and semiample line bundles is formulated as Corollary 3.27 in [IS10]. Its proof uses the well known intersection theory for smooth projective surfaces, cf. [Har77, Chapter V]. Furthermore, it is worthwhile to note that the direct Limage sheaf always splits into a direct sum of line bundles, namely  OXz .Dh / D u2 h \M OY .h .u// with no positivity assumptions on Dh . Since the higher direct image sheaves are not locally free in general, explicit higher cohomology group computations have so far proven unsuccessful.

10 Cox rings as affine T -varieties Assume X to be a complete normal variety with finitely generated divisor class group Cl.X/. The Cox ring of X is then given as the Cl.X /-graded abelian group M Cox.X / WD .X; OX .D// D2Cl.X/

which carries a canonical ring structure, see [HS10, Section 2]. In analogy to [HK00], but neither supposing X to be Q-factorial nor projective, we call X a Mori dream space (MDS) if Cox.X / is a finitely generated C-algebra.

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10.1 The Cox ring of a toric variety. well known quotient construction P n Š  The   Spec CŒz0 ; : : : ; zn  n V .z0 ; : : : ; zn / =C can be generalized to toric varieties TV.†/ for which the set of rays †.1/ generates NQ (cf. [Cox95]). The graded homogeneous coordinate ring CŒz0 ; : : : ; zn  for P n is replaced by the polynomial ring CŒz% j % 2 †.1/ whose variables correspond to the rays of † and whose grading is induced by the divisor class group Cl.TV.†//, which assigns the degree ŒD%  to the variable z% . Further, applying HomZ .; C  / to the short exact sequence from (7.1), one defines the algebraic group G WD ker .C  /†.1/ ! TN where TN WD N ˝Z C  D HomZ .M; C  /. Note that G acts naturally on C †.1/ and leaves V .B.†// invariant, where B.†/ ˚  denotes the irrelevant Q  CŒz% ˇj % 2 †.1/ ideal which is generated by zO WD %….1/ z% ˇ  2 † . The ideal B.†/ generalizes the standard irrelevant ideal hz0 ;: : : ; zn i  CŒz0 ; :: : ; zn  of projective space. Furthermore, it follows that TV.†/ D C †.1/ n V .B.†// =G is a good quotient. On the other hand, one can also approach the Cox ring via a more polyhedral point of view. As usual we denote the first non-trivial lattice point on a ray % 2 †.1/  NQ with the same letter %. Then we consider the canonical map ' W Z†.1/ ! N , e.%/ 7! %. It sends some faces (including the rays) of the positive orthant Q†.1/ 0 to cones of the fan †. Applying the functor TV, we obtain a rational map Spec CŒz% j % 2 †.1/! TV.†/. In particular, we recover the affine spectrum of CŒz% j % 2 †.1/ D Cox.TV.†// as the toric variety TV.Q†.1/ 0 /. Thus, the Cox ring of a toric variety gives rise to an affine toric variety itself, and the defining cone Q†.1/ 0 can be seen as a polyhedral resolution of the given fan †, since all linear relations among the rays have been removed.

10.2 The action of the Picard torus. Let X be an MDS and suppose that Cl.X / is torsion free. It follows that the total coordinate space Spec Cox.X / is a normal affine variety, and the Cl.X /-grading encodes an effective action of the so-called Picard torus T WD HomalgGr .Cl.X /; C  /. One could now ask for a description of Spec Cox.X / in terms of some p-divisor ECox on the so-called Chow quotient Y WD Spec Cox.X /==ch T which is defined as the normalization of the distinguished component of the inverse limit of the GIT quotients of Spec Cox.X /, cf. [AH06, Section 6]. This has been done in [AW], and in the case of smooth Mori dream surfaces, the result is as follows (cf. [AW, P Section 6]): Y D X and, up to shifts of the polyhedral coefficients, ECox D E X E ˝ E with E D fD 2 Eff.X /  ClQ .X / j .D  E/  1 and .D  E 0 /  0 for E 0 ¤ Eg where E, E 0 run through all negative curves in X . The common tailcone of the E equals Nef.X/, which is dual to Eff.X /, where the latter cone carries the degrees of Cox.X/. If X is a del Pezzo surface, the formula for E simplifies to E D 0E C Nef.X/  ClQ .X /. 10.3 Generators and relations for complexity-one torus actions. It is a fundamental problem to give a presentation of the Cox ring of an MDS in terms of generators and

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relations. The crucial idea to approach this problem for a T -variety TV./ is to relate its presentation to an appropriate quotient of TV./ by the torus action. This ansatz was pursued in [HS10] whose key result states that the Cox ring of TV./ is equal to a finitely generated algebra over the Cox ring of Y ı , where Y ı  Y is the image of the rational map coming from the composition of the rational p 1 W TV./ ! T V ./ with  W T V ./ ! Y . Let us now become more specific for complexity-one torus actions. For a point P 2 P 1 together with the set VP WD fD.P;v/  j v 2 P .0/g of all  vertical divisors lying over P , we define the tuple .P / WD .v/ j v 2 P .0/ : After applying an automorphism of P 1 , we may assume that P D f0; 1; c1 ; : : : ; cr g, with ci 2 C  (cf. (7.5)).

e

e

Theorem 40 ([HS10], Theorem 1.2). The Cox ring of TV./ is given by ı˝ ˛  C TD.P;v/ ; SD j P 2 P ; % 2 R T .0/ C ci T .1/ C T .ci / j i D 1; : : : ; r ; where T .P / WD

Q v2P .0/

TD .v/ and R is as in Definition 24. .P;v/

Example 41. Let us describe the Cox ring of P . F1 /, cf. Example 22. Using Theorem 40 and representatives .1; 1/, .1; 0/, and .0; 1/ for 1, 0, and 1, respectively, we see that Cox.P . F1 // D CŒT1 ; : : : ; T7 =.T1 T2 T3  T4 T5  T6 T7 /. 10.4 Polyhedral point of view for complexity-one torus actions. Similar to the viewpoint taken in (10.2) and keeping the setting from (10.3), one can describe Spec Cox.X/ via a p-divisor DCox . In this case, however, the given T -action on X carries over to Cox.X /, and by combining it with the action of the Picard torus, Spec Cox.X/ turns into a variety with an effective complexity-one action of a diagonizable group which in general involves torsion. Factoring out the latter gives rise to a finite abelian covering C ! P 1 (see (17.4)), so that DCox lives on a curve C of usually higher genus rather than on P 1 . Nonetheless, if the class group Cl.X / is torsion free C ! P 1 is the identity map. In this case, the p-divisor DCox describing Spec Cox.X / S utilizes the very same points P 2 P 1 as . Denoting by V WD P 2P P .0/, we define the compact polyhedra Pc WD convfe.v/=.v/ j v 2 P .0/g  QV[R where the e.v/ 2 ZV denote the canonical basis vectors and R is as usual. We similarly define the polyhedral cone Y V[R Pc C QR  WD Q0  0  Q0 : S2P

Proposition 42 ([AP], Theorem 1.2). Assume Cl.X / to be torsion free. Then the pdivisor DCox of Spec Cox.X / is, up to shifts of the polyhedral coefficients, given by P c 1 . P 2P P C  / ˝ ŒP  on P .

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For more details and the general result which also covers the case of a class group with torsion we refer to [AP, Theorem 4.2]. Comparing Proposition 42 with the analogous result in the toric setting (10.1), we see that it consists of a similar construction which resolves the linear dependencies among the elements of P .0/ 2 NQ and the rays % 2 R by assigning separate dimensions to each of them. Example 43. Once more we consider the threefold P . F1 //, cf. Example 22. The tailcone  WD tail.DCox / of the p-divisor DCox is then given as the cone over the product of a quadrangle (product of two intervals) and a triangle. This realization is induced by the three polytopes c1 (triangle), c0 (compact interval), and c1 (compact interval). The non-trivial coefficients of DCox are thus equal to Pc C  , P 2 f1; 0; 1g (up to a shift). It turns out that the polyhedral divisors ECox (as constructed in (10.2)) and DCox are related by upgrade and downgrade constructions via the exact sequence from above, cf. [IVb] and [Pet10, Section 3.5].

11 Invariant valuations and proper equivariant morphisms For a function field C.X / of a variety X=C, we consider discrete C-valuations of F . These are maps  W C.X / ! Q [ f1g fulfilling the properties (1) .f  g/ D .f / C .g/, (2) .f C g/  min..f /; .g/), (3) .C  / D 0, and .f / D 1 , f D 0. A center of a valuation is a point x 2 X such that .OX;x /  0 and .mX;x / > 0. By the valuative criterion for properness [Har77, Theorem 4.7], a scheme is complete if every discrete valuation has a unique center. Moreover, it is separated if there exists at most one center. 11.1 Completeness and properness for toric varieties. For a toric variety X D TV.†/, the function field of X equals the quotient field of CŒM  and a valuation  defines a linear form on M via v W u 7! .u /. Moreover, this gives a one-to-one correspondence between NQ and the discrete valuations on C.X /. Here, valuations with center on X correspond to elements in the support of †. Hence, a toric variety TV.†/ is complete if and only if its fan is complete, i.e. j†j D NQ . Moreover, let F W † ! †0 be a map of fans. Then F defines a proper morphism exactly when j†j D F 1 .j†0 j/. 11.2 Valuations on T -varieties and hypercones. The function field of a T -variety X D TV./ equals the quotient field of C.Y /ŒM . An invariant valuation corresponds via the relation .f  u / D .f / C hu; vi to a pair .; v/, where  is a valuation on C.Y / and v 2 NQ . In this situation we write simply  D .; v/.

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If  is a valuation with center y 2 Y , then we get a well defined group homomorphism  W CaDiv0 .Y / ! Q0 I D 7! .f /; where D D div.f / locally at y. This map extends to a homomorphism  W PolC .N;  / ˝Z0 CaDiv0 .Y / ! PolC .N;  /: Now we have the following Proposition 44 ([AHS08, Lemma 7.7]). A valuation .; v/ has a center on TV.D/ if and only if v 2 .D/. Remark 45. The criterion v 2 .D/ implies in particular that  has a center in the locus of D. Now, we turn to the case of complexity one. In this situation Y is a smooth complete curve and the non-negative multiples m  ordy of the vanishing orders at points y 2 Y are the only discrete valuations. For those  we thus have .D/ D m  Dy which is equal to the tailfan if m D 0. For an invariant valuation .; v/ with center on X D TV.D/, we must either have  0 and v 2 tail.D/ or  6 0 and .v; m/ 2 hDy  f1gi  NQ  Q. Thus, the set of invariant valuations with center on X can be identified with the disjoint union of the polyhedral cones htail.D/  f0g [ Dy  f1gi glued together along the common subsets tail.D/  f0g. Such an object is called a hypercone. It was introduced by Timashev in [Tim97] (in a version suitable to describe more generally reductive group actions of complexity one). Since the valuations with center already determine an affine variety, a hypercone gives an alternative description of a affine T -variety of complexity one. In this language a polyhedral divisor is nothing but a (hyper-)cut through a hypercone. 11.3 Complete T -varieties and proper morphisms. Consider a T -variety X D TV./ with  D fDi g1ir . The divisorial fan  provides us with an open affine torus invariant covering of X by the varieties Ui D TV.Di /, 1  i  r. As already mentioned at the start of (11), the valuative criterion of completeness states that a variety X is complete exactly when every valuation of C.X / has a unique center. For T -varieties it is in fact sufficient to restrict oneself to invariant valuations. Using the results of the previous section we obtain the following Theorem 46 ([AHS08, Theorem 7.5]). TV./ is complete if and only if every slice ./ WD f.D/ j D 2 g is a complete subdivision of NQ . As a relative version of this result we obtain a characterization of proper equivariant morphisms. Recall from (3.2) that an orbit dominating equivariant morphism X 0 ! X of T -varieties is given by a triple D .'; F; f/, consisting of a dominant morphism ' W Y 0 ! Y , a lattice homomorphism F W N 0 ! N and an element f 2 C.Y 0 / ˝ N .

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First we define the preimage of a p-divisor over Y and N via over Y 0 and N 1 D WD F 1 .'  D C div.f//:

as a polyhedral divisor

Let X 0 D TV. 0 / be a T -variety with  0 D fDi0 g1ir . Assume that map of polyhedral divisors Di0 ! D for every i .

defines a

Theorem 47. The morphism X 0 ! X D TV.D/ induced by is proper if and only if  0 WD f.Di0 /gi defines a complete polyhedral subdivision of . 1 .D// for every . Since the properness can be checked locally on the base, the above theorem can be used to determine whether an arbitrary orbit dominating equivariant morphism X 0 ! X is proper. Remark 48. It follows from [HI, Proposition 1.5] that it suffices to check the conditions of Theorems 46 and 47 just for slices of type y instead of for all slices  .

12 Resolution of singularities 12.1 Toric resolution of singularities. The only non-singular affine toric varieties are products of tori and affine spaces and are given by cones  spanned by a subset of a lattice basis. We will call such cones regular. Reducing to the sublattice N \ .Q   / we may assume that  is of maximal dimension. A cone  D ha1 ; : : : ; ar i is regular if and only if it is simplicial and its multiplicity m. / WD det.a1 ; : : : ; ar / equals ˙1. Since equivariant proper birational morphisms to an affine toric variety X D TV. / correspond to fans subdividing the cone , a resolution of X is given by a triangulation of  into regular cones. Indeed, such a triangulation always exists. It can be constructed by starting with an arbitrary triangulation and subsequently P refining non-regular cones D ha1 ; : : : ; ar i with rays along lattice elements aQ 2 i Œ0; 1/  ai . On the one hand, every non-regular cone contains such a lattice element. On the other, the resulting cones of theP stellar subdivision along aQ have strictly smaller multiplicities than . Indeed, if aQ D i ai we get m. i / D i m. /, where i is the cone spanned by aQ and all the generator of except ai . So after a finite number of steps we end up with a subdivision in regular cones. This combinatorial approach to resolving toric singularities was generalized by Varchenko in [Var76] to obtain (embedded) resolutions of non-degenerate Phypersurface singularities. Consider a regular cone !  MQ and a polynomial h D u2M ˛u u 2 CŒ! \ M  Š CŒx1 ; : : : ; xn . Now, the normal fan †h of the unbounded polyhedron _ r D conv.supp h C !/ refines the dual cone a compact face F r we P  D ! . For consider the restricted polynomial hF D u2F \M ˛u u . If all restricted polynomials define smooth hypersurfaces in the torus T D Spec CŒM , we call h non-degenerated. Theorem 49. If h is non-degenerated and ' W TV .†0 / ! TV .†h / is a toric resolution of singularities then the strict transform of V .h/ under the induced morphism TV.†0 / ! TV. / D An is an (embedded) log-resolution of the singularities of V .h/.

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45

Example 50. We consider the surface singularity X given by the equation h D x 2 C y 3 C z 5 . In a first step, we refine the positive orthant  D ! _ by replacing it with the normal fan †h of the Newton diagram of h. This Newton diagram and its normal fan are pictured in Figure 6; †h is obtained from the positive orthant  via a stellar subdivision along the ray .15; 5; 6/. In a second step we refine this fan and obtain †0 as pictured in Figure 7. The first step introduces a central exceptional divisor in X corresponding to the ray  D h.15; 5; 6/i, with the remaining singularities in the strict transform of X being toric. In the second step, the singularities of TV.†h / are resolved, which also resolves the (toric) singularities of the strict transform of X . z

y x Figure 6. The Newton diagram of h and its normal fan †h .

Remark 51. The approach of Varchenko and the notion of non-degeneracy is generalized in the context of tropical geometry, see [Tev07]. 12.2 Toroidal resolutions. We recall from [KKMSD73], [Dan91] that a complex normal variety X is toroidal at the closed point x 2 X if there exists an affine toric variety TV./ such that the germ .X an ; x/ is analytically isomorphic to the germ .TV./an ; x / with x being the unique fixed point in TV. /. If the above property holds for every x 2 X then X itself is called toroidal. More generally, let B  X be a closed subvariety. The pair .X; B/ is called toroidal if for every closed point x 2 X , there exists a toric variety TV. / together with a distinguished set of faces F . / ofS  such that the germ .X an ; B an I x/ is analytically an isomorphic to the germ .TV. / ; 2F ./ TV. /an I x /. Given a p-divisor on some Y , we consider a log-resolution ' of the pair .Y; supp D/. The pullback '  D leads to the same variety, i.e. X D TV.D/ D TV.'  D/, but by [LS, Proposition 2.6] we get a toroidal pair .Xz ; B/ consisting of the variety Xz D T V .'  D/ and the exceptional set B of the contraction morphism r W T V .'  D/ ! TV.'  D/.

e

e

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K. Altmann, N. O. Ilten, L. Petersen, H. Süß, and R. Vollmert

(0, 0, 1)

(3,2,2) (6,4,3) (9,6,4)

(1,1,1)

(12,8,5)

(4,3,2)

ρ

(2,1,1) (5,3,2)

(7,5,3) (10,7,4)

(8,5,3)

(2,2,1) (5,4,2)

(3,2,1)

(0, 1, 0) (1, 0, 0) Figure 7. Refinement of †h .

e

Hence, T V .'  D/ is called a toroidal resolution of X . Moreover, for any closed z Xz is formally locally isomorphic in a neighborhood of x to the toric point x 2 X, variety which is given by the so-called “Cayley cone” C.DZ1 ; : : : ; DZr / D h.tail.D//  0; DZ1  e1 ; : : : ; DZr  er i  NQ  Qr : Here, Z1 ; : : : ; Zr denotes a set of prime divisors intersecting transversely in the point .x/ 2 Y . Note that one can obtain a resolution of singularities from this construction by purely toric methods when applying them to the occurring Cayley cones. In particular, T V .D/ is smooth if and only if all the Cayley cones C.DZ1 ; : : : ; DZr / are regular. If Loc D is affine we have that T V .D/ D TV.D/ and thus obtain a smoothness criterion in this case. If Loc D is complete then, as in the toric situation, TV.D/ is smooth if and only if it is a product of toric and affine spaces [LS, Lemma 5.2]. Hence, D can be obtained by a toric downgrade, see Section 4. For general loci the situation seems to be more complicated.

e

e

Remark 52. The theory of toroidal embeddings [KKMSD73] associates with the toroidal resolution Xz ! X a fan that is glued from the Cayley cones described above. The same construction shows that p-divisors generalize Mumford’s P description of complexity-one torus actions [KKMSD73, Chapter 4, §1]: if D D DP ˝ P

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is a p-divisor on a curve, then the corresponding toroidal fan is obtained by gluing the cones C.DP / D h.tail.D//  0; DP  1i along their common face tail.D/ [Vol10]. This in turn coincides with Timashev’s description in terms of hypercones and hyperfans [Tim97], [Tim08]. Example 53. Given N D Z, Y D P 2 D Proj.CŒx; y; z/ and the two prime divisors Z1 D V .y/ and Z2 D V .x 2  zy/ we consider the p-divisor D D Œ1=2; 1/ ˝ Z1 C Œ1=3; 1/ ˝ Z2 : The corresponding affine variety is the hypersurface in A4 given by x1 x22  x32  x43 . To obtain a toroidal resolution we first need to blow up the intersection points of Z1 and Z2 or their strict transforms, respectively, in order to get to a normal crossing situation, which is pictured in Figure 8. E2 Z1

Z2

E1

Figure 8. Normal crossing situation after three blow ups.

Let us denote the morphism coming from the composition of these blowups by ' W Yz ! Y . For the pullback of D we obtain '  D D Œ1=2; 1/ ˝ Z1 C Œ1=3; 1/ ˝ Z2 C Œ1=6; 1/ ˝ E1 C Œ1=3; 1/ ˝ E2 :

e

Here, E1 and E2 denote the exceptional divisors. Now, T V .'  D/ ! X is a toroidal resolution. We will consider the intersection point Z1 \ E2 . The corresponding Cayley cone is illustrated in the left picture of Figure 9. It is spanned by .1; 0; 0/, .1; 2; 0/ and .1; 0; 3/. There is a canonical resolution spanned by the additional rays .1; 1; 1/, .1; 1; 0/, .1; 0; 1/ and .1; 0; 2/, cf. right-hand picture in Figure 9. For the other intersection points one has to refine the Cayley cone similarly. Let us now turn to the case of complexity one. Here, .Y; supp D/ is already smooth. Hence, we only have to consider T V .D/. It has only toric singularities which are given by the cones P D hDP  f1gi  N  Q and can be resolved by toric methods.

e

Example 54. On Y D P 1 we consider the p-divisor D D Œ1=2; 1/ ˝ 0 C Œ1=3; 1/ ˝ 1 C Œ1=5; 1/ ˝ 1: This is exactly the surface from Example 50. The corresponding cones 0 , 1 , and 1 and their toric resolutions are pictured in Figure 10. Note that the resolution obtained here is exactly the resolution from Example 50. Moreover, the cones 0 , 1 , and 1 can be identified with the facets hei ; i in the fan †h . Now, the subdivisions of these cones can be seen as induced by the subdivision of †h we have considered.

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e2

(1,0,3)

DE 2 ×e2

e1 N

DZ1 ×e1

(1,0,2) (1,1,1)

(1,0,1) (1,0,0)

(1,1,0)

(1,2,0)

Figure 9. Cayley cone C.DZ1 ; DE2 / and its refinement.

(1, 5)

(1, 3) (−1, 2)

(a) σ0

(b) σ∞

(c) σ1

Figure 10. The cones y and their refinements.

This observation is not a coincidence, but a result of the toric embedding coming from the Cox ring representation in Theorem 40. Indeed, in the case of hypersurfaces singularities with complexity one torus action these two resolution strategies are closely related.

13 (Log-)terminality and rationality of singularities 13.1 Canonical divisors and discrepancies in the toric case. Recall P from (8.1) that a canonical divisor on a toric variety X D TV. / is given by KX D   D . Since every line bundle trivializes on an affine toric variety, X is Q-Gorenstein of index `

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exactly when there is an element w 2 1` M such that hw; i D 1 for all rays  2  .1/ and ` 2 N is minimal with this property. Hence,  is the cone over a lattice polytope in the hyperplane Œhw; i D l with primitive vertices. Let † denote a triangulation of  which comes from a resolution of singularities. A ray  2 †.1/ n  .1/ then corresponds to an exceptional divisor with respect to the map TV.†/ ! TV. / and its discrepancy is given by discr D hw; i  1. Obviously, this value is always greater than 1 which implies log-terminality of toric varieties. Note that X is canonical if and only if trunc. / WD  \ fhw; i < 1g contains no lattice points apart from the origin. Furthermore, X is terminal if and only if trunc./ D  \ Œhw; i  1 contains only the cone generators as lattice points.

height=2

height=1

height=1

(a) non-canonical

(b) canonical

(c) terminal

Figure 11. Different types of toric singularities.

13.2 Rationality and log-terminality for p-divisors. Since toric singularities are log-terminal and thus rational it is sufficient to study a toroidal resolution of X , e.g. the contraction map T V .'  D/ ! TV.'  D/, for checking the rationality or logterminality of a variety. This observation leads to the results of [LS], which are summarized in this section.

e

Theorem 55 ([LS, Theorem 3.4]). TV.D/ has rational singularities if and only if H i .Loc D; O.D.u/// D 0 holds for all u 2 M \ tail.D/_ and all i > 0. Corollary 56. If TV.D/ has complexity one, i.e. Y is a curve, then TV.D/ has rational singularities if and only if (1) Loc D is affine, or (2) Y D P 1 and degbD.u/c  1 for all u 2 .tail.D//_ \ M . To obtain a nice characterization of log-terminality we need to restrict ourselves to sufficiently simple torus actions. A torus action on an affine variety is called a (very) good one, if there is a unique fixed point, which is contained in the closures of all orbits (and the GIT-chamber structure of X is trivial). Such a situation corresponds

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to a p-divisor D having a tailcone of maximal dimension, such that the locus of D is projective (and D.u/ is ample forP all u 2 relint  _ ). Fix a canonical divisor KY D Z cZ Z on Y . Recall from (8.1) that a T -invariant canonical divisor on T V .D/ and TV.D/, respectively, is given by X X E ..v/.1 C cZ /  1/D.Z;v/  KD

e



.Z;v/

e

where one is again supposed, if focused on TV.D/, to omit all prime divisors being contracted via p W T V .D/ ! TV.D/. Theorem 57 ([PS], Corollary 3.15). An affine T -variety TV.D/ with a good torus action is Q-factorial if and only if X .0/ .#DZ  1/ C #D0.1/ D dim N: rank Cl.Y / C Z

In particular, Y has a finitely generated class group. Remark 58. Note, that the left hand side of the equation in the theorem is always at least dim N due to the properness condition for D. We now turn to criteria for (log-)terminality. Due to the toroidal resolution from above one obtains the following result which generalizes the fact of log-terminality of toric varieties. Proposition 59 ([LS, Corollary 4.10]). Every Q-factorial variety X with torus action of complexity strictly smaller than codim.SingX /  1 is log-terminal. For a p-divisor we introduce a boundary divisor on its locus loc.Y /. Let Z be the maximal multiplicity .v/ of all vertices v 2 DZ . We define BD

X Z  1  Z: Z Z

Theorem 60 ([LS, Theorem 4.7]). A Q-factorial affine variety TV.D/ with a very good torus action is log-terminal if and only if the pair .loc.Y /; B/ is log-terminal and Fano. Remark 61. Here, for simplicity we restricted ourselves to the case of Q-factorial singularities. In [LS] the more general case of Q-Gorenstein singularities is considered.

14 Polarizations 14.1 Toric varieties built from polyhedra. Let r be a lattice polyhedron in MQ , i.e. its vertices (we assume that there is at least one) are contained in M . Then we

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can choose a finite subset F  r \ M such that F C .tail.r/ \ M / D r \ M . Moreover, let E  tail.r/ \ M be the Hilbert basis, cf. (2.1). Then, if ye (e 2 E) and zf (f 2 F ) denote the affine and projective coordinates, respectively, we define P .r/  C #E  PC#F 1 as the zero set of the equations Q Q Q Q i yei j zfj D k yek l zfl associated to relations P P P P i .ei ; 0/ C j .fj ; 1/ D k .ek ; 0/ C l .fl ; 1/ inside M ˚ Z. If r D  _ from (2.1), then this yields the affine variety P . _ / D TV./. examples are r D Œ0; 1  Q, Œ0; 12  Q2 , and  The easiest compact  conv .0; 0/; .1; 0/; .0; 1/  Q2 , yielding P 1 , P 1  P 1 , and P 2 , respectively. A nice mixed example r arises from adding Q20 to the line segment .1; 0/; .0; 1/, see Figure 12. With E D f.1; 0/; .0; 1/g and F D f.1; 0/; .0; 1/g we obtain P .r/  C 2  P 1 cut out by the single equation y.1;0/ z.0;1/ D y.0;1/ z.1;0/ . That is, P .r/ equals C 2 blown up in the origin.

(a) ∇

(b) N (∇)

Figure 12. From r  MQ to the toric variety TV.N .r//, cf. (14.1).

14.2 Maps induced from basepoint free divisors. We will present the relation between the toric varieties TV.†/ from (3.4) and those rather simple ones defined in (14.1). Let r be again a lattice polyhedron in MQ and denote by † WD N .r/ its inner normal fan, cf. (2.2). Then,  WD j†j and tail.r/ are mutually dual cones and there is a structure morphism TV.†/ ! TV. /. By (7.1), the polyhedron r corresponds to a basepoint free divisor which is globally generated by F  r \ M . If r \ M generates the abelian group M , then the associated morphism ˆr W TV.†/ ! P .r/

over TV. /  C E

is the normalization map. If r is replaced by a sufficiently large scalar multiple r WD N  r (for each vertex u 2 r, the set r  u has to contain the Hilbert basis of the cone generated by it), then it becomes an isomorphism. Thus, r is even an ample divisor. Moreover, if   r is a face, then we can use the face-cone correspondence from (2.2) to define orb./ WD orb.N .; r//  TV.†/. Note that orb./ D P ./  P .r/ if ˆr is an isomorphism.

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14.3 Coordinate changes. For projective toric varieties P .r/ D TV.†/ with † D N .r/, there are three sorts of coordinates hanging around. First, for every u 2 M , the element u 2 CŒM  we have seen   is a rational function on TV.†/. Second, in (10.1) that TV.†/ D C †.1/ n V .B.†/ /=G with G D ker .C  /†.1/ ! T . This leads to the “Cox coordinates” ya for a 2 †.1/. Finally, in (14.1) and (14.2) we have seen the projective coordinates zu for u 2 r \ M . What does the relation between these coordinates u , ya , and zu look like? 0 First, if u; u0 2 r \ M , then zu =zu0 D uu . In particular, if u0 is the ver0 tex corresponding to  2 †, i.e. if  D N .u0 ; r/, then uu 2 CŒ _ \ M  is a regular function on the open subset TV. /  TV.†/. If u 2 r \ M , then the asu sociated effective divisor r.u/   2 jOTV.r/ .r/j equals r.u/ D div. / C div.r/ D P a2†.1/ ha; ui  minha; ri . Hence, the two sets of homogeneous coordinates compare via Y yaha;uiminha;ri : zu D a2†.1/

Note that shifting r without moving u accordingly does change the exponents. They just measure the lattice distance of u from the facets of r. Finally, since the rational functions u are quotients of z coordinates, one obtains that Y yaha;ui : u D a2†.1/

This product is invariant under the G-action. 14.4 Divisorial polytopes. In (14.1) and (14.2) we have seen that polarized toric varieties correspond to lattice polyhedra. This can be generalized to complete complexityone T -varieties if we replace lattice polyhedra with so-called divisorial polytopes: Definition 62. Let Y be a smooth projective curve. A divisorial polytope ‰ on Y consists of a lattice polytope   MQ and a piecewise affine concave function (cf. Definition 34) X ‰D ‰P ˝ P W  ! DivQ Y; such that (1) For u in the interior of , deg ‰.u/ > 0 ; (2) For any vertex u of  with deg ‰.u/ D 0, ‰.u/ is trivial; (3) For all P 2 Y , the graph of ‰P is integral, i.e. has its vertices in M  Z. We construct a polarized T -variety from ‰ in a roundabout manner. For v 2 NQ and any point P 2 Y , set ‰P .v/ D minu2 .hv; ui  ‰P .u//; this gives us a collection of piecewise affine concave functions P on NQ . Now let P be the polyhedral subdivision of NQ induced by ‰P and take  D P ˝ P . Furthermore, let M be the subset of

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tail./ consisting of those  2 tail./ such that .deg ı‰/jF 0, where F  is the face where h; vi takes its minimum for all v 2  . As in remark 21, the sum of slices  together with markings M determines a T variety X‰ . Furthermore, ‰  D .‰P /P 2Y is a divisorial support function determining an ample Cartier divisor on X‰ as in (7.4) such that .‰  ; ‰ / D .‰; /. Theorem 63 ([IS], Theorem 3.2). The mapping ‰ 7! .X‰ ; O.D‰ // gives a correspondence between divisorial polytopes and pairs of complete complexityone T -varieties with an invariant ample line bundle. 14.5 Constructing divisorial polytopes. Given a polarized toric variety X deterz Q , and given some action of a codimensionmined by some lattice polytope r  M one subtorus T , one may ask what the corresponding divisorial polytope looks like. A downgrading procedure similar to that of (4.2) yields the answer: the inclusion of the subtorus T in the big torus once again gives an exact sequence 0

/Z

/M z

p

/M

/0

z of where M is the character lattice of T . We again choose a section s W M ,! M p. The downgrade of r is then the divisorial polytope ‰ W p.r/ DW  ! DivQ P 1 , where for u 2 , ‰0 .u/ D max.p 1 .u/  s.u//; ‰1 .u/ D min.p 1 .u/  s.u//: The corresponding variety X‰ is canonically isomorphic to X as a polarized T -variety. Example 64. Consider the polytope ˚ r D conv .1; 1; 1/; .1; 0; 1/; .0; 1; 1/; .0; 0; 1/; .1; 1; 0/; .1; 0; 0/; .0; 1; 0/; .0; 1; 0/.1; 0; 0/;  .1; 1; 0/.0; 0; 1/; .0; 1; 1/; .1; 0; 1/; .1; 1; 1/ pictured in Figure 13. Then P .r/ is a toric Fano threefold. We now consider the subtorus action corresponding to the projection Z3 ! Z2 onto the first two factors. Using the above downgrading procedure we get a divisorial polytope ‰ W  ! DivQ P 1 , where  is the hexagon pictured in Figure 14(a), and ‰0 and ‰1 are the piecewise affine functions with domains of linearity and values as pictured in Figure 14(b) and (c).

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Figure 13. r for a toric Fano threefold, cf. Example 64. 2

1

1

1

0

0

-1

1

0

0

1

0 1

0

1

1

0

1

-2 -2

-1

0

(a)

1

2

(b) Ψ0

(c) Ψ∞

Figure 14. A downgraded divisorial polytope, cf. Example 64.

14.6 Divisorial polytopes and the Proj construction. Similar to the toric case, we can construct a projective variety directly from ‰:   M  .Y; O.‰.u/// u : P .‰/ WD Proj Sym u2\M

Although the torus T acts on P .‰/, this action will in general not be effective. Note that the relationship between P .‰/ and X‰ is similar to that between P .r/ and TV.N .r// for a lattice polytope r, see [IS, Section 4]. Indeed, the ample divisor D‰ on X‰ determines a dominant rational map ˆ‰ W X‰ Ü P .‰/. ˆ‰ is a regular morphism if D‰ is globally generated, which is in particular the case if Y D P 1 . Furthermore, assuming that ˆ‰ is regular, ˆ‰ is the normalization map if it is birational.

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55

A sufficient condition for ˆ‰ to be birational is that ‰.u/ is very ample for some u 2  \ M , and the set fu 2  \ M j dim .Y; O.‰.u/// > 0g generates the lattice M . 14.7 The moment map. Consider an action of a torus T on P k given by weights ui 2 M , i.e. t:.x0 W : : : W xk / D .t u0 x0 W : : : W t uk xk /: Then a moment map for this action is given by m W P n ! MR ;

.x0 W : : : W xn / 7! P

X 1  jxi j2 ui : 2 jxi j

If X is any T -invariant subvariety  W X ,! P n ; then a moment map  for X is given by the composition  D m ı . In particular, if  is given by sections s0 ; : : : ; sk of some very ample line bundle L on X, then we have X 1  W X ! MR ; x 7! P  jsi .x/j2 ui : 2 jsi .x/j The image of this map is a rational polytope called the moment polytope. By the theorem of Atiyah, Guillemin, and Sternberg, this polytope is the convex hull of the images of the points of X fixed by T , see [GS82]. If X is toric, and L corresponds to some lattice polytope r  MQ , then .X / D r, see for example [Ful93], Section 4.2. This can be easily seen by considering the images of the T -fixed points. Indeed, here the weights ui are simply the elements of r \ MP , and the torus fixed points correspond to vertices of r. Setting j .x/ WD jsj .x/j2 =. jsi .x/j2 /, one can show that j .x/ D 0 unless x corresponds to uj . On the other hand, if X D X‰ for some divisorial polytope ‰ and L D O.D‰ /, then .X/ D . Indeed, let p W Xz ! X resolve the indeterminacies of the rational quotient map X ! Y to a regular map  W Xz ! Y . Then the fixed points of Xz in the fiber  1 .P / correspond exactly to the vertices of the graph of ‰P . By locally upgrading Xz to a toric variety in a formal neighborhood of  1 .P /, we get that j .x/ D 0 for a fixed point x unless it corresponds to a vertex of weight uj , in which case j .x/ might be non-zero. Since the fixed points of Xz map surjectively onto the fixed points of X , the claim follows. We can also easily determine the momentum polytopes of the individual components of each fiber  1 .P /. A component Z of  1 .P / corresponds to a facet of the graph of ‰P , and the fixed points of Z correspond exactly to the vertices of . By arguments similar to above, .Z/ is nothing other than the projection of to . Thus, the subdivision of  induced by the linearity regions of ‰P reflects the structure of the irreducible components of  1 .P /. This is similar to the situation presented in [Kap93, (1.2)] for Grassmannians, which is addressed in more generality in [Hu05].

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Example 65. Consider the divisorial polytope ‰ from Example 64. Then the moment polytope of X‰ is the hexagon of Figure 14(a). Furthermore, we can see the fibers of X‰ 7! P 1 . The general fiber is the toric del Pezzo surface of degree six corresponding to the aforementioned hexagon, whereas the fibers over 0 and 1 each consist of three intersecting copies of P 1  P 1 as can be seen from the subdivisions in Figures 14(a) and 14(b).

15 Toric intersection theory 15.1 Simplicial toric varieties. Let X D TV.†/ be a k-dimensional, complete toric variety induced from a simplicial fan. Thus, X has at most abelian quotient singularities and one can define (rational) intersection numbers among k divisors. This intersection product is uniquely determined by exploiting the following basic rules: (i) If, with the notation as in (2.1),  D ha1 ; : : : ; ak i is a full-dimensional cone of †, then .orb a1  : : :  orb ak / D 1= vol. / with vol. / WD det.a1 ; : : : ; ak /. (ii) If a1 ; : : : ; ak (not necessarily distinct) are not contained in a common cone of †, then .orb a1  : : :  orb ak / D 0. P (iii) The intersection product factors via the relation a2†.1/ a ˝ orb.a/ 0 of linear equivalence from (7.1). Example 66. In the case of a smooth, complete toric surface TV.†/ with †.1/ D fa1 ; a2 ; : : : g, one always has relations of the sort ai1  bi ai C aiC1 D 0 between adjacent generators. This leads to the self intersection numbers .orb.ai //2 D bi . In f2 ! C 2 is the blowup of the origin given by the fan † with †.1/ D particular, if C f.1; 0/; .1; 1/; .0; 1/g, then, despite the fact that it is not complete, one sees that .E 2 / D 1 with E WD orb .1; 1/. Let us assume that a0 D .1; 0/, ai D .pi ; qi /, with ai being numbered counterclockwise. Now, one can prove by induction that the self-intersection numbers bi and the generators ai are related by the following continued fraction formula Œb1 ; : : : ; bi  WD b1 

1 b2 

D

1

::

:

qi ; pi0

with 0 < pi0 < qi and pi0 pi .mod qi /:

15.2 Intersecting with Cartier divisors. For general, i.e. non-simplicial toric varieties, intersection numbers as in (15.1) do not make sense. However, one can always intersect cycles with Cartier divisors. Let r  MQ be a lattice polyhedron and † WD N .r/. Then, every face   r gives rise to two different geometric objects. First, in (7.1) we have constructed the Cartier divisor div./ on the toric variety TV.N .//. Afterwards, in (14.2), we have defined the cycle orb./  TV.†/. Note that, in accordance with (6.1), this cycle

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equals TV.N .//. Now, the fundamental but trivial observation in intersecting T invariant cycles with Cartier divisors is div.r/  orb./ D div./: Proof. According to (6.1), this formula is obtained by noting first that the vertices of r corresponding to cones   WD N .; r/ are exactly those from . Then, one restricts the corresponding monomials by shifting  into the abelian subgroup M WD   . This shift makes it possible to understand the above formula as a relation between divisor classes. The projective way of expressing the formula for div.r/ from the end of (7.3) is X div.r/ D dist.0; /  orb./

where  runs through the codimension-one faces (“facets”) of r and dist.0; / denotes the oriented (positive, if 0 2 r) lattice distance of the origin to the affine hyperplane containing . f2 ! C 2 of Example 67. Let r be the polyhedron from (14.1) describing the blowup C

the origin. Denoting the unique compact edge by e, we have that div.r/ D orb.e/ D E since 0 … r. Thus, .E 2 / D  div.r/  orb.e/ D div.e/ and the latter describes the ample divisor class of degree one, i.e. a point, on P .e/ D P 1 . Corollary 68. .div.r/k / D vol.r/  kŠ. Proof. We proceed by induction.

P .div.r/k / D .div.r/k1 /  dist.0; /  orb./ P D dist.0; /  .div./k1 / P D dist.0; /  vol./  .k  1/Š D k  vol.r/  .k  1/Š:

15.3 A result for complexity-one T -varieties. We already know from (14.4) that polytopes have to be replaced by divisorial polytopes when proceeding from toric varieties to complexity-one T -varieties. To do intersection theory on a complete complexity-one T -variety TV./ we thus have to come up with a natural extension of the volume function. Definition 69. For a function h W  ! DivQ Y we define its volume to be XZ  vol h WD hP volMR : 

P

We associate a mixed volume to functions h1 ; : : : ; hk by setting V .h1 ; : : : ; hk / WD

k X iD1

.1/i1

X

vol.hj1 C    C hji /:

1j1 :::ji k

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Proposition 70 ([PS], Proposition 3.31). Let  be a complete f-divisor on the curve Y and let Dh be a semiample Cartier divisor on TV./. Setting n WD dim TV./, the top self-intersection number of Dh is given by .Dh /n D nŠ vol h : Moreover, assuming that h1 ; : : : ; hn define semiample Cartier divisors Dhi , we have that .Dh1    Dhn / D nŠ V .h1 ; : : : ; hn /: This assertion follows from the fact that .Dh /n is equal to limk!1 knŠn dim .kDh / and the explicit description of the global sections from (9.2). Note that this statement is a special instance of [Tim00, Theorem 8]. P 15.4 Intersection graphs for smooth C  -surfaces. Let  D P P ˝ P together with deg D ; be a complete f-divisor in the lattice N D Z on a curve Y of genus g. Let fP P1 ; : : : ; Pr g be the P support of  and i WD Pi . In this situation we also write  D P P ˝ P D riD1 i ˝ Pi . The slices i are complete subdivisions of Q into bounded and half-bounded intervals. This data gives rise to a C  -surface X D TV./. Here, we only consider the case of smooth surfaces (for a criterion on smoothness see (12.2)). Since we assume that deg D ;, T V ./ D TV./. By the results on invariant prime divisors from (6.3), we will find two horizontal prime divisors on X , one for each ray in the tail fan fQ0 ; 0; Q0 g. In the case of a C  -surface a horizontal prime divisor is simply a curve consisting of fixed points. Moreover, these curves turn out to be isomorphic to Y ; we will denote them by FC and F . In addition to these curves we get vertical prime p divisors corresponding to boundary points qijij 2 i , i D 1; : : : ; r and j D 1; : : : ; ni .

e

p

p

i.j C1/ > qijij DW vij . These curves are closures of Here we assume, that vi.j C1/ WD qi.j C1/  maximal orbits of the C -action, and we will denote them by Dij . Now, similar to the observations in [LS], it turns out that in a neighbourhood of Di1 ; : : : ; Diri , X is locally formally isomorphic to the toric surface spanned by the rays .1; 0/; .pi1 ; qi1 /; : : : ; .pi ni ; qi ni /; .1; 0/ in a neighbourhood of the invariant prime divisors. In particular, the mutual intersection behaviour of the Di1 ; : : : ; Di ni is exactly the same as that of the invariant divisors of the toric surface. Hence, two curves Dij and Dlm intersect (transversely) exactly when i D l and jm  j j D 1. A curve p Dij intersects F C if and only if qijij is a maximal boundary point and it intersects F if and only if it is a minimal boundary point, i.e. j D 1. In addition, we obtain the self-intersection number   X  X vi1 Di1  F D vi1 c D .F /2 D ..F C div.1 //  F / D

i

and similarly cC D .FC /2 D 

X i

vi ni :

i

The geometry of T -varieties

59

By the considerations in Example 66, the self-intersection numbers bji D .Dij /2 can p be related to the boundary points vij D qijij using the formulae qi1 D 1;

i i Œb1i ; : : : ; b.j 1/  WD b1 

b2i

1 

D

1

::

:

qij : pij0

Hence, we obtain an intersection graph of the following form:

F

HIJK ONML b11   rrr  rrrr c  :: PQRS WVUT Œg
HIJK ONML b21

:: : HIJK ONML b2r

1 ONML HIJK bn 1 << LLL <<< LLL<< L cC :: WVUT PQRS Œg : r rrr rrr   r ONML HIJK bn r

FC

By Orlik and Wagreich [OW77], these graphs correspond to deformation classes of C  -surfaces with two curves of consisting of fixed points. We indeed obtain families of surfaces having these intersection graphs by varying Y together with the point configuration P1 ; : : : ; Pr and by adding algebraic families of degree zero divisors to . We can also reverse the procedure to obtain P an f-divisor out of an intersection graph. Choose any integral vi1 D pi1 satisfying i vi1 D c . The remaining vij are then determined by the bji via the above continued fractions. Choosing any genus g curve Y and pairwise different points P1 ; : : : ; Pr 2 Y then determines an f-divisor. The open choices of the Y , of P1 ; : : : ; Pr 2 Y , and of particular vi1 reflect the fact that the intersection graph only determines the topological type of X . Remark 71. The Neron–Severi group of a C  -surface is generated by the curves F ˙ , Dij , D0 . Here the curve D0 corresponds to the vertex of a trivial coefficient of  and its intersection behaviour is given by D02 D 0, D0  F ˙ D 1 and D0  Dij D 0. The matrix of the intersection form is encoded by our graph and by the intersection behaviour of D0 . The relations between the generators are given as the kernel of this matrix. Now Section 8 provides a representative of the canonical class X K D F C  F   Dij C .r C 2g  2/  D0 : ij

Remark 72. In order to obtain arbitrary C  -surfaces, Orlik and Wagreich considered a so-called canonical equivariant resolution of the surface. This resolution is always of the above type. In our notation this is the same as considering the minimal resolution of T V ./ for an arbitrary C  -surface given as TV./.

e

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Recall from the downgrade procedure in Section 4.2 that TV./ is toric if Y D P 1 and the support of  consists of at most two points (i.e. .Y; supp.D// is a toric pair). By the above, such an f-divisor corresponds to a circular intersection graph with g D 0.

16 Deformations 16.1 Deformations of affine toric varieties. An equivariant deformation of a T variety X is a T -equivariant pullback diagram X

/X

 0

 /S



where  is a flat map. The torus T acts on X , X, and S , and the T -action on the total space X induces the given action on the special fiber X D  1 .0/. A deformation of X over CŒ"=."2 / such that T acts on " with weight r 2 M is called homogeneous of degree r. This induces an M -grading on the module TX1 of first order deformations. We fix now some primitive degree r 2 M , along with a cosection s  W N ! Nr WD N \ r ? . Non-primitive degrees can also be handled at the cost of more complex notation. Consider now an affine toric variety X D TV. /. Using the toric description of

X [Dan78],  determines a complex that allows one to compute TX1 .r/ and, if X is non-singular in codimension 2, TX2 .r/ [Alt94], [Alt97a]. TX1 .r/ may also be described as a vector space of Minkowski summands of the polyhedron r WD s  . \fr D 1g/: the set of scalar multiples of Minkowski summands of a polyhedron  forms a cone C./, with Grothendieck group V ./ WD C./  C./. For example, if  is a parallelogram, C./ is two-dimensional, spanned by the edges of . Then TX1 .r/ may be described by augmenting V .r / with information about possible non-lattice vertices of r [Alt00, Theorem 2.5]. Definition P 73. A Minkowski decomposition of a polyhedron  is a Minkowski sum  D i of polyhedra i with common tailcone. It is admissible if for each face of , at most one of the corresponding faces face.i ; u/ does not contain lattice points. For degrees r 2  _ , admissible decompositions r D 0 C    C l allow one to construct toric deformations, where X is a toric variety and the embedding X ,! X is a morphism of toric varieties. Similarly to the Cayley cone in (12.2), we define the cone Q in Nz WD Nr ˚ ZlC1 generated by i  fei g, 0  i  l. Taking X D TV.Q /, i 0 the binomials Œ0;e  Œ0;e define a map  W X ! C l . Theorem 74 ([Alt00, Theorem 3.2]). (1) X ! C l is a toric deformation of X . (2) The corresponding Kodaira–Spencer map C l ! TX1 .r/ maps ei to the class of the Minkowski summand i 2 C.r /  V .r /.

The geometry of T -varieties

61

Example 75. We consider deformations of the cone over the rational normal curve from Example 1 with r D Œ0; 1. The non-trivial Minkowski decompositions of r D convf. 12 ; 1/; . 12 ; 1/g correspond to the decompositions of the interval Œ 12 ; 12  D Œ 12 ; 0 C Œ0; 12  D f 12 g C Œ0; 1. For the first decomposition, we get the cones (generated by the columns of) 0 1 0 1 1 2 0 0 2  2 0 0 12 Q D @ 1 1 0 0 A and Q _ D @ 0 0 1 1A ; 1 1 0 0 0 0 1 1 with Hilbert basis E D fŒe; 0; 1 j 2  e  0g [ fŒe; 1; 0 j 0  e  2g. The equations for TV.Q / and those corresponding to the second decomposition are 1 0

y0 y1 y2 0 y y1 y2 y3 rank 0  1 and rank @y1 y20 y3 A  1; y1 y2 y3 y4 y2 y3 y4 yielding two one-parameter deformations with deformation parameters s D y2  y20 of degree r. These two one-parameter deformations generate T 1 .r/. This is in fact Pinkham’s famous example of a singularity whose versal base space consists of two irreducible components; we see curves from both components via the above toric deformations.

[r=1]

−1

0

1

= =

(a) σr

−1

0

1

−1

0

1

+ +

−1

0

1

−1

0

1

(b) Two decompositions.

Figure 15. Toric deformations of the cone over the rational normal curve.

This construction may be generalized to degrees r 62  _ by deforming the toric variety associated with D  \ fr  0g [Alt00], but the total spaces are no longer toric. The resulting families arise naturally in the language of polyhedral divisors as explained in (16.2), or by Mavlyutov’s approach using Cox rings [Mav]. It is even possible to describe the homogeneous parts of the versal deformation if X is non-singular in codimension 2. The universal Minkowski summand Cz .r / is a cone lying over C.r /. The associated map of toric varieties TV.Cz .r // ! TV.C.r // x where the scheme M x is determined by C.r /  allows one to obtain a family X ! M, V .r /. x is the versal deformation of X in degree r. Theorem 76 ([Alt97b], [AK]). X ! M If X is an isolated Gorenstein singularity, then T 1 is concentrated in the Gorenstein degree r0 , so X is the versal deformation.

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16.2 Equivariant deformations of affine T -varieties. Suppose that X is some affine T -variety. By restricting the torus action to the subtorus Tr WD ker.r W T ! C  /, the study of deformations in degree r may be reduced to the study of invariant deformations of the Tr -variety X . Thus, in the following, we replace T by Tr , and consider equivariant deformations in degree r D 0. In particular, the torus T acts trivially on the base, and acts on every fiber. Such families may be described by families of polyhedral divisors: the total space X over B corresponds to a p-divisor E on Z ! B that restricts to p-divisors Es D EjZs with Xs D TV.Es /. See Figure 16 for a simple example. D 00

D 01 D∞ t 0

(a) Z = P1 × A1 → A1

Et D = E0

Δ00 Δ10 0

t

Δ00 +Δ10

Δ∞ ∞ Δ∞

(b) Restrictions of E.

Figure 16. One-parameter toric deformation as deformation of p-divisors.

Note that when two prime divisors on Z restrict to the same prime divisor on a fiber Zs , their polyhedral coefficients are added via Minkowski summation. If we wish to fix some central fiber X D TV.D/, we may obtain invariant deformations of X in two ways. First of all, we may simply move prime divisors on the base of Y . Secondly, for more interesting deformations, we may split up some of the polyhedral coefficients into Minkowski sums. Requiring that X is the fiber X0 implies that these Minkowski decompositions must be admissible. Concretely, we say that a deformation of .Y; fDi g/ is a deformation Z ! B of Y together with a collection of prime divisors fEi;j g such that no Ei;j contains fibers of Z ! B and such that each Ei;j restricts to Di in Y . Given li -parameter Minkowski P decompositions DDi D ji of the coefficients DDi , we define a polyhedral divisor E on Z by setting EEi;j D ji . Theorem 77 ([IVa, Section 2]). If Y D P 1 and the given Minkowski decompositions are admissible, then there exists a deformation .Z; fE Pi;j g/ of .Y; fDi g/, where Z is li , such that E is a p-divisor, the trivial family over some open subset of C l , l D and X D TV.E/ ! C l is an invariant deformation of X . The fibers of this family are T -varieties TV.Es /, where Es is the restriction of E to the fiber Zs . Example 78. Suppose X D TV. / is toric, and consider some degree r. By downgrading the torus action to the torus Tr D Nr ˝ C  , we obtain the p-divisor D D r ˝ f0g C r ˝ f1g on P 1 . A Minkowski decomposition of r gives rise to an invariant deformation TV.E/ of the Tr -variety X . This deformation is equal to the corresponding deformation obtained in (16.1), whether r 2  _ or not. The case of a

The geometry of T -varieties

63

one-parameter deformation is illustrated in Figure 16. Note that r D ; if and only if r 62  _ , so for r 2  _ , D1 has coefficient ;, and E is actually defined on AlC1 . In particular, the deformations of Example 75 are Tr -varieties with divisors E 1 D Œ 12 ; 0 ˝ D 0 C Œ0; 12  ˝ D 1 ; E 2 D  12 ˝ D 0 C Œ0; 1 ˝ D 1 on A2 D Spec CŒx; s, with D 0 D V .x/ and D 1 D V .sx/. Here, s is the deformation parameter. 16.3 Deformations of complete toric varieties. Consider now a rational complexity-one T -variety X defined by a set  of p-divisors as in section 4.4 on Y D P 1 . It is possible to construct invariant deformations of X by constructing invariant deformations of TV.D/ for all D 2  subject to certain compatibility conditions, see [IVa, Section 4]. This construction can in particular be used to obtain deformations of non-affine toric varieties. There is an especially nice result for the case of complete and smooth toric varieties. Let † be a complete fan in NQ describing a smooth toric variety X , and let r be as before. For any ray  2 †.1/ with h; ri D 1, let  .r/ be the graph embedded in NQ \ fr D 1g with vertices consisting of rays 2 †.1/ n  fulfilling h ; ri > 0. Two vertices 1 , 2 of  .r/ are connected by an edge if they generate a cone in †. After applying any cosection s  W N ! Nr mapping  to the origin, we can consider  .r/ to actually be embedded in NQ \ r ? . Let .r/ denote the set of all rays  of † such that  .r/ is not empty. Theorem 79 ([Ilt11], Proposition 2.4). There is an isomorphism M H 0 . .r/; C/=C H 1 .X; TX /.r/ Š 

describing first-order infinitesimal deformations of X . Now fix some  2 .r/ and choose some connected component C of  .r/. For each  2 †, define a Minkowski decomposition of r via ´ r C tail.r /; r \ C D ;; r D tail.r / C r ; r \ C ¤ ;: Theorem 80 ([IVa], Section 4). (1) The toric deformations of TV. / for  2 † constructed from the above decompositions of r glue to an equivariant deformation .C; ; r/ of X D TV.†/. (2) Letting C range over all connected components of  .r/, the images of the .C; ; r/ under the Kodaira–Spencer map span H 0 . .r/; C/=C.

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Example 81. We consider the fourth Hirzebruch surface F4 D ProjP 1 .O ˚ O.4//. This is a toric variety, and a fan † with TV.†/ D F4 is pictured in Figure 17(a). Considering r D Œ0; 1 and  the ray through .0; 1/, we get that  .r/ consists of two points. Thus, dim TF14 .r/ D 1. A one-parameter deformation spanning this part of TF14 can be constructed from the Minkowski decomposition pictured in Figure 17(b); the lines f0 g and f1 g picture the 0th and 1st summands dilated by a factor of two. This Minkowski decomposition comes from choosing the connected component C D f1=3g of  .r/. Note that the general fiber of this deformation is the second Hirzebruch surface F2 D ProjP 1 .O ˚ O.2//. Indeed, the general fiber is given by a collection of pdivisors on P 1 with slices f0 g and f1 g. Performing a toric upgrade yields the fan in Figure 17(c), which is a fan for F2 . In these lattice coordinates, TF14 also has one dimensional components in degrees Œ1; 1 and Œ2; 1. The general fibers of the corresponding homogeneous deformations are P 1  P 1 and F2 , respectively.

[r = 1]

3

3

2

2

ρ

1

f Δ0 g

− 13

0

1

Σ ∩ [r = 1]

0

-1 -1

0

(a) Fan for F 4 .

1

f Δ1 g

0

-1

0

1

(b) A Minkowski decomposition.

-1

0

1

(c) Fan for F 2 .

Figure 17. Deforming F4 to F2 .

17 Related constructions 17.1 A non-toric view on p-divisors. We return P to the setting of (2.3) and consider affine T -varieties. Let us assume that D D i i ˝ Zi is a p-divisor on Y . Note that, in contrast to (2.3), we have symmetrized the notation of the tensor factors, i.e. i 2 PolC Q .N;  / and Zi 2 CaDiv0 .Y /. In the current section, we moreover adopt another convention differing from (2.3). Setting Y WD Loc.D/, this variety is no longer complete. Nevertheless, it is projective over Y0 WD Spec .Y; OY /. Thus, we can now assume that the polyhedra i are non-empty.

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65

P If  WD i i (which equals deg D from (5.1) if Y is a complete curve), then N ./ is a fan in MQ which refines the polyhedral cone  _ . Note that we usually would not construct a toric variety from these data since the fan is given in the “wrong” space MQ instead of NQ . However, we make an exception here and define W WD P ./ D TV.N .// ! TV. _ / DW W0 : This is a projective map, and P the polyhedra i can be interpreted as semiample divisors Ei on W . Thus, D D i Ei ˝ Zi becomes a “double divisor”, i.e. an element of CaDivQ .W / ˝Z CaDivQ .Y /. 17.2 Symmetrizing equivalences between p-divisors. The equivalence relations on p-divisors mentioned just before Theorem 4 in (2.3) can also be expressed in the symmetric language of (17.1). First, the operation of pulling back D via a modification ' W Y 0 ! Y can be contrasted with pulling back D via a toric modification W W 0 ! W . The latter corresponds to giving a subdivision of the fan N ./; both Ei and  Ei correspond to the same polyhedron i . Second, let us recall from (2.3) the notion of a principal polyhedral divisor .aC /˝ div.f / with a 2 N and f 2 C.Y / . In our new setting it is equal to div.a /˝div.f /, i.e. the latter can be understood as a “double principal divisor”. 17.3 Interpreting evaluations. P Recall from (2.3) that elements u 2 M were associated to an evaluation D.u/ D i minhi ; ui  Zi 2 CaDivQ .Y /. To interpret this construction within the language of (17.1), we have to understand the former characters u of T as germs of curves in W , i.e. as maps u W .C; 0/ ! W . Note that the scalar minhi ; ui equals the multiplicity of u Ei at the origin or, likewise, the intersection number .Ei  u OC 1 / in W . Hence, the sheaf OY .D/ on Y corresponds P to i .Ei  U/Di where U WD ˚u u OC 1 is a quasi-coherent sheaf on W with onedimensional support. The idea behind the preceding construction is that T V .D/ ! Y looks like a fibration which becomes degenerate over each divisor Zi  Y , and the polyhedron i tells us what the degeneration looks like. Translating these polyhedra into divisors Ei on a toric variety W is linked to the fact that the fibers of T V .D/ ! Y are also toric. We hope that this might serve as a general pattern to describe degenerate fibrations of a certain type in a much broader context. We conclude this section with a construction pointing into this direction.

e

e

17.4 Abelian Galois covers of P 1 . Let A be a finite, abelian group. Every divisor D 2 DivA0 P 1 WD A ˝Z Div0 P 1 of degree 0 can then be understood as a linear map A ! Div0Q=Z P 1 , u 7! D.u/ where A WD Hom.A; Q=Z/ denotes the dual group. Choosing arbitrary lifts Du 2 Div0Q P 1 and, afterwards, rational functions

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K. Altmann, N. O. Ilten, L. Petersen, H. Süß, and R. Vollmert

fu;v D fv;u 2 C.P 1 / which satisfy Du C Dv C div.fu;v / D DuCv ; we may, after possibly correcting the functions fuv by suitable constants, assume that fu;vCw fv;w D fu;v fuCv;w . Then we can define a multiplication via OP 1 .Du / ˝ OP 1 .Dv / ! OP 1 .DuCv /;

1 f ˝ g 7! fg fu;v ;

which provides the sheaf O.D/ WD

M

OP 1 .Du /:

u2A

with an associative and commutative OP 1 -algebra structure. Up to isomorphism, the latter does not depend on the choices we have made. Finally, the Galois covering associated to D is defined as the relative spectrum C.D/ WD SpecP 1 O.D/. Proposition 82 ([AP, Theorem 3.2]). The relative spectrum C.D/ yields a Galois covering  W C.D/ ! P 1 whose ramification points are contained in supp D. The ramification index of P 2 C equals the order of the D-coefficient of .P / inside A. Example 83. If D 2 DivZ P 1 is an effective divisor with nj deg D, then the associated well-known cyclic n-fold covering of P 1 is given, via the above recipe, by understanding D as an element of Div0Z=nZ P 1 .

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N. O. Ilten and R. Vollmert, Deformations of rational T -varieties. J. Algebraic Geom. 21 (2012), 531–562.

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N. O. Ilten and R. Vollmert, Upgrading and downgrading torus actions. arXiv:1103.4010v1 [math.AG].

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M. M. Kapranov, Chow quotients of Grassmannians. I. In I. M. Gel0 fand Seminar, Adv. Soviet Math. 16, Amer. Math. Soc., Providence, RI, 1993, 29–110.

[KKMSD73] G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings I. Lecture Notes in Math. 339, Springer-Verlag, Berlin 1973. [Lie10b]

A. Liendo, Affine T -varieties of complexity one and locally nilpotent derivations. Transform. Groups 15 (2010), no. 2, 389–425.

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A. Liendo, Ga -actions of fiber type on affine T -varieties. J. Algebra 324 (2010), no. 12, 3653–3665.

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A. Liendo, Roots of the affine Cremona group. Transform. Groups 16 (2011), no. 4, 1137–1142.

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A. Liendo and H. Süß, Normal singularities with torus actions. arXiv:1005.2462v2 [math.AG].

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A. Mavlyutov, Deformations of toric varieties via Minkowski sum decompositions of polyhedral complexes. arXiv:0902.0967v3 [math.AG].

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P. Orlik and P. Wagreich, Algebraic surfaces with k*-action. Acta Math. 138 (1977), 43–81.

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L. Petersen, Line bundles on T -varieties and beyond. PhD thesis, Freie Universität Berlin, 2010.

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KlausAltmann, Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany E-mail: [email protected] Nathan Owen Ilten, Max Planck Institut für Mathematik, PF 7280, 53072 Bonn, Germany E-mail: [email protected] Lars Petersen, Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany E-mail: [email protected] Hendrik Süß, Institut für Mathematik, LS Algebra und Geometrie, Brandenburgische Technische Universität Cottbus, PF 10 13 44, 03013 Cottbus, Germany E-mail: [email protected] Robert Vollmert, Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany E-mail: [email protected]

Introduction to equivariant cohomology in algebraic geometry Dave Anderson

Contents 1

Lecture 1: Overview . . . . . . . . . . . . . . . . . . . 1.1 The Borel construction . . . . . . . . . . . . . . . 1.2 Approximation spaces . . . . . . . . . . . . . . . . 1.3 Functorial properties . . . . . . . . . . . . . . . . 1.4 Fiber bundles . . . . . . . . . . . . . . . . . . . . 1.5 Two notions . . . . . . . . . . . . . . . . . . . . . 2 Lecture 2: Localization . . . . . . . . . . . . . . . . . . 2.1 Restriction maps . . . . . . . . . . . . . . . . . . 2.2 Gysin maps . . . . . . . . . . . . . . . . . . . . . 2.3 First localization theorem . . . . . . . . . . . . . . 2.4 Equivariant formality . . . . . . . . . . . . . . . . 2.5 Integration formula (Atiyah–Bott–Berline–Vergne) 2.6 Second localization theorem . . . . . . . . . . . . 3 Lecture 3: Grassmannians and Schubert calculus . . . . 3.1 Pre-history: Degeneracy loci . . . . . . . . . . . . 3.2 The basic structure of Grassmannians . . . . . . . 3.3 Fixed points and weights . . . . . . . . . . . . . . 3.4 Schubert classes in HT X . . . . . . . . . . . . . . 3.5 Double Schur functions . . . . . . . . . . . . . . . 3.6 Positivity . . . . . . . . . . . . . . . . . . . . . . 3.7 Other directions . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Lecture 1: Overview A general principle of mathematics holds that one should exploit symmetry to simplify a problem whenever possible. A common manifestation of symmetry is the action of a Lie group G on a topological space X – and when one is interested in understanding the cohomology ring H  X , the equivariant cohomology HG X is a way of exploiting this symmetry. 

Partially supported by NSF Grant DMS-0902967.

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Topologists have long been interested in a sort of converse problem: given some topological (or cohomological) information about X, what can one say about the kinds of group actions X admits? For example, must there be fixed points? How many? It was in this context that Borel originally defined what is now called equivariant cohomology, in his 1958–1959 seminar on transformation groups [Bo]. The goal of these lectures is to give a quick introduction to equivariant cohomology in the context of algebraic geometry. We will review the basic properties of HG X and give some examples of applications. 1.1 The Borel construction. Let G be a complex linear algebraic group, and let X be a complex algebraic variety with a left G-action. The construction Borel introduced in [Bo] goes as follows. Find a contractible space EG with a free (right) G-action. (Such spaces exist, and are universal in an appropriate homotopy category; we will see concrete examples soon.) Now form the quotient space EG G X WD EG  X=.e  g; x/  .e; g  x/: Definition 1.1. The equivariant cohomology of X (with respect to G) is the (singular) cohomology of EG G X : HG X WD H  .EG G X /: (We always use singular cohomology with Z coefficients in these notes.) The idea behind the definition is that when X is a free G-space, we should have HG X D H  .GnX /. To get to this situation, we replace X with a free G-space of the same homotopy type. From a modern point of view, this is essentially the same as taking the cohomology of the quotient stack ŒGnX  (see, e.g., [Be]). General facts about principal bundles ensure that this definition is independent of the choice of space EG; for instance, BG D EG=G is unique up to homotopy. When X is a point, EG G fptg D BG usually has nontrivial topology, so  HG .pt/ ¤ Z! This is a key feature of equivariant cohomology, and HG .pt/ D H  BG may be interpreted as the ring of characteristic classes for principal G-bundles. Other functorial properties are similar to ordinary cohomology, though. (In fact, HG ./ is a generalized cohomology theory, on a category of reasonable topological spaces with a left G-action.) Example 1.2. Let G D C  . The space EG D C 1 X f0g is contractible, and G acts freely, so BG D EG=G D P 1 . We see that HC  .pt/ D H  P 1 ' ZŒt ; where t D c1 .OP 1 .1// is the first Chern class of the tautological bundle.

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1.2 Approximation spaces. The spaces EG and BG are typically infinite-dimensional, so they are not algebraic varieties. (This may partly account for the significant lag before equivariant techniques were picked up by algebraic geometers.) However, there are finite-dimensional, nonsingular algebraic varieties Em ! Bm D Em =G which serve as “approximations” to EG ! BG. This works because of the following lemma: Lemma 1.3. Suppose Em is any (connected) space with a free right G-action, and H i Em D 0 for 0 < i < k.m/ (for some integer k.m/). Then for any X , there are natural isomorphisms H i .Em G X / ' H i .EG G X / DW HGi X; for i < k.m/. Example 1.4. For G D C  , take Em D C m X f0g, so Bm D P m1 . Since Em is homotopy-equivalent to the sphere S 2m1 , it satisfies the conditions of the lemma, with k.m/ D 2m  1 in the above lemma. Note that k.m/ ! 1 as m ! 1, so any given computation in HG X can be done in H  .Em G X /, for m  0. We have Bm D P m1 , so Em ! Bm is an intuitive choice for approximating EG ! BG. Example 1.5. Similarly, for a torus T ' .C  /n , take Em D .C m X f0g/n ! .P m1 /n D Bm : We see that HT .pt/ D HT ..P 1 /n / D ZŒt1 ; : : : ; tn , with ti D c1 .Oi .1//. (Here Oi .1// is the pullback of O.1/ by projection on the i th factor.) The above example is part of a general fact: For linear algebraic groups G and H , one can always take E.G  H / D EG  EH . Indeed, G  H acts freely (on the right) on the contractible space EG  EH . ı Example 1.6. Consider G D GLn , and let Em WD Mmn be the set of full rank m  n matrices, for m > n. This variety is k.m/-connected, for k.m/ D 2.m  n/. Indeed, ı Mmn is the complement of a closed algebraic set of codimension .m  1/.n  1/ in Mmn , and it is a general fact that i .C n X Z/ D 0 for 0 < i  2d  2 if Z is a Zariski closed subset of codimension d . See [Fu2, §A.4]. It follows that the maps

Em ! Bm D Gr.n; C m / approximate EG ! BG, in the sense of Lemma 1.3. We have HG .pt/ D ZŒe1 ; : : : ; en ; where ei D ci .S / is the ith Chern class of the tautological bundle over the Grassmannian.

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Since any linear algebraic group G embeds in some GLn , the above example gives a construction of approximations Em that works for arbitrary G. Example 1.7. The partial flag manifold Fl.1; 2; : : : ; nI C m / parametrizes nested chains of subspaces E1  E2      En  C m , where dim Ei D i . There is also an infinite version, topologized as the limit taking m ! 1. If B  GLn is the subgroup of upper-triangular matrices, we have Em

ı Mmn

 Bm

 Fl.1; 2; : : : ; nI C m /;

so BB is the partial (infinite) flag manifold Fl.1; 2; : : : ; nI C 1 /. Remark 1.8. The idea of approximating the infinite-dimensional spaces EG and BG by finite-dimensional ones can be found in the origins of equivariant cohomology [Bo, Remark XII.3.7]. More recently, approximations have been used by Totaro and Edidin–Graham to define equivariant Chow groups in algebraic geometry. 1.3 Functorial properties. Equivariant cohomology is functorial for equivariant '

f

 G 0 and a map X  ! X 0 such that f .g  x/ D maps: given a homomorphism G ! '.g/  f .x/, we get a pullback map f  W HG 0 X 0 ! HG X; 0

by constructing a natural map E G X ! E0 G X 0 . There are also equivariant Chern classes and equivariant fundamental classes: • If E ! X is an equivariant vector bundle, there are induced vector bundles Em G E ! Em G X . Set ciG .E/ D ci .Em G X / 2 HG2i X D H 2i .Em G X /; for m  0. • When X is a nonsingular variety, so is Em G X . If V X is a G-invariant subvariety of codimension d , then Em G V Em G X has codimension d . We define ŒV G D ŒEm G V  2 HG2d X D H 2d .Em G X /; again for m  0. (Any subvariety of a nonsingular variety defines a class, using e.g. Borel–Moore homology.)

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In fact, the Chern classes could be defined directly as ciG .E/ D ci .EG G E/, but for ŒV G one needs the approximation spaces. (Of course, one has to check that these definitions are compatible for different Em ’s.) In the special case X D pt, an equivariant vector bundle E is the same as a representation of G. Associated to any representation, then, we have characteristic classes ciG .E/ 2 HG2i .pt/. Example 1.9. Let La D C be the representation of C  with the action z  v D zav where a is a fixed integer. Then 

Em C La ' OP m1 .a/ 

as line bundles on Bm D P m1 , so c1C .La / D at 2 ZŒt . (This also explains our choice of generator for HC  .pt/ D ZŒt : we want t to correspond to the standard representation, L1 .) Example 1.10. Let T D .C  /n act on E D C n by the standard action. Then ciT .E/ D ei .t1 ; : : : ; tn / 2 HT .pt/ D ZŒt1 ; : : : ; tn ; where ei is the i-th elementary symmetric function. To see this, note that Em T E ' O1 .1/ ˚    ˚ On .1/ as vector bundles on Bm D .P m1 /n . Problem 1.11. Let T be the maximal torus in GLn , and let E be an irreducible polynomial GLn -module. The above construction assigns to E its equivariant Chern classes ciT .E/, which are symmetric polynomials in variables t1 ; : : : ; tn . What are these polynomials? Since they are symmetric polynomials, we can write X a s .t /; ciT .E/ D 

where s are the Schur polynomials (which are defined in §3 below). A theorem of Fulton and Lazarsfeld implies that the integers a are in fact nonnegative, as was observed by Pragacz in [P, Corollary 7.2]. This provides further motivation for the problem: we are asking for a combinatorial interpretation of the coefficients a . 1.4 Fiber bundles. The formation of E G X is really an operation of forming a fiber   bundle with fiber X . The map X !  pt becomes E G X !  B (via the first projection); ordinary cohomology H  X is an algebra over H  .pt/ D Z (i.e., a ring), while HG X

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is an algebra overHG .pt/; and one can generally think of equivariant geometry as the geometry of bundles. From this point of view, many statements about equivariant cohomology are essentially equivalent to things that have been known to algebraic geometers for some time – for instance, the Kempf–Laksov formula for degeneracy loci is the same as a “Giambelli” formula in HT Gr.k; n/. Example 1.12 (Equivariant cohomology of projective space). Let T D .C  /n act on C n in the usual way, defining an action on P n1 D P .C n /. This makes OP .C n / .1/ a T -equivariant line bundle. Write  D c1T .OP n1 .1//. Claim. We have HT P n1 ' ZŒt1 ; : : : ; tn Œ=. n C e1 .t / n1 C    C en .t // Q D ZŒt1 ; : : : ; tn Œ=. niD1 . C ti //: Proof. Pass from the vector space C n to the vector bundle E D Em T C n on Bm . We have Em T P n1 ' P .E/

and

Em T OP n1 .1/ ' OP .E / .1/;

all over Bm . The claim follows from the well-known presentation of H  P .E/ over H  B, since ei .t/ D ciT .C n / D ci .E/

and  D c1T .OP n1 .1// D c1 .OP .E / .1//;

as in Example 1.10. 1.5 Two notions. There are two general notions about equivariant cohomology to have in mind, especially when G is a torus. The first notion is that equivariant cohomology determines ordinary cohomology. From the fiber bundle picture, with the commutative diagram  / E G X X   pt 

 / B;

HG X is an algebra over HG .pt/, and restricting to a fiber gives a canonical map HG X ! H  X , compatible with HG .pt/ ! H  .pt/ D Z. In nice situations, we will have: HG X ! H  X is surjective, with kernel generated by the kernel of HG .pt/ ! Z. The second notion is that equivariant cohomology is determined by information at the fixed locus. By functoriality, the inclusion of the fixed locus  W X G ,! X gives a restriction (or “localization”) map  W HG X ! HG X G . In nice situations, we have:

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 W HG X ! HG X G is injective. Example 1.13. Both of these notions can fail, even when G is a torus. For example, take G D X D C  , where G acts on itself via left multiplication. One the one hand, 

HC1  .C  / D H 1 ..C 1 X f0g/ C C  / D H 1 .C 1 X f0g/ D 0; but on the other hand, H 1 .C  / D H 1 .S 1 / D Z, so the first notion fails. Since the action has no fixed points, the second cannot hold, either. However, we will see that the “nice” situations, where both notions do hold, include many interesting cases. When the second notion holds, it provides one of the most powerful techniques in equivariant theory. To get an idea of this, suppose X has finitely many fixed points; one would never expect an injective restriction map  in ordinary cohomology, by degree reasons! Yet in many situations, all information about HT X is contained in the fixed locus. This will be the topic of the next lecture.

2 Lecture 2: Localization From now on, we will consider only tori: G D T ' .C  /n . Since it comes up often, it is convenient introduce notation for the equivariant cohomology of a point: ƒ D ƒT D HT .pt/ ' ZŒt1 ; : : : ; tn : 2.1 Restriction maps. If X is a T -space and p 2 X is a fixed point, then the inclusion p W fpg ! X is equivariant, so it induces a map on equivariant cohomology p W HT X ! XT .fpg/ D ƒ: Example 2.1. Let E be an equivariant vector bundle of rank r on X , with Ep the fiber at p. Then p .ciT .E// D ciT .Ep /, as usual. Now Ep is just a representation of T , say with weights (characters) 1 ; : : : ; r . That is, Ep ' C r , and t  .v1 ; : : : ; vr / D .1 .t /v1 ; : : : ; r .t /vr / for homomorphisms i W T ! C  . So ciT .Ep / D ei .1 ; : : : ; r / is the ith elementary symmetric polynomial in the ’s. In particular, the top Chern class goes to the product of the weights of Ep : p .cr .E// D 1    r : Example 2.2. Consider X D P n1 with the standard action of T D .C  /n , so .t1 ; : : : ; tn /  Œx1 ; : : : ; xn  D Œt1 x1 ; : : : ; tn xn : The fixed points of this action are the points pi D Œ0; : : : ; 0; 1; 0; : : : ; 0 „ ƒ‚ … i

for i D 1; : : : ; n:

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The fiber of the tautological line bundle O.1/ at pi is the coordinate line C  "i , so T acts on O.1/pi by the character ti , and on O.1/pi by ti . In the notation of Example 1.9, O.1/pi ' L ti and O.1/pi ' Lti . Setting  D c1T .O.1//, we see that pi  D ti : Exercise 2.3. Show that the map of ƒ-algebras ƒŒ=

n Y

. C ti / ! ƒ˚n ;

 7! .t1 ; : : : ; tn /;

iD1 

is injective. (By 1.12 and 2.2, this is the restriction map HT P n1  ! LExamples   n1 T HT .P / D HT .pi /.) 2.2 Gysin maps. For certain kinds of (proper) maps f W Y ! X, there are Gysin pushforwards f W HT Y ! HTC2d X; as in ordinary cohomology. Here d D dim X  dim Y . We will use two cases, always assuming X and Y are nonsingular varieties. 1. Closed embeddings. If  W Y ,! X is a T -invariant closed embedding of codimension d , we have  W HT Y ! HTC2d X . This homomorphism has the following properties: (a)  .1/ D  ŒY T D ŒY T is the fundamental class of Y in HT2d X . (b) (Self-intersection)   .˛/ D cdT .NY =X /  ˛, where NY =X is the normal bundle. 2. Integral. For a complete (compact) nonsingular variety X of dimension n, the map  W X ! pt gives  W HT X ! HT2n .pt/. Example 2.4. Let T act on P 1 with weights 1 and 2 , so t  Œx1 ; x2  D Œ1 .t /x1 ; 2 .t /x2 /: Let p1 D Œ1; 0, p2 D Œ0; 1 as before. Setting  D 2  1 , the tangent space Tp1 P 1 has weight : t  Œ1; a D Œ1 .t /; 2 .t /a D Œ1; .t /a: Similarly, Tp2 P 1 has weight . So p1 Œp1 T D c1T .Tp1 P 1 / D , and p2 Œp2 T D . (And the other restrictions are zero, of course.) From Example 2.2, we know p1  D 1 D   2 and p1  D 2 D   1 , so Œp1 T D  C 2 in HT P 1 .

and

Œp2 T D  C 1

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Exercise 2.5. More generally, show that if T acts on P n1 with weights 1 ; : : : ; n , then Y Œpi T D . C j / j ¤i

in

HT P n1 .

Example 2.6. If p 2 Y X , with X nonsingular, and p a nonsingular point on the (possibly singular) subvariety Y , then p ŒY T

D

cdT .Np /

D

d Y

i

iD1

in ƒ D HT .p/, where the i are the weights on the normal space Np to Y at p. 2.3 First localization theorem. Assume that X is a nonsingular variety, with finitely many fixed points. Consider the sequence of maps M M   ƒ D HT X T  ! HT X  ! HT X T D ƒ: p2X T

p2X T

L L The composite map   W ƒ! ƒ is diagonal, and is multiplication by cnT .Tp X / on the summand corresponding to p. Theorem 2.7. Let S ƒ be a multiplicative set containing the element Y cnT .Tp X /: c WD p2X T

(a) The map

S 1  W S 1 HT X ! S 1 HT X T

. /

is surjective, and the cokernel of  is annihilated by c. (b) Assume in addition that HT X is a free ƒ-module of rank at most #X T . Then the rank is equal to #X T , and the above map . / is an isomorphism. Proof. For (a), it suffices to show that the composite map S 1 . ı / D S 1  ıS 1  is surjective. This in turn follows from the fact that the determinant Y cnT .Tp X / D c det. ı  / D p

becomes invertible after localization. For (b), surjectivity of S 1  implies rank HT X  #X T , and hence equality. Finally, since S 1 ƒ is noetherian, a surjective map of finite free modules of the same rank is an isomorphism.

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2.4 Equivariant formality. The question arises of how to verify the hypotheses of Theorem 2.7. To this end, we consider the following condition on a T -variety X : (EF) HT X is a free ƒ-module, and has a ƒ-basis that restricts to a Z-basis for H  X . Using the Leray–Hirsch theorem, this amounts to degeneration of the Leray–Serre spectral sequence of the fibration E T X ! B. A space satisfying the condition (EF) is often called equivariantly formal, a usage introduced in [GKM]. One common situation in which this condition holds is when X is a nonsingular projective variety, with X T finite. In this case, the Białynicki-Birula decomposition yields a collection of T -invariant subvarieties, one for each fixed point, whose classes form a Z-basis for H  X . The corresponding equivariant classes form a ƒ-basis for HT X restricting to the one for H  X , so (EF) holds. Moreover, since the basis is indexed by fixed points, HT X is a free ƒ-module of the correct rank, and assertion (b) of Theorem 2.7 also holds. Corollary 2.8. The “two notions” from §1.5 hold for a nonsingular projective T variety with finitely many fixed points: HT X  H  X and HT X ,! HT X T : Remark 2.9. Condition (EF) is not strictly necessary to have an injective localization map S 1 HT X ! S 1 HT X T . In fact, if one takes S D ƒ X f0g, no hypotheses at all are needed on X : this map is always an isomorphism (though the rings may become zero); see [H, §IV.1]. On the other hand, this phenomenon is peculiar to torus actions. For example, the group B of upper-triangular matrices acts on P n1 with only one fixed point. However, since B admits a deformation retract onto the diagonal torus T , the ring HB P n1 D HT P n1 is a free module over ƒB D ƒT . There can be no injective map to HB .P n1 /B D HB .pt/ D ƒ, even after localizing. (The difference is that HB .X X X B / is not necessarily a torsion ƒB -module when B is not a torus.) 2.5 Integration formula (Atiyah–Bott–Berline–Vergne) Theorem 2.10. Let X be a compact nonsingular variety of dimension n, with finitely many T -fixed points. Then X p ˛  ˛ D cnT Tp X T p2X

for all ˛ 2 HT X . Proof. Since  W S 1 HT X T ! HT X is surjective, it is enough to assume ˛ D .p / ˇ, for some ˇ 2 HT .p/ D ƒ. Then the LHS of the displayed equation is  ˛ D

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 .p / ˇ D ˇ. (The composition HT .p/ ! HT X ! ƒ is an isomorphism.) The RHS is X q .p / ˇ p .p / ˇ D D ˇ; cnT Tq X cnT Tp X T q2X

using the self-intersection formula for the last equality. Example 2.11. Take X D P n1 , with the standard action of T via character t1 ; : : : ; tn , and let  D c1T .O.1//. Then one computes ´ 0 if k < n  1; by degree: HT2k2.n1/ .pt/ D 0I k  . / D 1 if k D n  1; by ordinary cohomology: On the other hand, using the localization formula, we obtain  . k / D

n X iD1

.ti /k ; j ¤i .tj  ti /

Q

yielding a nontrivial algebraic identity! Remark 2.12. Ellingsrud and Strømme [ES] used this technique, with the aid of computers, to find the answers to many difficult enumerative problems, e.g., the number of twisted cubics on a Calabi–Yau three-fold. As an illustrative exercise, one could compute the number of lines passing through to four given lines in P 3 . (Use localization for the action of T ' .C  /4 on the Grassmannian Gr.2; 4/.) A more challenging problem is to compute number of conics tangent to five given conics in P 2 , via localization for the action of T ' .C  /2 on the space of complete conics; see, e.g., [Br2, p. 15]. 2.6 Second localization theorem. A remarkable feature of equivariant cohomology is that one can often characterize the image of the restriction map  W HT X ! HT X T , realizing HT X as a subring of a direct sum of polynomial rings. To state a basic version of the theorem, we use the following hypothesis. For characters  and 0 appearing as weights of Tp X , for p 2 X T , assume: . / if  and 0 occur in the same Tp X they are relatively prime in ƒ. Condition . / implies that for each p 2 X T and each  occurring as a weight for Tp X , there exists a unique T -invariant curve E D E;p ' P 1 through p, with Tp E ' L . By Example 2.4, E T D fp; qg, and Tq E has character . Theorem 2.13. Let X be a nonsingular variety with X T finite, and assume . / holds. Then an element M .up / 2 ƒ D HT X T p2X T

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lies in  HT X if and only if, for all E D E;p D E;q , the difference up  uq is divisible by . As with the other localization theorems, the idea of the proof is to use the Gysin map and self-intersection formula, this time applied to the compositions HT X T ! HT X  ! HT X ! HT X  ! HT X T ; where X  is a union of invariant curves E for a fixed character . For a detailed proof, see [Fu2, §5]. The theorem is originally due to Chang and Skjelbred [CS]. It was more recently popularized in an algebraic geometry context by Goresky, Kottwitz and MacPherson in [GKM]. The utility of this characterization is that it makes HT X a combinatorial object: the ring can be computed from the data of a graph whose vertices are the fixed points X T and whose edges are the invariant curves E D E;p D E;q .

3 Lecture 3: Grassmannians and Schubert calculus 3.1 Pre-history: Degeneracy loci. Let X be a Cohen–Macaulay variety, and let E be a rank n vector bundle on X, admitting a filtration 0 D E0  E1      En D E;

where rank Ei D i:

Let F be a rank r vector bundle, and let ' W E ! F be a surjective morphism. Given a partition  D .r  1  2      nr  0/, the associated degeneracy locus is defined as '.x/

D .'/ D fx 2 X j rank.Eri Ci .x/ ! F .x//  r  i for 1  i  n  rg X: Since these schemes appear frequently in algebraic geometry, it is very useful to have a formula for their cohomology classes. Such a formula was given by Kempf and Laksov [KL], and independently by Lascoux [L]: P Theorem 3.1 (Kempf–Laksov). Set k D n  r. When codim D D jj WD i , we have ŒD  D  .c.F  E// WD det.ci Cj i .i // ˇ ˇ c1 .1/ c1 C1 .1/    ˇ ˇ ˇ c 1 .2/ c2 .2/ 2 D ˇˇ :: :: ˇ : : ˇ ˇ c kC1 .k/   k in H  X, where cp .i / D cp .F  Eri Ci /.

c1 Ck1 .1/ :: : :: : ck .k/

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

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Here the notation c.A  B/ D c.A/=c.B/ means the formal series expansion of .1 C c1 .A/ C c2 .A/ C    /=.1 C c1 .B/ C c2 .B/ C    /, and cp is the degree p term. The proof of this theorem starts with a reduction to the Grassmannian bundle  W Gr.k; E/ ! X . Since ' is surjective, the subbundle K D ker.'/ E has rank k, and it defines a section W X ! Gr.k; E/ such that  S D K. (Here S   E is the tautological rank k bundle on Gr.k; E/.) In fact, the theorem is equivalent to a formula in equivariant cohomology. The universal base for rank n bundles with filtration is BB; see Example 1.7. Consider the following diagram: / Gr.k; E/ Gr.k; E/ q8 g qqq q  qq   qqqq f / BB: X Writing E1      En D E for the tautological sequence on BB D Fl.1; 2; : : : ; nI C 1 /, the map f is defined by Ei D f  Ei for 1  i  n. The map g is defined by F D g  F , where E ! F is the universal quotient on Gr.k; E/. Now a formula for D in H  X can be pulled back from a universal formula for a corresponding locus  in H  Gr.k; E/ D HB Gr.k; C n / D HT Gr.k; C n /. We will see how such a formula can be deduced combinatorially, using equivariant localization. Remark 3.2. Some extra care must be taken to ensure that the bundles Ei and F are pulled back from the algebraic approximations Bm ; see, e.g., [G1, p. 486]. 3.2 The basic structure of Grassmannians. The Grassmannian X D Gr.k; C n / is the space of k-dimensional linear subspaces in C n and can be identified with the quotient ı Mnk =GLk of the set of full-rank n by k matrices by the action of GLk by right multiplication. The groups T ' .C  /n  B  GLn act on X by left multiplication. For any k-element subset I  f1; : : : ; ng, we denote by UI the set of all k-dimensional linear subspaces of C n whose projection on the subspace spanned by the vectors fei j i 2 I g, where fe1 ; : : : ; en g is the standard basis for C n . It follows that UI ' C k.nk/ and UI is open in X . Indeed, UI can be identified with the set of k  .n  k/ matrices M whose square submatrix on rows I is equal to the k  k identity matrix. For I D f2; 4; 5g, we have 82 39

> ˆ ˆ > ˆ6 > ˆ > 1 0 0 7 ˆ > ˆ 6 7 > <6 7=



6 7 UI D 6 7 : ˆ ˆ 6 0 1 0 7> > ˆ ˆ > 4 0 0 1 5> ˆ > ˆ > : ;



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In particular, dim X D k.n  k/. The topology of Gr.k; C n / is easily studied by means of the decomposition into Schubert cells: Each point p 2 X has an “echelon” form, that is, it can be represented by a full rank n  k matrix such as 2 3



6 1 0 0 7 6 7 6 0 7 6 7 6 0 1 0 7: 7 6 4 0 0 1 5 0 0 0 We denote the set of rows with 1’s by I (so in the example above, I D f2; 4; 5g) and call it the pivot of the corresponding subspace. For any k-element subset I  f1; : : : ; ng, the set Iı of points with pivot I forms a cell, isomorphic to an affine space of dimension equal to the number of stars in the matrix. These are called Schubert cells, and they give a cell decomposition of X . Summarizing, we have closed

open

Iı ,! UI ,! X: Note that Iı and UI are T -stable. 3.3 Fixed points and weights. Let pI 2 UI be the origin, that is, the point corresponding to the subspace spanned by fei j i 2 I g. Working with matrix representatives, the following basic facts are easy to prove: sees that   • The T -fixed points in X are precisely the points pI . In particular, #X T D kn . • The weights of T on TpI X D TpI UI ' UI are ftj  ti j i 2 I; j … I g. • The weights of T on TpI Iı are ftj  ti j i 2 I; j … I; i > j g. • The weights of T on NI =X;pI are ftj  ti j i 2 I; j … I; i < j g. Example 3.3. With I D f2; 4; 5g, one sees that t4  t3 is a weight on TpI Iı : 2 6 6 6 z6 6 6 4

0 1 0 0 0 0

0 0 a 1 0 0

0 0 0 0 1 0

3

2

7 6 7 6 7 6 7D6 7 6 7 6 5 4

0 z2 0 0 0 0

0 0 0 0 z3 a 0 z4 0 0 z5 0 0

3

2

7 6 7 6 7 6 7 6 7 6 7 6 5 4

The other weights can be determined in a similar manner.

0 1 0 0 0 0

0 0 z3 a z4 1 0 0

0 0 0 0 1 0

3 7 7 7 7: 7 7 5

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3.4 Schubert classes in HT X . The closure I WD Iı of a Schubert cell is called a Schubert variety. It is a disjoint union of all Schubert cells ıJ for J  I with respect to the Bruhat order: J I

iff

j1  i1 ; j2  i2 ; : : : ; jk  ik :

Since the Schubert cells have even (real) dimension, it follows that the classes of their closures form bases for H  X and HT X : M M Z  ŒI  and HT X D ƒ  ŒI T : H X D I

I

HT X

In particular, is free over ƒ, of the correct rank, so the localization theorems apply. Let us record two key properties of Schubert classes. From the description of weights, and from Example 2.6, it follows that Y .tj  ti /: (1) pI ŒI T D i2I j …I i<j

Moreover, since pJ 2 I if and only if J  I we see that pJ ŒI T D 0

if J 6 I:

(2)

It turns out that the Schubert classes are unique solutions to a corresponding interpolation problem: Proposition 3.4 (Knutson–Tao [KT], Fehér–Rimányi [FR]). Relations (1) and (2) determine ŒI T as a class in HT X . 3.5 Double Schur functions. Every k-element subset I  f1; : : : ; ng can be represented in a form of a Young diagram: one first draws a rectangle with k rows and n  k columns, then draws a path from the upper-right to the lower-left corner; at i -th step, we go downwards if i 2 I and to the left otherwise. Counting the number of boxes in each row, we obtain a partition  D .I / D .1  2      k  0/. Example 3.5. With n D 7 and k D 3, the subset I D f2; 4; 5g corresponds to the partition  D .3; 2; 2/, whose Young diagram is below:

A semistandard Young tableau (SSYT for short) on a Young diagram  is a way of filling the boxes of  with numbers from f1; : : : ; kg such that they are weakly increasing along rows (going to the right) and strictly increasing along columns (going downwards). We write SSY T ./ for the set of all SSYT on the diagram .

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Definition 3.6. The double Schur function associated to  is a polynomial in two sets of variables, x D .x1 ; x2 ; : : :/ and u D .u1 ; u2 ; : : :/, defined by the formula X Y   s .xju/ D xS.i;j /  uS.i;j /Cj i : S2SSY T ./ .i;j /2

Here S.i; j / is the .i; j /-entry of the tableau S , using matrix coordinates on the diagram . Example 3.7. There are 8 semistandard Young tableaux on the diagram  =

1 1 2

1 1 3

1 2 2

1 2 3

1 3 2

1 3 3

2 2 3

;

2 3 3 ;

so the double Schur function is s .xju/ D .x1  u1 /.x1  u2 /.x2  u1 / C .x1  u1 /.x1  u2 /.x3  u2 / C .x1  u1 /.x2  u3 /.x2  u1 / C .x1  u1 /.x2  u3 /.x3  u2 / C .x1  u1 /.x3  u4 /.x2  u1 / C .x1  u1 /.x3  u4 /.x3  u2 / C .x2  u2 /.x2  u3 /.x3  u2 / C .x2  u2 /.x3  u4 /.x3  u2 /: It is not obvious from the definition, but in fact s .xju/ is symmetric in the x variables. A nice discussion of some properties of these functions can be found in [Ma, 6th Variation]; see also [CLL] for a more recent study. A crucial fact for us is that they solve the same interpolation problem as the Schubert classes: Proposition 3.8 (Molev–Sagan [MoS], Okounkov–Olshanski [OO]). Set ui D tnC1i ;

tj D tij ;

where I D fi1 <    < ik g is the subset corresponding to . Then Y .tj  ti / s .t  ju/ D i2I j …I i<j

and s .t  ju/ D 0

if 6 :

One can deduce a “Giambelli” formula for the Schubert classes. When  is the partition corresponding to a subset I , let us write  D I for the corresponding Schubert variety. Now observe that the stars in the echelon matrix form of ı naturally fit into the complement of the diagram  inside the k  .n  k/ rectangle. Therefore the P codimension of  is equal to jj D kiD1 i , the number of boxes in . Moreover, J  I with respect to the Bruhat order if and only if  as diagrams, where is the partition corresponding to J .

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Corollary 3.9. Let S be the tautological subbundle on X D Gr.k; C n /, let x1 ; : : : ; xk be the equivariant Chern roots of the dual bundle S _ , and let t1 ; : : : ; tn be the standard weights of T D .C  /n . Using the substitution from Proposition 3.8, we have Œ T D s .xju/ in HT X. Proof. Using Propositions 3.4 and 3.8, it suffices to observe that the weights of SjpJ are t1 ; : : : ; tk if is the partition corresponding to J , so s .t  ju/ is equal to the restriction J s .xju/. Finally, to relate this formula to the Kempf–Laksov formula, consider the sequence of vector bundles on X '

 Q; E1 ,! E2 ,!    ,! En D C n ! where the Ei are trivial bundles, spanned by fe1 ; : : : ; ei g, and Q D C n =S is the universal quotient bundle. We claim that  D D .'/: This is a standard fact, and the main point is that D .'/ is irreducible; see, e.g., [Fu1]. One way to see this it as follows. The group B of upper-triangular matrices acts on X, and it preserves the Ei , so it also acts on each D .'/. On the other hand, from the matrix representatives, it is easy to see that ı is the B-orbit of pJ (for J corresponding to , as usual). One checks that the conditions defining D are satisfied by pJ if and only if J  I , and it follows that D is the union of the cells ı for . Remark 3.10. A determinantal formula for s .xju/ can be proved directly by algebraic means; see [Ma, 6.7]. Using this, one obtains a new proof of the Kempf–Laksov formula. A similar proof that the double Schur functions represent Schubert classes is given in [Mi, §5]. 3.6 Positivity. Since the Schubert classes form a ƒ-basis for HT X , we can write X c .t /Œ T ; (3) Œ T  Œ T D .t / 2 ƒ D ZŒt1 ; : : : ; tn , homogeneous of degree jj C j j  for some polynomials c  j j. Under the map HT X ! H  X , one recovers the structure constants c D c .0/ for ordinary cohomology. In fact, the integers c are nonnegative – using the Kleiman–

Bertini theorem, they count the number of points in a transverse intersection of generic translates of three Schubert varieties. (There are also many combinatorial formulas for these numbers, which are called the Littlewood–Richardson coefficients.) A remarkable fact about the equivariant coefficients c .t / is that they also exhibit a certain positivity property:

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.t / lies in Z0 Œt2  t1 ; t3  t2 ; : : : ; tn  tn1 . Theorem 3.11. The polynomial c

This is a special case of a theorem of W. Graham [G2], who used a degeneration argument. We will sketch a different proof based on a transversality argument, from [A]. Sketch of proof. First note that there are equivariant Poincaré dual classes, given by the opposite Schubert varieties. Specifically, let Ezi be the span of fen ; en1 ; : : : ; enC1i g, and consider the sequence 'Q

Ez1 ,! Ez2 ,!    ,! Ezn D C n !  Q: z  to be the degeneracy locus D Q .'/ Q for this sequence, where Q is the partition Define   whose shape (rotated 180 degrees) is the complement to  inside the k  .n  k/ z rectangle. Equivalently, let Bz be the subgroup of lower triangular matrices; then  ı z z is the closure of the cell  obtained as the B-orbit of pI . z From the description as a B-orbit closure, it is easy to see that the intersection z  consists of the single point pI , and is transverse there. Applying the integration  \  formula, we see that z  T / D ı  .Œ T  Œ in ƒ, and applying this to both sides of (3), z T /: .t / D  .Œ T  Œ T  Œ c

The idea of the proof is find a subvariety Z in the approximation space Em T X z T , for a special choice of approximation whose class is equal to Œ T  Œ T  Œ Em ! Bm . Pushing forward the class of such a Z yields an effective class in H  Bm , which corresponds to c .t /. To set this up, fix an isomorphism T ' .C  /n using the following basis for the character group: t1  t2 ; t2  t3 ; : : : ; tn1  tn ; tn : (The reason for this choice will become clear later.) Now for m  0, take Bm D .P m /n , and write Mi D Oi .1/ for the pullback via the ith projection, as in Example 1.5. Identifying H  Bm with ƒ D ZŒt1 ; : : : ; tn , we have c1 .Mi / D ti  tiC1 for 1  i  n  1, and c1 .On .1// D tn . Note that every effective class in H  Bm is a nonnegative linear combination of monomials in t2  t1 ; : : : ; tn  tn1 ; tn , so if we find Z as above, the positivity theorem will be proved. (In fact, the whole setup may be pulled back via the projection onto the first n  1 factors of Bm D .P m /n , so that tn does not contribute, and the class lies in the claimed subring of ƒ.) To find Z, we construct a group action on the approximation space. First some notation. Set Li D Mi ˝ MiC1 ˝    ˝ Mn , so c1 .Li / D ti . Using the standard action of T on C n , we have Em T C n D L1 ˚    ˚ Ln D E as vector bundles

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on Bm . Moreover, the flags used to define the Schubert varieties are replaced with Ei D L1 ˚    ˚ Li and Ezi D Ln ˚    ˚ LnC1i , and  D Em T 

z  D Em T  z and 

are corresponding degeneracy loci. L The key observation is that the vector bundle End.E/ D i;j Lj_ ˝ Li has global sections in “lower-triangular matrices”, since when i  j , the line bundle Lj_ ˝ Li D _ Mj_ ˝    ˝ Mi1 is globally generated. The bundle of invertible endomorphisms Aut.E/  End.E/ is a group scheme over Bm , and its group of global sections 0 maps surjectively onto Bz by evaluation at each fiber. Including the action of the group G D .PGLmC1 /n acting transitively on the base Bm , it follows that the opposite z  are the orbits for a connected group  D G Ì 0 acting on Schubert bundles  Em T X. An application of the Kleiman–Bertini theorem guarantees that there is an element z is a proper intersection, so it has codimension  2  such that Z D  \    \  T z  D jj C j j  j j C dim X in Em  X . By construction, ŒZ D Œ   Œ   Œ T T T z  , so we are done. Œ   Œ   Œ Remark 3.12. A similar argument works to establish positivity for any homogeneous space X D G=P , so one can recover the general case of Graham’s theorem. An intermediate approach is to observe that the group  has a unipotent subgroup with finitely many orbits on Em T X , so a theorem of Kumar and Nori [KN] guarantees positivity. .t / Remark 3.13. From the point of view of degeneracy loci, the coefficients c are universal structure constants. That is, for a fixed degeneracy problem E1 ,! '    ,! En D E !  F on an arbitrary (Cohen–Macaulay) variety X , the subring of H  X generated by the classes ŒD .'/ has structure constants c .u/, where ui D c1 .Ei =Ei1 / is the pullback of ti by the classifying map X ! BB. The positivity theorem therefore implies a corresponding positivity for intersections of degeneracy loci on any variety.

3.7 Other directions. A great deal of recent work in algebraic geometry involves equivariant techniques, either directly or indirectly. Without pretending to give a complete survey of this work, we conclude by mentioning a small sampling of these further applications. Generalized Schubert calculus. In the case of the Grassmannian, there are combi .t /, due to Knutson and Tao [KT] natorial formulas for the equivariant coefficients c and Molev [Mo]. However, the question remains open for other homogeneous spaces G=P , even in the case of ordinary cohomology! It can happen that the extra structure present in equivariant cohomology simplifies proofs; such is the case in [KT]. Some

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formulas for ordinary cohomology of cominiscule G=P were given by Thomas and Yong [TY]. Degeneracy locus formulas. Extending the Kempf–Laksov formula, there are formulas for (skew-)symmetric degeneracy loci, due to Józefiak–Lascoux–Pragacz [JLP], Harris–Tu [HT], Lascoux–Pragacz [LP], and others. These are equivalent to equivariant Giambelli formulas in the Lagrangian and orthogonal Grassmannians, which are homogeneous spaces Sp2n =P and SO2n =P . Working directly in equivariant cohomology, Ikeda et al. have given equivariant Giambelli formulas for these spaces [IMN]. Thom polynomials. The theory of Thom polynomials is one of the origins of equivariant techniques in algebraic geometry; see, e.g., [FR]. These are universal polynomials related to singularities of mappings, and can be interpreted as equivariant classes of certain orbit closures: for an algebraic group G acting linearly on a vector space V , one has ŒG  vT in HT .V / ' ƒ. They are difficult to compute in general, and have been studied recently by Fehér–Rimányi [FR], Kazarian [K], and Pragacz–Weber [PW], among others. An important special case is where G is the product of two groups of upper-triangular matrices, acting on the space of n  n matrices; a detailed combinatorial study of this was carried out by Knutson–Miller [KM]. Ackowledgements. I learned much of what I know about equivariant cohomology from William Fulton, and the point of view presented here owes a debt to his lectures on the subject. I am grateful to the organizers of IMPANGA for arranging the excellent conference in which these lectures took place. These notes were assembled with the help of Piotr Achinger, and I thank him especially for assistance in typing, researching literature, and clarifying many points in the exposition. I also thank the referee for valuable input and careful reading.

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Dave Anderson, Department of Mathematics, University of Washington, Seattle, WA 98195, U.S.A. E-mail: [email protected]

Recent developments and open problems in linear series Thomas Bauer, Cristiano Bocci, Susan Cooper, Sandra Di Rocco, Marcin Dumnicki, Brian Harbourne, Kelly Jabbusch, Andreas L. Knutsen, Alex Küronya, Rick Miranda, Joaquim Roé, Hal Schenck, Tomasz Szemberg, and Zach Teitler

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2 Original problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.1 Asymptotic effectivity (B. Harbourne) . . . . . . . . . . . . . . . . . 94 2.2 Semi-effectiveness (B. Harbourne) . . . . . . . . . . . . . . . . . . . 95 2.3 Stability of speciality (T. Szemberg) . . . . . . . . . . . . . . . . . . 96 2.4 Regularity for generic monomial zero-schemes (J. Roé) . . . . . . . . 97 2.5 Bounding cohomology (B. Harbourne) . . . . . . . . . . . . . . . . . 98 2.6 Algebraic fundamental groups and Seshadri numbers (J.-M. Hwang) . 98 2.7 Blow-ups of P n and hyperplane arrangements (H. Schenck) . . . . . 99 2.8 Bounds for symbolic powers (Z. Teitler) . . . . . . . . . . . . . . . . 101 3 Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.1 Relating h0 and h1 on surfaces . . . . . . . . . . . . . . . . . . . . . 102 3.2 Speciality on blow-ups of P 2 . . . . . . . . . . . . . . . . . . . . . . 105 3.3 Bounded negativity . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.4 Partial proof of Conjecture 3.3.5: Very weak bounded negativity . . . 108 3.5 Partial proof of Conjecture 3.3.4: Weak bounded negativity . . . . . . 110 3.6 Bounded Negativity Conjecture and Seshadri constants . . . . . . . . 111 3.7 The Weighted Bounded Negativity Conjecture . . . . . . . . . . . . . 113 3.8 Bounded negativity for reducible curves . . . . . . . . . . . . . . . . 114 3.9 Imposing higher vanishing order at one point . . . . . . . . . . . . . 116 3.10 Geometrization of Dumnicki’s method [12] . . . . . . . . . . . . . . 118 3.11 Linear systems connected to hyperplane arrangements . . . . . . . . . 125 3.12 Limitations of multiplier ideal approach to bounds for symbolic powers 126 Appendix. Logarithmic differentials and the Miyaoka–Yau inequality . . . . . 127 A.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 1 A.2 The Miyaoka–Yau inequality for X .log C / . . . . . . . . . . . . . . 129 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

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1 Introduction In the week of October 3–9, 2010, the Mathematisches Forschungsinstitut at Oberwolfach hosted the Mini-Workshop “Linear Series on Algebraic Varieties.” These notes contain a variety of interesting problems which motivated the participants prior to the event, and examples, results and further problems which grew out of discussions during and shortly after the workshop. Many arguments presented here are scattered in the literature or constitute “folklore.” It was one of our aims to have a usable and easily accessible collection of examples and results.1

2 Original problems We begin with a list of problems which were suggested by the participants for the Mini-Workshop. This list was discussed for three months before the workshop began. 2.1 Asymptotic effectivity (B. Harbourne). Let S D fp1 ; : : : ; pr g be distinct points in P N , over an algebraically closed ground field k of arbitrary characteristic. Let f W X ! P N be the morphism obtained by blowing up p1 ; : : : ; pr , and denote the exceptional divisors by E1 ; : : : ; Er . Let H D f  .OP N .1// and let L.d; m/ D dH  m.E1 C    C Er /. Waldschmidt [56] introduced and showed the existence of the following quantity: a.S; m/ e.S/ WD lim m!1 m ˚  0 where a.S; m/ WD min d W h .X; L.d; m// > 0 . It follows from the proof that me.S/ 6 a.S; m/ for all m > 1. Problem 2.1.1. Develop computational or conceptual methods for evaluating, estimating or bounding e.S/. It is trivial to show that a.S; m/ D rm and hence e.S / D r when N D 1. It is an open problem in general to compute e.S/ when N > 1. For example, for r > 9 generic points pi 2 P 2 , a still popen conjecture of Nagata [40, Conjecture, p. 772] is equivalent to having e.S/ D r. Using complex analytic methods, Waldschmidt [56] and Skoda [47] showed that a.S; 1/ 6 e.S /: N

(1)

1 Cooper’s participation was supported by the “US Junior Oberwolfach Fellows” joint NSF-MFO program under NSF grant DMS-0540019. Partial support during this project is kindly acknowledged as follows: Bocci by Italian PRIN funds; Di Rocco by Vetenskapsådets grant NT:2006-3539; Küronya by DFG-Forschergruppe 790 “Classification of Algebraic Surfaces and Compact Complex Manifolds” and the OTKA grants 77476 and 77604 by the Hungarian Academy of Sciences; Roé by Spanish Ministerio de Educación y Ciencia, grant MTM2009-10359; Schenck by NSF 07–07667, NSA 904-03-1-0006; Szemberg by MNiSW grant N N201 388834

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Given k > 0, by [28, Theorem 1.1(a)], for any finite subset S  P N and all m > 1 we have a.S; k C 1/ a.S; m.N C k// 6 kCN m.k C N / and hence a.S; k C 1/ a.S; k C 1/ 6 e.S/ 6 : kCN kC1 Taking k D 0 recovers the Waldschmidt-Skoda bound (1). Taking large values of k gives one a way of computing arbitrarily accurate estimates of e.S / [24], but computing a.S; k/ for large k is difficult to do. Chudnovsky [8] conjectured (and proved for N D 2) the stronger bound Conjecture 2.1.2.

a.S; 1/ C N  1 6 e.S /: N   (For the proof when N D 2, reduce to the case that r D aC1 , where a D a.S; 1/, 2 and use the fact that then L.a; 1/ is nef.) Two examples are known where equality in (2.1.2) holds: when the points lie in a hyperplane, or when S is a star configuration [4] (i.e., given a set of s hyperplanes in P N such   that at most N of these hyperplanes meet at any single point, S is the set of r D Ns points at which N of the hyperplanes meet). Problem 2.1.3. If for some m we have a.S; 1/ C N  1 a.S; m/ D m N is it true that S is either contained in a hyperplane or is a star configuration? What if e.S/ D

a.S; 1/ C N  1 ‹ N

Bocci and Chiantini [3] show for N D 2 that a.S;2/ D 2 either a set of collinear points or a star configuration.

a.S;1/C1 2

implies that S is

2.2 Semi-effectiveness (B. Harbourne). Again let the ground field k be an algebraically closed field of arbitrary characteristic. Definition 2.2.1. Let X be an algebraic variety and L a line bundle on X . We say that L is semi-effective, if there exists n > 0 such that h0 .nL/ > 0. Let p1 ; : : : ; pr be distinct points in P N . Let f W X ! P N be obtained by blowing up p1 ; : : : ; pr with the exceptional divisors being E1 ; : : : ; Er . Let D D dH  m1 E1      mr Er ; where H is the pullback via f of a general hyperplane. The following question was raised by M. Velasco and D. Eisenbud (in an email from Velasco to Harbourne, November 2009).

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Problem 2.2.2. Is there a way to determine if D is semi-effective? For a specific problem consider D D 13L  5E1  4E2      4E10 for generic points p1 ; : : : ; p10 2 P 2 . M. Dumnicki and J. Roé have independently shown that this system is not semi-effective (see Section 3.10). But there are infinitely many more similar examples for which semi-effectivity is still not known. For example, given generic points pi 2 P 2 , consider D D 111L  36E1  35E2      35E10 or more generally D D dL  .m C 1/E1  mE2      mE10 , where d D .b 2 C a2 /=2, m C 1 D ba, m D .b 2  a2 /=6 and where 0 < a < b are odd integers satisfying .a C 3b/2  10b 2 D 6. Note that D 2 D 0 but D cannot be reduced by Cremona transformations. According to the SHGH Conjecture [46], [23], [20], [27], we expect that none of these D are semi-effective. P Limits like those of Waldschmidt are relevant to Problem 2.2.2. Let a. riD1 mi pi / P be the least t such that h0 .tL  i mi Ei / > 0 and define P X  a.t i mi pi / P mi pi WD r lim : e t t i mi i

P P P mi is semi-effective Then D D dL  riD1P Pif e. i mi pi /=r < d= i mi , and it is > d= i mi . It is not clear whether D is seminot semi-effectivePif e. i mi pi /=r P effective when e. i mi pi /=r D d= i mi . But, if the SHGH Conjecture is true, this boundary case is precisely the situation of the examples in the preceding paragraph. 2.3 Stability of speciality (T. Szemberg). Let p1 ; : : : ; pr be points in the projective plane. Let f W X ! P 2 be the blow-up of p1 ; : : : ; pr with exceptional divisors E1 ; : : : ; Er . Let H D f  .OP 2 .1//, and let D D dH  m1 E1      mr Er : Assume that the divisor D is special (i.e., D is effective with h1 .X; D/ > 0). Is it then true that nD is special for all n > 1‹ A somewhat more demanding problem is to determine whether the asymptotic cohomology function hO 1 as defined in [33] is positive. This is not true if the points are arbitrary (or less than P10?). Indeed, let p1 ; : : : ; p9 be intersection points of two cubics and let L D 3H  9iD1 Ei be the anti-canonical pencil. Then h0 .nL/ D n C 1;

h1 .nL/ D n and

h2 .nL/ D 0;

so that all nL are special but all asymptotic cohomology functions vanish.

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2.4 Regularity for generic monomial zero-schemes (J. Roé). Let I  kŒx; y be a monomial ideal, and let Z D Spec kŒx; y=I . Let n D dimk kŒx; y=I be the length of Z, which we assume to be finite (in this case, I is .x; y/-primary, and Z is supported at the origin). For each (irreducible smooth projective) surface S defined over k there is an irreducible constructible subset HilbI S of Hilbn S whose closed points are the subschemes Y of S isomorphic to Z. Of course, if I D .x; y/m then Z is just an m-fold point, and HilbI S Š S is the set of points of S taken with multiplicity m. If I D .y 2 ; yx 2 ; x 3 / then Z is a cusp scheme, which means that curves containing Z have at least a cusp at the point supporting Z (i.e., they have a cusp or a more special singularity, and generically it is a cusp). Since the cusp scheme marks the tangent direction to the cusp, HilbI S is in this case naturally isomorphic to the (projectivized) tangent bundle of S . Other monomial ideals correspond to other singularity “types.” Problem 2.4.1. Describe HilbI S. Is it locally closed? What adjacencies are there between such subsets of the Hilbert scheme? As HilbI S is irreducible, it makes sense to consider general (or very general, or generic) subschemes Y of S isomorphic to Z. For each divisor D we have an exact sequence 0 ! Y ˝ OS .D/ ! OS .D/ ! OY .D/ ! 0 inducing the usual exact sequence in cohomology. Problem 2.4.2. What can we say (or conjecture) about the cohomology of Y ˝ OS .D/? To be more precise, it would be nice to have conditions on S , I and D implying that h0 D maxf0; h0 .OS .D//  ng, in which case h1 and h2 are easily computed by Riemann–Roch and duality. Assuming the previous problem is understood, one can then further ask about the base locus of global sections of Y ˝OS .D/, and ask if they cut out the scheme Z. Such questions are of interest in the construction of curves with imposed singularities, and have been studied mainly for schemes of small multiplicity (the multiplicity of Z is the maximum m such that I  .x; y/m ). Note that multiplicity 1 schemes are curvilinear and well known; multiplicity 2 monomial schemes are also quite well understood [43]. The same kind of questions arise as auxiliary problems for the induction arguments of differential Horace methods [15], even when one is primarily interested in linear series defined by ordinary multiple points. Note also that the scheme defined by I D .y; x r /m is a specialization (or collision) of r distinct m-fold points; thus the dimension of the linear series H 0 .Y ˝ OS .D// is by semicontinuity a bound for the dimension of the linear series determined by a set of r general m-fold points. This gives a link with the Nagata conjecture [40] and the SHGH conjecture [46], [23], [20], [27]. More precisely:

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Conjecture 2.4.3. Let S D P 2 and I D .y; x r /m , where r and m are natural numbers with r > 9. Then for all d > 0, and general Y 2 HilbI S , ´ ! !μ d C2 mC1 0 r : H .Y ˝ OP 2 .D// D max 0; 2 2 Conjecture 2.4.4. Let S D P 2p and I D .y; x r /m , where r and m are natural numbers with r > 9. Then for all d 6 m r, and general Y 2 HilbI S , H 0 .Y ˝OP 2 .D// D 0. Conjecture 2.4.3 implies the uniform Harbourne–Hirschowitz conjecture, and Conjecture 2.4.4 implies Nagata’s conjecture. It is also clear that Conjecture 2.4.3 implies Conjecture 2.4.4. A conjecture in terms of monomial ideals implying the general Harbourne–Hirschowitz conjecture (without uniformity assumptions on the multiplicities) can be stated similarly; we skip it here to avoid introducing the necessary notations, which would lengthen this section unnecessarily, and refer to Hirschowitz’s description of the “collisions de front” in [26] instead. 2.5 Bounding cohomology (B. Harbourne). There are various equivalent versions of the SHGH Conjecture [46], [23], [20], [27]. Here is one: Conjecture 2.5.1 (SHGH). Let C  X be a prime divisor where X ! P 2 is the blow-up of generic points p1 ; : : : ; ps . Then h1 .X; OX .C // D 0. Problem 2.5.2. How can we remove the assumption about the points being generic in the SHGH Conjecture? The following conjecture arose out of discussions between Harbourne, J. Roé, C. Ciliberto and R. Miranda. [NB: Corollary 3.1.2 gives a counterexample. See also Proposition 3.1.3.] Conjecture 2.5.3. Let X be a smooth projective surface (either rational or assume the characteristic is 0). Then there exists a constant cX such that for every prime divisor C we have h1 .X; C / 6 cX h0 .X; C /. The SHGH Conjecture is that cX D 0 when X is obtained by blowing up generic points of P 2 . 2.6 Algebraic fundamental groups and Seshadri numbers (J.-M. Hwang). Denote by  y1 .Y / the algebraic fundamental group of an irreducible variety Y . Following [32, Definition (2.7.1)], we say that a projective manifold X has large algebraic fundamental x ! X , the group if for every irreducible variety Z  X and its normalization  W Z image of the induced homomorphism on the algebraic fundamental groups x ! y1 .Z/ y1 .X /  W 

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is infinite. This is equivalent to saying that the algebraic universal cover of X does not contain a complete subvariety. The proof of [32, Lemma 8.2] gives the following. Proposition 2.6.1. Let N be a positive number. Let X be a projective manifold with large algebraic fundamental group and let L be an ample line bundle on X . Then there exists a finite étale cover p W X 0 ! X such that any irreducible subvariety W in X 0 satisfies .p  L/dim.W /  W > N . One can then ask the following: Problem 2.6.2. Let X be a projective manifold with large algebraic fundamental group and let L be an ample line bundle on X . (1) Given a positive number N , does there exist a finite étale cover p W X 0 ! X such that the Seshadri number of p  L at any point is bigger than N ? (2) Given a positive number N , does there exist a finite étale cover p W X 0 ! X such that denoting by pQ W X 0  X 0 ! X  X the self-product of p and D 0  X 0  X 0 the diagonal, the Seshadri number of pQ  .p1 L ˝ p2 L/ along D 0 is bigger than N? (3) If the answer to (1) or (2) is negative or unclear, what is the condition on the fundamental group of X to guarantee a positive answer? 2.7 Blow-ups of P n and hyperplane arrangements (H. Schenck). Let AD

d [

V .˛i /  P 2

iD1

be a union of lines in P 2 , Y the singular locus, and  W X ! P 2 the blow-up at Y . Let P R D CŒy1 ; : : : ; yd , and for each linear dependency ƒ D jkD1 cij ˛ij D 0 on the lines of A, let k X cij .yi1    yyij    yik /: fƒ D j D1

The ideal I generated by the fƒ is called the Orlik–Terao ideal, and the quotient C.A/ D R=I is called the Orlik–Terao algebra; of course, C.A/ can be defined for arrangements in higher dimensional spaces. For a real, affine arrangement A, Aomoto conjectured a relationship between C.A/ and the topology of Rn n A, which Orlik and Terao proved in [42]. Example 2.7.1. Consider A D V .x1 x2 x3 .x1 C x2 C x3 /.x1 C 2x2 C 3x3 //  P 2 : Since ˛1 C ˛2 C ˛3  ˛4 D 0, y2 y3 y4 C y1 y3 y4 C y1 y2 y4  y1 y2 y3 2 I:

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The five lines meet in ten points, and every subset of four lines gives a similar relation, one of which is redundant. Thus, I is generated by four cubics, which turn out to be the maximal minors of a matrix of linear forms. This means that I has a Hilbert–Burch resolution, and V .I / is a surface of degree six in P 4 ; a computation shows V .I / has five singular points. Consider the divisor 10 X DA D 4E0  Ei iD1

on X, where X is the blow-up of P 2 at the ten points of Y , Ei are the exceptional curves over the singular points, and E0 is the proper transform of a line. Then DA is nef but not ample; the lines of the original arrangement are contracted to points, and I is the ideal of X in Proj.H 0 .DA //. The example above is representative of the general case. In [45], it is shown that if 'A W X ! P .H 0 .DA /_ /; then C.A/ is the homogeneous coordinate ring of 'A .X / and 'A is an isomorphism on   .P 2 n A), contracts the lines of A to points, and blows up Y . The motivation for studying C.A/ arises from its connection to topology. In [41], Orlik and Solomon determined the cohomology ring of a complex, affine arrangement nC1 complement n A: A D H  .M; Z/ is the quotient of the exterior algebra V d M DC E D .Z / on generators P e1 ; : : : ; ed in degree 1 by the ideal generated by all elements of the form @ei1 :::ir WD q .1/q1 ei1    e iq    eir , for which codim Hi1 \  \Hir < r. Since A is a quotient of an exterior algebra, multiplication by an element a 2 A1 gives a degree one differential on A, yielding a cochain complex .A; a/: .A; a/ W

0

/ A0

a

/ A1

a

/ A2

a

/ 

a

/ A`

/0:

P The first resonance variety R1 .A/ consists of points a D diD1 ai ei $ .a1 W    W ad / in P .A1 / Š P d 1 for which H 1 .A; a/ ¤ 0. Conjectures of Suciu [49] relate the fundamental group of M to R1 .A/. Falk showed that R1 .A/ may be described in terms of combinatorics, and conjectured that R1 .A/ is a subspace arrangement, which was shown by Cohen–Suciu ([9]) and Libgober–Yuzvinsky ([36]). The paper [45] describes a connection between combinatorics of R1 .A/ which give rise to factorizations of DA and corresponding determinantal equations in I . Problem 2.7.2. Do the results for lines in P 2 generalize to higher dimension? For example, if A  P n , then [45] shows that the Castelnuovo–Mumford regularity of C.A/ is bounded by n. Can an explicit description of the graded Betti numbers of C.A/ be given in terms of the geometry of A?

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2.8 Bounds for symbolic powers (Z. Teitler). Let X be a non-singular variety of dimension n defined over the complex numbers and let Z  X be a reduced subscheme of X with ideal sheaf I D IZ  OX . The p th symbolic power of I , denoted I .p/ , is the sheaf of all function germs vanishing to order > p at each point of Z. The inclusion I p  I .p/ is clear, or in other words I r  I .m/ for r > m, but it is not clear when inclusion in the other direction, I .m/  I r , holds; m > r is necessary but not sufficient in general. Related to a result of Swanson [50], Ein–Lazarsfeld–Smith used multiplier ideals to prove that if every component of Z has codimension 6 e in X then I .re/  I r for all r 2 N; in particular I .rn/  I r [13]. This was subsequently proved in greater generality by Hochster–Huneke using the theory of tight closure [28]. The big height of a radical ideal I , denoted bight.I /, is the maximum codimension of a component of V .I /. Harbourne raised the question whether it is possible to give an improvement of the form I .m/  I r whenever m > f .r/, for some function f .r/ 6 re, for all radical ideals of big height e. Bocci–Harbourne [4] showed if  > 0 is such that for all radical ideals I , I .m/  I r whenever m >  r, then  > n. Furthermore, if I .m/  I r holds whenever m >  r for all radical ideals I with bight.I / D e, then  > e. This shows that the function f .r/ D re appearing in the Ein–Lazarsfeld–Smith result cannot be decreased by lowering the coefficient e D bight.I /. Harbourne asked whether a constant term can be subtracted, that is whether I .m/  r I holds whenever m > f .r/ D er  k for all radical ideals of big height e, for some k. Values k > e do not work (the containment fails if I is a complete intersection and – rather trivially – if r D 1 or e D 1). On the other hand, examples studied by Bocci–Harbourne suggest that k D e  1 might work. Harbourne made the following: Conjecture 2.8.1 ([2, Conjecture 8.4.3]). For radical ideals I of big height e, I .m/  I r whenever m > re  .e  1/. A weaker result was observed by Takagi–Yoshida [52] (see also [53] for an expository account) who showed that if ` < lct.I ./ / then I .m/  I r whenever m > re  `. Here lct.I ./ / is the log canonical threshold of the graded system of ideals I ./ ; in particular this is always 6 e (so ` 6 e  1). Harbourne–Huneke have asked if an improvement is possible on the other side of the inclusion: Instead of asking for I .m/  I r , they raise the following: Problem 2.8.2. Suppose .R; m/ is a regular local ring of dimension n and I  R is an ideal with bight.I / D e. Then do the following hold? 1. I .m/  mrnr I r for m > rn. 2. I .m/  mrnr.n1/ I r for m > rn  .n  1/. 3. I .m/  mrer I r for m > re. 4. I .m/  mrer.e1/ I r for m > re  .e  1/.

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3 Progress In this part we show several solutions to the original problems, present examples which are closely related to the problems, and provide some evidence either for positive or negative answers or forcing reformulation of the original statements. 3.1 Relating h0 and h1 on surfaces. We show that Conjecture 2.5.3 is false in general. We claim (see Corollary 3.1.2) that there exists a surface X such that for an arbitrary positive integer c there exists a reduced, irreducible curve C  X with h1 .X; OX .C // > c  h0 .X; OX .C //: It turns out that an example of Kollár (taken from [35, Example 1.5.7]) provides a counterexample to Conjecture 2.5.3. We recall briefly the construction, in which we will closely follow the exposition mentioned above. We consider an elliptic curve E without complex multiplication, and take the abelian surface Y WD E  E to be our starting point. Divisors and the various cones on Y are well understood. The Picard number .Y / equals 3, and N 1 .X /R has the fairly natural basis consisting of the classes of F1 , F2 (the fibres of the two projection morphisms), and that of the diagonal   E  E. The intersection form on Y is given by the numbers .F12 / D .F22 / D .2 / D 0;

.F1  F2 / D .F1  / D .F2  / D 1:

It is known (for details see for example [35, Section 1.5.B]) that the nef and pseudoeffective cones coincide, and a class C D a1 F1 C a2 F2 C b is nef if and only if .C 2 / > 0 and .C  H / > 0 for some ample class H . In coordinates we can express this as a1 a2 C a1 b C a2 b > 0 and a1 C a2 C b > 0 by choosing the ample divisor F1 C F2 C  for H . For every integer n > 2 set An WD nF1 C .n2  n C 1/F2  .n  1/: It is immediate to check that 1. .A2n / D 2, and 2. .An  .F1 C F2 // D n2  2n C 3 > 0. Kollár now sets R WD F1 C F2 , and picks a smooth divisor B 2 j2Rj (which exists because 2R is base point free by the Lefschetz theorem [34, Theorem 4.5.1]) to form the double cover f W X ! Y branched along B. Let Dn WD f  An .

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Proposition 3.1.1. With notation as above, h1 .X; OX .nDn // > n3  2n2 C 3n  1:   Proof. We will estimate h1 X; OX .nDn / from below with the help of the Leray spectral sequence. It is a standard fact that     E2p;q WD H p Y; Rq f OX .nDn / H) H pCq X; OX .nDn / : We are interested in the case p C q D 1, in which case the involved terms are     H 0 Y; R1 f .OX .nDn / and H 1 Y; f OX .nDn / : Of these the second term survives unchanged to the E1 term, and so we obtain an inclusion     H 1 Y; f OX .nDn / ,! H 1 X; OX .nDn / :   It is H 1 Y; f OX .nDn / that we will estimate from below. By [35, Proposition 4.1.6] on the properties of cyclic coverings, one has f OX D OY ˚ OY .R/; which implies f .OX .nDn // D OY .nAn / ˚ OY .nAn  R/ via the projection formula. It follows that       H 1 Y; f .OX .nDn // D H 1 Y; OY .nAn / ˚ H 1 Y; OY .nAn  R/ : We can determine the second term of the sum from the Riemann–Roch theorem. On the abelian surface Y D E  E one has .OY / D 0 and KY D OY , hence Riemann–Roch has the particularly simple form .OY .nAn  R// D

1 .nAn  R/2 : 2

We compute .nAn  R/2 D n2 A2n  2n.An  R/ C R2 D 2n2  2n.n2  2n C 3/ C 2 D 2n3 C 6n2  6n C 2 D 2.n  1/3 < 0; therefore neither nAn  R nor its negative is effective, resulting in   H 0 Y; OY .nAn  R/ D 0

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and

    H 2 Y; OY .nAn  R/ D H 0 Y; OY .R  nAn / D 0:

Hence we can conclude that     h1 X; OX .nDn / > h1 Y; OY .nAn  R/ D n3  2n2 C 3n  1 > 0: Corollary 3.1.2. With notation as above, for an arbitrary positive integer c there exists a prime divisor C on X such that     h1 X; OX .C / > c  h0 X; OX .C / : Proof. This is in fact a corollary of the proof of the Proposition 3.1.1. For n > 2 the linear system jnAn j is globally generated by the Lefschetz theorem [34, Theorem 4.5.1] and it is not composed with a pencil. The same holds true for jnDn j, therefore the base-point free Bertini theorem implies that the general element of jnDn j is reduced and irreducible. Let Cn be such an element. Then     1 h0 X; OX .Cn / D h0 Y; OY .nAn / D ..nAn /2 / D n2 2 by the Riemann–Roch theorem. On the other hand,   h1 X; OX .Cn / > n3  3n2 C 3n  1 > c  n2 for large enough n. The surface X studied above is of general type. It still could be true that Conjecture 2.5.3 holds when restricted to rational surfaces, in any characteristic. For some evidence in this direction we now prove a particularly strong form of the conjecture in the case of smooth projective rational surfaces with an effective anticanonical divisor, and hence in particular for smooth projective toric surfaces. Proposition 3.1.3. Let X be a smooth projective rational surface having an effective anticanonical divisor D 2 j  KX j. Then there exists a constant cX such that for every prime divisor C  X we have h1 .X; C / 6 cX . In fact, the maximum value of h1 .X; C / is either 0 or 1, or it occurs when C is a component of D. Proof. Let C be a prime divisor on X . Clearly, we can assume that C is not a component of D. Thus KX  C D D  C > 0, and if KX  C D 0, then C is disjoint from D. Suppose KX  C > 0. If C 2 > 0, then h1 .X; C / D 0 by [22, Theorem III.1(a, b)]. If C 2 < 0, then 0 > C 2 D 2pa .C /  2  KX  C > 2pa .C /  2 by adjunction, hence pa .C / D 0 and C is smooth and rational with C 2 D 1. Consider 0 ! OX ! OX .C / ! OC .C / ! 0: 2

1

.?/

Since X is rational we have h .X; OX / D h .X; OX / D 0. Since C is smooth and rational with C 2 D 1, we have h1 .C; OC .C // D 0. Thus h1 .X; OX .C // D 0. We are left with the case that D  C D KX  C D 0, hence C is disjoint from D. Thus OC .C / D KC by adjunction, so h1 .C; OC .C // D 1, and therefore from .?/ we have h1 .X; C / D 1.

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3.2 Speciality on blow-ups of P 2 . In this section we look at the speciality of nef linear systems on blow-ups of P 2 . We try to formulate a statement, which is valid without assuming that the points we blow up are in general position. The examples presented here suggest that one needs to reformulate the problem stated in Section 2.3. The first example shows that multiples of a special effective nef linear system might no longer be special. Example 3.2.1. This example is based on the existence of very ample but special linear systems. Let p1 ; : : : ; p25 be transversal intersection points of two smooth curves of degree 5. Let f W X ! P 2 be the blow-up of the plane at these 25 points. By [19, p. 796], for 1 6 r 6 2 and any m > 0, the linear system L D .5m C r/H  m.E1 C    C E25 /; where H is the class of the line and Ei are the exceptional divisors of f , is very ample and special, but Serre vanishing implies that some multiple of L is no longer special. Let X 0 ! X be the blow-up of X at an additional point, and let L0 be the pullback to X 0 of L. Then we have an example of a linear system L0 which is special and nef but not ample and for which sL0 is non-special for s  0. The second example shows that speciality might persist. Example 3.2.2. Let C and D be smooth plane curves of degree d > 3 intersecting transversally in p1 ; : : : ; pd 2 . Let f W X ! P 2 be the blow-up of the plane at these points. The linear system d2 X L D dH  Ei iD1

is special (again because its virtual dimension is negative). The same remains true for all multiples of L. This follows from the restriction sequence 0 ! mL ! .m C 1/L ! .m C 1/LjC ! 0: Indeed, by Serre duality we have h2 .mL/ D 0 for all m > 1. Also, as L is a pencil of disjoint curves, we have LjC D OC . Then taking the long cohomology sequence of the restriction sequence we have the mapping    ! H 1 .X; .m C 1/L/ ! H 1 .C; OC / ! 0: The assumption d > 3 guarantees that C is non-rational, hence h1 .C; OC / D g.C / > 0. The next example shows that speciality may increase linearly while the number of global sections remains fixed. Example 3.2.3. Let C  P 2 be a quartic curve with a simple node at p0 and smooth otherwise. Such a curve exists by a simple dimension count. Then we can take

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12 points p1 ; : : : ; p12 on C in such a way that C is the unique quartic passing through p0 with multiplicity 2 and through p1 ; : : : ; p12 with multiplicities all equal to 1. Let f W X ! P 2 be the blow-up of the plane at the points p0 ; : : : ; p12 . We consider the linear system L D 4H  2E0  E1      E12 on X . By a slight abuse of notation we write C also for the proper transform of C on X . It is a smooth curve of genus 2. If the points p1 ; : : : ; p12 are generic enough, then mLjC has no global sections for all m > 1 and by Riemann–Roch on C we have h1 .C; mLjC / D 1. Using again the restriction sequence we get h0 .X; mL/ D 1 and h1 .X; mL/ D m for all m > 1: These examples suggest the following problem. Problem 3.2.4. Let X be the blow-up of P 2 at r distinct points (r is arbitrary and also the position of the points is arbitrary but we require the points to be distinct) and let L be an effective nef divisor on X . Then is it true that either a) there exists m > 1 such that h1 .mL/ D 0 (and this m should be 1 if the points are in general position); or b) there is a (not necessarily irreducible) curve C on X such that jL  C j ¤ ¿, pa .C / > 0 and L  C D 0? 3.3 Bounded negativity. In the course of the discussions, we investigated the following problem, see [21, Conjecture 1.2.1]. Conjecture 3.3.1 (Bounded negativity). Let X be a smooth projective surface in characteristic zero. There exists a positive constant b.X / bounding the self-intersection of reduced, irreducible curves on X , i.e., C 2 > b.X / for every reduced, irreducible curve C  X . The restriction to characteristic zero is in general essential (see Example 3.3.3) and of course so is the hypothesis that the curves be reduced, but it is not necessarily essential that they be irreducible. See Section 3.8 for further discussion. One situation where bounded negativity holds is when the anti-canonical divisor is Q-effective (see [21, I.2.3]): Proposition 3.3.2. Let X be a smooth projective surface such that for some integer m > 0 the pluri-anti-canonical divisor mKX is effective. Then there exists a positive constant b.X/ such that C 2 > b.X / for every irreducible curve C on X . Proof. As mKX is effective, there exist only finitely many irreducible curves C such that KX  C < 0. Hence apart from these finitely many prime divisors, we have KX  C > 0, in which case by the adjunction formula C 2 D 2pa  2  KX  C > 2pa  2 > 2:

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Note that the hypothesis of Q-effectivity holds for instance on toric surfaces. Another case where the conjecture is clearly true is when KX is nef (with the same argument as that in the proof of Proposition 3.3.2). The following example (see also [21, Remark 1.2.2]) shows that the restriction on the characteristic in Conjecture 3.3.1 cannot be avoided in general (but perhaps it can be avoided by restricting X to be, for example, rational). Example 3.3.3 (Exercise V.1.10, [25]). Let C be a smooth curve of genus g > 2 defined over a field of characteristic p > 0 and let X be the product surface X D C  C . The graph q of the Frobenius morphism defined by taking q D p r -th powers is a smooth curve of genus g and self-intersection q2 D q.2  2g/. With r going to infinity, we obtain a sequence of smooth curves of fixed genus with self-intersection going to minus infinity. In view of this example it is interesting to ask if at least one of the following is true. Conjecture 3.3.4 (Weak bounded negativity). Let X be a smooth projective surface in characteristic zero and let g > 0 be an integer. There exists a positive constant b.X; g/ bounding the self-intersection of curves of geometric genus g on X , i.e., C 2 > b.X; g/ for every reduced, irreducible curve C  X of geometric genus g (i.e., the genus of the normalization of C ). Conjecture 3.3.5 (Very weak bounded negativity). Let X be a smooth projective surface in characteristic zero and let g > 0 be an integer. There exists a positive constant bs .X; g/ bounding the self-intersection of smooth curves of geometric genus g on X, i.e., C 2 > bs .X; g/ for every irreducible smooth curve C  X of geometric genus g. It would be interesting to know if a bound of this type extends to a family of surfaces, i.e., if the following question has an affirmative answer. Problem 3.3.6. Let f W Y ! B be a morphism from a smooth projective threefold Y to a smooth curve B such that the general fibre is a smooth surface. Is there a constant b.Y; g/ such that C 2 > b.Y; g/ for all vertical curves C  Y (i.e., f .C / D a point) of geometric genus g? (Here the self-intersection is computed in the fibre of f containing C .) We show that to a large extent both Conjectures 3.3.4 and 3.3.5 are true. The key ingredient of the proofs is the following vanishing result, [14, Corollary 6.9]. We recall the basic properties of log differential forms in Appendix 3.12.

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Theorem 3.3.7 (Bogomolov–Sommese Vanishing). Let X be a smooth projective variety defined over an algebraically closed field of characteristic zero, L a line bundle, C a normal crossing divisor on X . Then a .log C / ˝ L1 / D 0 H 0 .X; X

for all a < .X; L/. Corollary 3.3.8. Let X be a smooth projective surface defined over an algebraically 1 closed field of characteristic zero, C a normal crossing divisor on X . Then X .log C / contains no big line bundles. 1 Proof. Note that L being big means .X; L/ D 2. An inclusion L ,! X .log C / gives 1 0 1 rise to a non-trivial section of H .X; X .log C / ˝ L /, which vanishes according to Theorem 3.3.7.

Remark 3.3.9. This kind of result was first observed by Bogomolov for the cotangent bundle of a surface itself: 1 is a line bundle, then L is not big. If L  X

(2)

We refer to [54, Proposition 2.2] for a nice and detailed proof. Remark 3.3.10. Theorem 3.3.7 and statement (2) are known to be false in positive characteristic, see [31, Remark 7.1]. 3.4 Partial proof of Conjecture 3.3.5: Very weak bounded negativity. Here we prove Conjecture 3.3.5 on surfaces with Kodaira dimension .X / > 0. Let X be a surface as in Theorem 3.3.7 and let F be a coherent sheaf of rank r on X. Recall that the discriminant .F / is defined as .F / WD 2rc2 .F /  .r  1/c1 .F /2 : If F is a rank 2 vector bundle, then this reduces to .F / D 4c2 .F /  c1 .F /2 : The interest in the discriminant of a vector bundle stems in part from the following useful numerical criterion for stability of vector bundles on surfaces. Recall first Definition 3.4.1. Let F be a rank 2 vector bundle on a smooth projective surface X . We call F unstable if there exist line bundles A and B on X and a finite subscheme Z  X (possibly empty) such that the sequence 0 ! A ! F ! B ˝ Z ! 0 is exact and the Q-divisor

1 P WD A  c1 .F / 2

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satisfies the conditions

P2 > 0

and

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P H >0

for all ample divisors H on X . Remark 3.4.2. The conditions in the definition imply that P is a big divisor. The fundamental result of Bogomolov [5] is the following numerical criterion Theorem 3.4.3 (Bogomolov). Let F be a rank 2 vector bundle on a smooth projective surface X. If .F / < 0, then F is unstable. As a corollary we prove very weak bounded negativity for smooth curves on surfaces with .X/ > 0. Proposition 3.4.4. Let X be a smooth projective surface with .X / > 0 and let C  X be a smooth curve of genus g.C /. Then C 2 > c12 .X /  4c2 .X /  4g.C / C 4: Proof. We consider the following exact sequence which arises via an elementary trans1 formation of X along 1C (see [29, Example 5.2.3]) 1 1 .log C / ˝ C ! X ! 1C ! 0: 0 ! W WD X

By [29, Proposition 5.2.2] we have 1 c1 .W / D .KX C C /  2C D KX  C and c2 .W / D c2 .X / C deg.1C /  KX  C:

Hence

.W / D 4c2 .X /  c12 .X / C 4g  4 C C 2 :

If we assume that C 2 < c12 .X /  4c2 .X /  4g.C / C 4, then .W / < 0 and W is unstable. According to Definition 3.4.1 there exists then a line bundle A  W such that 1 1 A  c1 .W / D A C .C  KX / is big. 2 2 It follows that 1 .log C / A C C  X and since .X/ > 0 and C is effective  1 1 1 1 A C C D A C C  KX C KX C C 2 2 2 2 is big as well. However, this contradicts the Bogomolov–Sommese Vanishing (Theorem 3.3.7).

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Remark 3.4.5. Note that a statement as in the above Proposition cannot hold on ruled surfaces, i.e., there is no lower bound on C 2 with C smooth depending only on the genus of C and Chern numbers of X . Indeed, all Hirzebruch surfaces Fn D P .OP 1 ˚ OP 1 .n// have the same invariants c12 .Fn / D 8, c2 .Fn / D 4 and each Fn contains a smooth rational curve of self-intersection n. This observation is of course not a counterexample either to Conjecture 3.3.1 or to Conjecture 3.3.5. It merely means that the bound for C 2 on ruled surfaces must depend on something else. A reasonable possibility, in the case where X is obtained by blowing up r points of a surface Y when b.Y / exists, is to try to show that b.X / can be defined in terms of b.Y / and r. 3.5 Partial proof of Conjecture 3.3.4: Weak bounded negativity. The main auxiliary ingredient in this part is the logarithmic version of the Miyaoka–Yau inequality. Theorem 3.5.1 (Logarithmic Miyaoka–Yau inequality). Let X be a smooth projective surface and C a smooth curve on X such that the adjoint line bundle KX C C is Q-effective, i.e., there is an integer m > 0 such that h0 .m.KX C C // > 0. Then 1 1 .log C // 6 3c2 .X .log C //; c12 .X

equivalently .KX C C /2 6 3 .c2 .X /  2 C 2g.C //. We refer to Appendix 3.12 for the proof. Note that our statement of this inequality is not the most general, but it suffices for our needs. We also need the following elementary lemma. Lemma 3.5.2. Let X be a smooth projective surface, C  X a reduced, irreducible curve of geometric genus g.C /, P 2 C a point with multP C > 2. Let W Xz ! X be the blow-up of X at P with the exceptional divisor E. Let Cz D  .C /  mE be the proper transform of C . Then the inequality Cz 2 > c12 .Xz /  3c2 .Xz / C 2  2g.Cz / implies

C 2 > c12 .X /  3c2 .X / C 2  2g.C /:

Proof. We have C 2 D Cz 2 C m2 ; c12 .X / D c12 .Xz / C 1; c2 .X / D c2 .Xz /  1 and g.C / D g.Cz /: Hence C 2 D m2 C Cz 2 > m2 C c12 .Xz /  3c2 .Xz / C 2  2g.Cz / D m2 C c12 .X /  1  3c2 .X /  3 C 2  2g.C / > c12 .X /  3c2 .X / C 2  2g.C /:

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Proposition 3.5.3. Let X be a smooth projective surface with .X / > 0. Then for every reduced, irreducible curve C  X of geometric genus g.C / we have C 2 > c12 .X /  3c2 .X / C 2  2g.C /:

(3)

Proof. The idea is to reduce the statement to a smooth curve and use Theorem 3.5.1. We blow up f W Xz D XN ! XN 1 !    ! X0 D X resolving step-by-step the singularities of C . The proper transform of C in Xz is then a smooth irreducible curve Cz . Applying Lemma 3.5.2 recursively to every step in the resolution f we see that it is enough to prove inequality (3) for C smooth. This follows easily from the logarithmic Miyaoka–Yau inequality 3.5.1. Note that our assumption .X / > 0 implies that KXz C Cz is Q-effective. Hence 1 1 .log C // 6 3c2 .X .log C // c12 .X/ C 2C  .KX C C /  C 2 D c12 .X D 3c2 .X /  6 C 6g.C /:

By adjunction C  .KX C C / D 2g.C /  2 and rearranging terms we arrive at (3). Remark 3.5.4. Note that Proposition 3.5.3 applies in particular to smooth curves and provides in general a better bound than that in Proposition 3.4.4. We do not pursue the optimality problem here, and we found it instructive to provide two possible proofs for the Very Weak Bounded Negativity Conjecture. 3.6 Bounded Negativity Conjecture and Seshadri constants. We next point out an interesting connection between bounded negativity and a question on Seshadri constants posed by Demailly in [10, Question 6.9]: Problem 3.6.1. Is the global Seshadri constant ".X / WD inf f".L/ L 2 Pic.X / ampleg positive for every smooth projective surface X ? At present, this is unknown. In fact, it is unknown whether for every fixed x 2 X the quantity ".X; x/ D inf f".L; x/ L 2 Pic.X / ampleg is always positive. The latter, however, would be a consequence of the Bounded Negativity Conjecture: Proposition 3.6.2. If the Bounded Negativity Conjecture is true, then ".X; x/ > 0 for every smooth projective surface X in characteristic zero, and every x 2 X .

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The proof below actually gives an effective lower bound on ".X; x/: If Y D Blx .X / is the blow-up of X at x, then 1 ".X; x/ > p b.Y / C 1 So if we knew that the constant b.Y / is the same for every one-point blow-up of X (or at least bounded from below by a constant that is independent of x), then we would get a lower bound on ".X /. Proof of the proposition. Let C  X be an irreducible curve of multiplicity m at x, and let Cz  Y be its proper transform on the blow-up Y of X in x. Then C 2  m2 D .f  C  mE/2 D Cz 2 > b.Y /: p Consider first the case where m 6 b.Y /. Then LC 1 LC >p >p m b.Y / b.Y / p In the alternative case, where m > b.Y /, we have C 2 > m2  b.Y / > 0 and hence, using the Index Theorem, we get s r p p LC b.Y / b.Y / 1 L2 C 2 : > 1 > > 1 Dp 2 m m m b.Y / C 1 b.Y / C 1 An alternative argument for the proof of the proposition goes as follows. Suppose that ".X; x/ D 0. Then there is a sequence of ample line bundles Ln and curves Cn such that Ln  Cn ! 0; mn where mn denotes the multiplicity of Cn at x. We use now that the blow-up Y D Blx .X / has bounded negativity, so that for the proper transform Czn of Cn we have .Czn /2 > b.Y /2 : But .Czn /2 D .f  Cn  mn E/2 D Cn2  m2n , which, upon using the Index Theorem, tells us that .Ln  Cn /2 : m2n  b.Y / 6 Cn2 6 L2n So we see that  Cn2 .Ln  Cn /2 1 Ln  C n 2 b.Y / 6 6  6 : 1 m2n m2n m2n L2n mn Now, the left hand side of this chain of inequalities tends to one, whereas the right hand side goes to zero, a contradiction.

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3.7 The Weighted Bounded Negativity Conjecture. The following conjecture is yet another variant of the Bounded Negativity Conjecture 3.3.1: Conjecture 3.7.1 (Weighted bounded negativity). Let X be a smooth projective surface in characteristic zero. There exists a positive constant bw .X / such that C 2 > bw .X /.H  C /2 for every irreducible curve C  X and every big and nef line bundle H satisfying H  C > 0. The interest in this conjecture is due to the fact that this conjecture is sufficient to imply the conclusion of the previous proposition, by the following result. Proposition 3.7.2. If the Weighted Bounded Negativity Conjecture is true, then ".X; x/ > 0 for every smooth projective surface X in characteristic zero, and every x 2 X . Proof. As above, there are two proofs, and one of these yields the effective lower bound 1 ".X; x/ > p ; bw .Y / C 1 where Y D Blx .X / is the blow-up of X at x. We first look at this proof. Let C  X be an irreducible curve of multiplicity m at x and L an ample line bundle on X. Let Cz  Y be the proper transform of C on Y . Since f  L is big and nef and f  L  Cz D L  C > 0, the Weighted Bounded Negativity Conjecture on Y yields the existence of a constant bw .Y / such that C 2  m2 D .f  C  mE/2 D .Cz /2 > bw .Y /.f  L  Cz /2 D bw .Y /.L  C /2 : Then, using the Index Theorem, we obtain  L2 C 2 .L  C /2  .L  C /2 2 1  b ; > > L .Y / w m2 m2 m2 that is,

 L  C 2 

 1 C bw .Y /L2 > L2 :

m This yields LC > m

s

as asserted.

L2 D 1 C bw .Y /L2

s 1 L2

1 > C bw .Y /

s

1 1 ; Dp 1 C bw .Y / 1 C bw .Y /

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The second version of the proof goes as follows: Suppose that ".X; x/ D 0. Then there is a sequence of ample line bundles Ln and curves Cn such that Ln  C n !0 mn where mn denotes the multiplicity of Cn at x. Let f W Y D Blx .X / ! X be the blow-up of X at x and denote by Czn the proper transform of Cn . Since f  Ln is big and nef and f  Ln  Czn D Ln  Cn > 0, the Weighted Bounded Negativity Conjecture on Y yields the existence of a constant bw .Y / such that .Czn /2 > bw .Y /.f  Ln  Czn /2 D bw .Y /.Ln  Cn /2 : But .Czn /2 D .f  Cn  mn E/2 D Cn2  m2n , which yields .Ln  Cn /2 Cn2  1 > b .Y / : w m2n m2n Combining with the Index Theorem, we obtain  L  C 2 C 2  L  C  2 1 n n n n 6 n2 6  2: mn mn mn Ln

1  bw .Y /

But the left hand side of this chain of inequalities tends to one, whereas the right hand side tends to zero, a contradiction. Note that the second proof does not give an effective lower bound for ".X; x/. 3.8 Bounded negativity for reducible curves. We now consider a reducible version of Conjecture 3.3.1: Conjecture 3.8.1 (Reducible bounded negativity). Let X be a smooth projective surface in characteristic zero. Then there exists a positive constant b 0 .X / bounding the self-intersection of reduced curves on X , i.e., C 2 > b 0 .X / for every reduced (but not necessarily irreducible) curve C  X . Conjecture 3.3.1 implies Conjecture 3.8.1 in a very explicit way. Proposition 3.8.2. Let X be a smooth projective surface (in any characteristic) for which there is a constant b.X / such that C 2 > b.X / for every reduced, irreducible curve C  X. Then C 2 > .X /b.X /db.X /=2e for every reduced curve C  X , where .X/ is the Picard number of X (i.e., the rank of the Néron–Severi group NS.X / of X ).

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Proof. Let C D C1 C    C CrP be a sum of distinct curves Ci on X . By reindexing we may write this sum as C D ij Cij , where for each i , the sets fCij gj are linearly independent in the Néron–Severi group, and for i < i 0 , the span of fCij gj contains the span of fCi 0 j gj . P For each i, let Bi D j Cij and let ˇ be the number of elements Bi . Then, P Bi2 > j Cij2 > .X /b.X /, where the first inequality is because all of the cross terms are non-negative, and the second because Cij2 > b.X / but there can be at most .X/ linearly independent elements in the Néron–Severi group. Also note for any distinct prime divisors A1 ; : : : ; An ; A such that the Ai are linearly independent in NS.X / and A2 < 0 with A in the span (in NS.X /) of the Ai , we have A  Ai > 0P for some i , P and hence .A1 C    C An /  A > 0. This is because we can write A C s as As D t a t A t for non-negative rational coefficients as and a t , with the sums over s and t running over disjoint subsets of f1; : : : ; ng. Since A ¤ Ai for all i , we have A  Ai P > 0 for all i , but if PA  Ai D 0 for all i , then we would have 2 0 > A D A  .A C s as As / D A  . t a t A t / D 0. In particular this shows for i < i 0 that Bi  Ci 0 j > 0. Let u D min.ˇ; db.X /=2e/. We will now show that C 2 > .B1 C    C Bu /2 . The result follows from this, since .B1 C    C Bu /2 > B12 C    C Bu2 > u .X /b.X / > .X/b.X/db.X /=2e. If u D ˇ, then C D B1 C    C Bu , and clearly C 2 > .B1 C    C Bu /2 . Otherwise, C D B1 C    C Bu C D1 C    C D t , where D1 ; : : : ; D t are the terms of the sum C P D C1 C  CCr not alreadyP subsumed by B1 C  CBu . But C 2 > .B1 C  CBu /2 C i 2.B1 C  CBu /Di C i Di2 , and P Bi Dj > 0 for each 0 j > 0), so  C i and P j (by our observation above regarding B i i i 2.B1 C    C Bu /  P P Di C i Di2 > i .2u C Di2 / > i .2u  b.X // > 0. Thus C 2 > .B1 C    C Bu /2 , as claimed. It seems unlikely that the bound given in Proposition 3.8.2 is ever sharp. For a blowup X of P 2 at n generic points, we expect by the SHGH Conjecture that C 2 > 1 for any reduced, irreducible curve C  X . The bound given in Proposition 3.8.2 would then be C 2 > .n C 1/ for reduced but possibly reducible curves C . It is easy to produce reduced examples with C 2 D n; for example, take C D E1 C    C En . On the other hand, one cannot reach C 2 D .n C 1/. Indeed, the only possibility would be to have n C 1 disjoint .1/ curves. As the Picard number of the blow-up is .n C 1/, this would contradict the Index Theorem. Thus it is of interest to give specific examples of reduced curves with C 2 as negative as possible. Clearly the negativity of C 2 can grow with .X /, so it makes sense to normalize C 2 . We will consider several examples involving blow-ups of P 2 at n points, with the goal of finding values of C 2 =n that are as negative as possible. Example 3.8.3. Let X be the blow-up of P 2 at n collinear points, and let C be the proper transform of the line containing the points. Then C 2 =n is approximately 1. Here C is a prime divisor.

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Example 3.8.4. Consider a general map f W P 1 ! P 2 of degree d . Let C  X be the proper transform of the image of P 1 (which has degree by  d )where X is obtained 2 blowing up the singular points of f .P 1 / (there are n D d 1 =n is nodes). Thus C 2 approximately 2. Here C is again a prime divisor. Example 3.8.5. Let X be the blow-up of the points of intersection of s > 2 general lines in P 2 ; thus n D 2s . Let C be the proper transform of the union of the lines. Then C 2 D s.s  2/, so C 2 =n is approximately 2. Example 3.8.6. Assume the ground field k is algebraically closed of characteristic p. Let q be some power of p > 0. Blow up the points of P 2 with coordinates in the finite field Fq . Let C be the proper transform of the union of all lines through pairs of the points blown up. There are n D q 2 C q C 1 points, and also q 2 C q C 1 lines, and q C 1 .qC1/2 .q 2 CqC1/ D lines pass through each of the points. Thus C 2 D .q 2 CqC1/2p 2 2 q.q C q C 1/, so C =n D q, which is approximately  n. Remark 3.8.7. This last example raises the question of how negative C 2 can be if C is the proper transform of a union of lines. This makes contact with interesting problems studied by combinatorists. For example, let S  P 2 be a set of n distinct points, and let ƒ D fL1 ; : : : ; Ll g be a set of l distinct lines. Let M.S; ƒ/ be the incidence matrix; its rows correspond to the points, its columns correspond to the lines, and an entry is either 1 or 0 according to whether the corresponding point lies or does not lie on the corresponding line. Let jM j be the sum of the entries of M . If X is obtained by blowing up the pointsP of S, and C D L01 C    C L0l , where L0i is the proper transform of Li , then C 2 > i .L0i /2 D l  jM j, with equality if S contains all of the points where any two of the lines meet.   It is easy to get a coarse bound on C 2 =n. Clearly, l 6 n2 , and jM j 6 ln, so     C 2 =n > n2 .1  n/=n >  n2 . For points with real coefficients and lines defined over the reals, the Szemerédi–Trotter Theorem [51] gives an order of magnitude estimate jM j 6 O..ln/2=3 C l C n/. There also are Szemerédi–Trotter type results over finite fields; for example, see Bourgain–Katz–Tao [6] and Vinh [55]. 3.9 Imposing higher vanishing order at one point. We next study the polynomial interpolation problem on the projective plane. We denote by L.d I m1 ; m2 ; : : : ; mn / the linear system of homogeneous polynomials of degree d passing through n generic points with multiplicities at least m1 ; m2 ; : : : ; mn . We want to study L.d I m1 C 1; m2 ; : : : ; mn /, provided L.d I m1 ; m2 ; : : : ; mn / 6D ¿. Let  W X ! P 2 the blow-up of n generic points; let Ei D  1 P.pi / be the exceptional divisors and H D   L. Consider the divisor D D dH  mi Ei . We can identify L.d I m1 ; m2 ; : : : ; mn / D H 0 .OX .D// and

L.d I m1 C 1; m2 ; : : : ; mn / D H 0 .OX .D  E1 //:

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Thus, the starting point for our problem is to analyze the cohomology of the exact sequence 0 ! OX .D  E1 / ! OX .D/ ! OD .DjE1 / ! 0: As a matter of fact, passing to the long exact cohomology sequence we have 

 H 0 .O.DjE1 // 0 ! H 0 .O.D  E1 // ! H 0 .O.D// ! ! H 1 .O.D  E1 // ! H 1 .O.D// ! 0 The image of  is a linear system on E1 of degree m1 . Example 3.9.1. Consider D D 4H  E1  2E2      2E5 , then 

 H 0 .OP 1 .1// ! H 1 .4I 25 / ! H 1 .4I 1; 24 / : H 0 .4I 25 / ! H 0 .4I 1; 24 / ! k k k k k 1 2 2 1 0 It is known that L.4I 25 / is a special system, since its expected dimension is 1 but there is an element in it: the quartic 2C , where C is the conic passing through the five points. Thus H 1 .4I 25 / D 1, and in this case  does not have maximal rank. Example 3.9.2. As a slight modification of the above example we next consider D D 6H  2E1  3E2      3E5 . Then 

 H 0 .OP 1 .2// ! H 1 .6I 35 / ! H 1 .6I 2; 34 / : H 0 .6I 35 / ! H 0 .6I 2; 34 / ! k k k k k 1 2 3 3 1 This shows that also in this case the rank of  is not maximal. The above examples give hints about the behavior of L.d I m1 C 1; m2 ; : : : ; mn /, given L.d I m1 ; m2 ; : : : ; mn / 6D ¿. We formulate the following: Conjecture 3.9.3. Let X be the blow-up of P 2 in r general points, D an effective  divisor on X and E a .1/-curve. For the restriction map jDj !  jDjE j one has: 1. All base points in the image of  come from a base curve in jDj. 2. If jDj has no fixed component, then  has maximal rank. If points in special position are allowed, then it is not hard to find counterexamples to this statement. However, we would like to consider the analogous question when E is a .n/-curve, with n > 1. We denote by L.d I m1 ; : : : ; mr ; mrC1 ; : : : ; mn / the linear system of curves of degree d with points of multiplicities m1 ; : : : ; mn , the first r of which are collinear, but which are otherwise general. The discussions led to the following problem:

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Problem 3.9.4. Is it true that, if d > m1 C    C mr , then L.d I m1 ; : : : ; mn / 6D ¿ if and only if L.d I m1 ; : : : ; mr ; mrC1 ; : : : ; mn / 6D ¿? This statement seems quite strong. If true, the induction arguments common in approaches to the SHGH conjecture as [26], [7], [15] would be significantly simplified, which might even lead to a proof. However we want to remark that some genericity condition is necessary here as well; in particular it is not possible to allow the existence of many negative curves even if they meet the system non-negatively. For example, if we pick four general lines, and four general points on each of them, there is a quartic, if not a pencil, through the 16 points, which is not the case for general points. It is easy to see that we can generalize this counterexample just considering d > 3 lines and d points on each line. We do not have, at the moment, any counterexample to the statement with only two or three lines. 3.10 Geometrization of Dumnicki’s method [12]. Let D  N 2 be a finite set, such that D D P \ N 2 , where P is a convex polygon with integer vertices. We denote by L.D/ the linear series on the affine plane A2 spanned by monomials in D (we identify a point .k; l/ 2 N 2 with the monomial x k y l ). It can also be viewed as the complete linear series associated to the polarized toric variety .XD ; LD / defined by the polygon D, i.e., L.D/ D P .jLD j/: We also write L.D; m1 ; : : : ; mr /p1 ;:::;pr for the subsystem of L.D/ consisting of polynomials vanishing at the given r smooth points with multiplicities at least m1 ; : : : ; mr . One then defines L.D; m1 ; : : : ; mr / D

min

p1 ;:::;pr 2XD

fL.D; m1 ; : : : ; mr /p1 ;:::;pr g:

Consider a partition D D D1 [ D2 [    [ Dr determined by lines (not through the vertices). Then the following holds: Fact 1. If L.Di ; mi / D ¿, for all i D 1; : : : ; r, then L.D; m1 ; : : : ; mr / D ¿.   Fact 2. Given D  N 2 such that jDj D mC1 , then L.D; m/ ¤ ¿ if and only 2 if there exists a nonzero polynomial F 2 QŒx; y with deg.F / D m  1, such that F .a; b/ D 0 for all .a; b/ 2 D. From Fact 2 (and Bézout’s Theorem), we immediately obtain the following Fact 3. If, for D  N 2 , there exist m horizontal (resp. vertical) lines `1 ; : : : ; `m such that m [ `k ; #.D \ `k / 6 k for k D 1; : : : ; m; D kD1

then L.DI m/ D ¿. As an example, we prove that the divisor 13H  5E1  4E2      4E10 is not semi-effective (see Definition 2.2.1 and Problem 2.2.2). For a fixed n, we consider the set D D f.x; y/ 2 N 2 W x C y 6 13ng  N 2 . We will cut D with nine lines into ten

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subsets D1 ; : : : ; D10 . The equations of lines are given by the following functions (for small " > 0): f1 W x  y C 5n  1 C "; f2 W x  9n  1 C "; f3 W y  9n  1 C "; f4 W x  y  5n  1 C "; f5 W x C y  5n  1 C "; f6 W 3x  y  15n  1 C "; f7 W 3x C y C 3n  1 C "; f8 W .2n C 1/x  2ny  2n2  5n  1 C "; f9 W x C 3y  21n C ":

D3

f3

D5 f5 D9 f7

f9

D7 D10

f8

D8 D1

f6

D6

f4

f1

f2

D4

D2

Figure 1. Subdivision of D.

The sets D1 ; : : : ; D10 are defined inductively by Dj D .D n .D1 [    [ Dj 1 // \ f.x; y/ W fj .x; y/ > 0g;

for j D 1; : : : ; 9;

and D10 D Dn.D1 [  [D9 /. Due to Fact 1, it is enough to show that L.D1 I 5n/ D ¿ and L.Dk I 4n/ D ¿ for k D 2; : : : ; 10. For each Dk we will proceed using Fact 3,

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and this is more or less a straightforward computation. We present it for D1 (for D2 ; : : : ; D5 it is very similar) and D6 (for D7 ; : : : ; D10 it is also very similar). The set D1 is given by equations x  y C 5n  1 C " > 0, x > 0, y > 0 and it is in fact a simplex (triangle) with 5n lattice points along the bottom border line, so the assumptions of Fact 3 are satisfied.

Figure 2. The set D1 for n D 3.

The set D6 is given by 3x  y  15n  1 C " > 0, x C y  13n 6 0, and x  y  5n  1 C " < 0.

Figure 3. The set D6 for n D 3.

Let .x; y/ 2 D6 . Observe that for x > 9n we would have y < 4n, so y > 4n and x  y  5n  1 > 9n  4n  5n  1 > 1, hence x  y  5n  1 > 0, a

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contradiction. For x 6 5n we would have y 6 3x  15n  1 6 1, a contradiction. Hence x 2 Œ5nC1; 9n, so D6 lies on at most 4n vertical lines. From the first and the last defining inequality we easily obtain that for .x; y/ 2 D6 , x 2 Œ5nC1; 7n we must have y 2 Œx 5n; 3x 15n1. But for x D 5nC1 we have exactly two lattice points in the interval Œx5n; 3x15n1 D Œ1; 2. If x increases by one then #Œx5n; 3x15n1 increases by two, so we have at most 2; 4; : : : ; 4n  2; 4n points on 2n vertical lines `2 D fx D 5n C 1g; `4 D fx D 5n C 2g; : : : ; `4n D fx D 7ng. Similarly we show that on lines `4n1 D fx D 7n C 1g; `4n3 D fx D 7n C 2g; : : : ; `1 D fx D 9ng we have at most 4n  1; 4n  3; : : : ; 1 points. Facts 1 and 2 can be understood from the point of view of toric degenerations, and as a result we can get a more geometric proof of non-semi-effectivity of 13L  5E1  4E2      4E10 . The translation of Fact 1 into a toric statement is based on the construction of a projective toric variety from any polytope D (defined as an intersection of half-spaces) in a space M as follows. Let Q be a face of D. Define a cone CQ in the dual space N by CQ D fv 2 N j hv; p  qi > 0 for all p 2 D and q 2 Qg As Q varies over all of the faces of D, one gets a fan of cones FD , which defines a toric variety XD (see [18], Section 1.5). Furthermore, this toric variety comes equipped with an ample line bundle LD (see [18], Section 3.4, page 72). For example, if the polytope is the interval Œ0; d  in one-dimensional space, then the variety is the projective line, and the line bundle has degree d . Similarly, if the polytope is the usual triangle with vertices .0; 0/, .0; d /, .d; 0/ in the plane, then the variety is the projective plane and the line bundle has degree d . The construction works up to a point with unbounded polytopes, too. The one that is useful for us is to take a polytope D and cross it with the positive reals. This gives a new polytope, D 0 , and the associated toric variety is XD  A1 , the product of the original toric variety for D with the affine line. The line bundle is just the pullback of the line bundle on XD . A more interesting example of an unbounded polytope can be used to see a degeneration. Take the set of three line segments: .0; 0/ to .1; 1/; .1; 1/ to .2; 1/; and .2; 1/ to .3; 0/. Take as the (unbounded) polytope all points .x; y/ which lie on or above these line segments: 8 9 if 0 6 x 6 1 then y > xI > ˆ < = ˇ D D .x; y/ ˇ 0 6 x 6 3I if 1 6 x 6 2 then y > 1I : ˆ > : ; if 2 6 x 6 3 then y > x  3 Each of the four vertices of D gives four maximal cones of the fan in the dual space, and these are bounded by the rays .1; 0/ and .1; 1/; .1; 1/ and .0; 1/; .0; 1/ and .1; 1/; .1; 1/ and .1; 0/. The associated toric variety is the blow-up of P 1  A1 at two points of the central fiber, giving a degeneration of P 1 to a chain of three P 1 ’s. As

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for the line bundle, it restricts to degree 3 on the general fiber, and degree one on each component of the special fiber. This then realizes the degeneration of the twisted cubic curve to a chain of three lines. What we see in this example is the 3-Veronese of P 1 , defined by the polytope which is just the interval Œ0; 3, and a ‘subdivision’ of the interval suitably used to create the degeneration. In order to properly construct the degeneration, we had to create the ‘lifting’ of the subdivision to one dimension higher. In general, one may in fact use a similar construction. One needs a similar lifting of the subdivision of the ‘bottom’ faces of an unbounded polytope. The key ingredient is to have that the faces of the unbounded polytope lie exactly above the sub-polytopes. This is the idea of a regular subdivision. S Suppose D is a polytope, and S D Di is a subdivision of D. Suppose that we have a real-valued continuous function F on D, which is linear on each Di . One says that the function F is strictly S-convex if it is a convex function, with the additional property that for any subpolytope Di of S , if Li is the linear function for the subpolytope, extended to the entire space, then F .p/ > Li .p/ for all p in D n Di . A subdivision S of D is called regular if D has a strictly S-convex function. One may consult [57], Chapter 5, for additional detail on these ideas. Now suppose we have a polytope D in Rn , defining a toric variety XD , as above. Suppose we have a subdivision S of D, which is regular; let F be a strictly S -convex function on D. Define the unbounded polytope P .F / in RnC1 by  ˚ P .F / D .x; y/ 2 P  R1 j y > F .x/ This polytope P .F / defines a toric variety Y , which is a suitable blow-up of XD A1 in the central fiber, and exhibits a degeneration of XD to a union of toric varieties defined by the subpolytopes Di . There is a more elementary way of seeing the degeneration, embedded in projective space. Suppose that D is a polygon (with lattice points for vertices of course) whose set of lattice points is fmi g. These lattice points correspond to monomials in the variables in the usual way: the lattice coordinates are the exponents of the monomials. If there are k C 1 monomials, this defines a mapping g W P 2 ! P k by sending a point Œx to the point Œx mi , and XD is the image of this mapping. Now suppose we have this subdivision S , with a strictly S-convex function F as above. If y is a coordinate on A1 , this data defines a new mapping G W P 2  .A1  0/ ! P k by sending a pair .Œx; y/ to the point Œy F .mi / x mi . For a fixed y not equal to zero, this is simply a scaling of the original map g, and so the P 2 fibers are all mapped to XD . We want to know what happens in the limit as y approaches zero. Of course, the values of F may well be either negative, or all positive, and so we cannot take the limit with the above formula.

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To see what happens, consider one of the subpolytopes Di , and the corresponding linear function L D Li . Vary the above formula by multiplying through by y L . This will have the effect that as y approaches zero, a limiting value will be available in the projective space. The simplest example of this construction is to take a polytope D and divide it into two pieces, by a hyperplane. Suppose that this hyperplane does not pass through any of the lattice points. Then one can start with a convex function H that is linear on the two sides of the hyperplane. Form the construction above, and then take the convex hull of the graph of H , using the lattice points. This will produce two ‘primary’ subpolytopes as faces of this convex hull, containing the two subsets of the lattice points, on the two sides of the hyperplane. It will also contain a set of ‘secondary’ subpolytopes that cross the boundary of the hyperplane. The construction then produces a toric degeneration, into two ‘primary’ toric varieties, and a set of ‘secondary’ ones. If we want to use this construction to understand an interpolation problem à la Dumnicki, we put the limits of the fat points on the two ‘primary’ varieties, and none on the secondary ones. If the interpolation problems on the two ‘primary’ varieties give empty linear systems, then the linear system will be empty in the limit, and therefore empty on the general fiber (by semicontinuity). In Dumnicki’s example, with the triangle subdivided into ten subpolytopes, one easily sees that this subdivision can be achieved iteratively, by making a single-hyperplane subdivision, nine times. This should give the result, and provide a ‘toric’ interpretation for the example. The proof of Fact 2 mainly involves an interpolation matrix M.D/. LetS.XD ; LD / be the polarized toric surface defined by the polygon D: Recall that XD D v2Vert.D/ Uv where Uv denotes an affine neighborhood around the fixed point on XD corresponding to v: By placing one vertex of the polytope at 0 one chooses an affine patch around the corresponding fixed point so that M Ahs˛ i; where s˛ D x a y b : H 0 .Uv ; LD jUv / D ˛D.a;b/2D

Let Jm .D/ D Jm .LD / be the mth jet sheaf, i.e., the coherent sheaf defined locally at smooth points p; as Jm .LD /p D H 0 .LD ˝ OXD =mpmC1 /; where mp denotes the maximal ideal at p: The mth evaluation map at a smooth point p is defined as the map k;p W H 0 .XD ; LD / ! Jm .LD /p i Cj

si @ that assigns k;p .si / D .si .p/; : : : ; @x j @x i .p/; : : : /16iCj 6m to a basic section si ; for a choice of local coordinates .x; y/ in a neighborhood Uv around p: The map is in fact independent of the choice of Uv and I m.k;p / D Opm is referred to as the mth osculating space at p: Recall that the dimension of Opm at a generic smooth point, e.g., at the generic point in the torus T Š .C  /2  XD ; is constant and it is called the generic osculation dimension.

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For a smooth point p the dimension is dim.L.D; m/p / D dim.XD ; LD /  dim.Opm1 /: We denote by 1 D .1; 1/ the generic point of the torus. Because dim.O1m / > dim.Opm / for all other smooth points p; dim.L.D; m// D dim.L.D; m/1 / and L.D; m/ D ¿

if and only if

dim O1m1 D dim.XD ; LD /:

Let dim.H 0 .XD ; LD // D jDj D N: The map m1;1 W H 0 .XD ; LD / ! Jm1 .LD /1 is represented by the N  mC1 matrix M.D/; whose rows are indexed by the deriva2 tives, until order m  1, with respect to x and y, of the si evaluated at 1 while the columns are indexed by monomials in D. Thus, for example, the .i; j / entry in M.D/ is the derivative, in row i, of the monomial in column j . 0 1 1 1 ::: 1 B a1 a2 : : : aN C B C b b : : : bN C M.D/ D B 2 B 1 C @a1 b1 a2 b2 : : : aN bN A ::: ::: ::: ::: where D D f.a1 ; b1 /; : : : ; .aN; bN /g:  In the case when jDj D mC1 the matrix is a square matrix and thus one has 2 L.D; m/ D ¿ if and only if det.M.D// 6D 0, from which the existence of a polynomial F 2 QŒx; y with deg.F / D m1, such that F .a; b/ D 0 for all .a; b/ 2 D is derived. It is important to remark here that the proof gives an interpretation of the lattice points given by D as points in A2 . We observed that the set of derivatives used can be substituted by any set E D f@˛x1 @yˇ1 ; : : : ; @˛xN @yˇN g which (interpreting .˛i ; ˇi / as points in N 2 ) is closed under downward and leftward moves; in the literature such sets E are known as staircases (see [15]) and they are used to define monomial ideals I.E/ D .x ˛ y ˇ /.˛;ˇ /62E . In the method above one then has to use a modified jet sheaf where mpmC1 is replaced by the translation I.E/p of I.E/ to p by the action of the torus. Denoting by L.D; E/ the subsystem of L.D/ consisting of sections which belong to the monomial ideal .E/1 supported at the generic point, we obtain a Generalized Fact 2. Given D  N 2 such that jDj D jEj, then L.D; E/ ¤ ¿ if and only if there exists a nonzero polynomial F 2 QŒx; y containing only polynomials of E, such that F .a; b/ D 0 for all .a; b/ 2 D. Such a generalization allows to deal with the Hermite interpolation schemes of tree type considered by Lorentz (see [37], Section 3), which is actually a specialized case of the general monomial interpolation problem suggested as Problem 2.4.2. Note

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that in the original problem the coordinates in which the scheme is monomial can be “deformed” by the immersion of the scheme in the surface, whereas here they are torically fixed. Thus when L.D; E/ D ¿ it follows that L.D; Y / D ¿ for general schemes Y isomorphic to Spec kŒx; y=IE , but not conversely. 3.11 Linear systems connected to hyperplane arrangements. We keep the notation introduced in Section 2.7. Results of Falk–Yuzvinsky [16] show that certain combinatorial objects known as weak .k; m/-multinets give rise to components of R1 .A/, and [45] shows that a weak .k; m/-multinet corresponds to a divisor A on X with h0 .A/  D 2.  If the weak multinet is actually a net, then DA D A C B, with h0 .B/ > km  mC1 : 2 This decomposition then gives rise to determinantal equations (and syzygies) in the Orlik–Terao ideal. In particular, since h0 .A/ D 2, if h0 .B/ D m, then I contains the two by two minors of a matrix of linear forms, which has an Eagon–Northcott resolution, which has a very explicit description. The next example shows that not all linear first syzygies arise from components of R1 .A/. Example 3.11.1. The arrangement below is obtained by deleting a line from the Maclane arrangement (Example 10.7 of [49]). A @ 6 5 A@ A @ @ A @ 4 @ A @ A @ @ @3 A @ 2 A@ @ AA 1 0 Figure 4. The M8 arrangement.

The graded Betti numbers for the Orlik–Terao algebra are given below. total 0 1 2

1 1 – –

8 – 7 1

36 – 1 35

56 – – 56

35 8 – – – 35 8

The entry in position .i; j / is dimC Tor R i .C.A/; C/iCj , so there are seven quadratic generators for I , and a single linear first syzygy. On the other hand, for this arrangement, R1 .A/ consists only of local components. An analysis shows that we may choose a basis for the quadratic component I2 so that the linear first syzygy involves only five of the seven minimal quadratic generators.

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Letting J denote the ideal generated by these five elements, we compute that J has the following Betti diagram. total 0 1 2 3

1 1 – – –

5 – 5 – –

12 – 1 11 –

10 – – 10 –

2 – – 1 1

The corresponding variety is a Cohen–Macaulay surface; intersecting with a generic P 5 yields a smooth, projectively normal curve of genus seven and degree eleven. This curve appears in [48] as a counterexample to several conjectures in algebraic geometry. A sole linear first syzygy cannot arise from a decomposition of DA : such a decomposition would yield at least one additional linear first syzygy. 3.12 Limitations of multiplier ideal approach to bounds for symbolic powers. It is natural to ask whether the strategy of [13], using asymptotic multiplier ideals, can be used to prove Conjecture 2.8.1 or any of the improved versions in Problem 2.8.2, or whether another strategy is needed. A weaker version of 2.8.1 was shown by Takagi– Yoshida using tight closure methods [52], but (as observed in [53]) the same result follows from the asymptotic multiplier ideal approach of [13]. On the other hand it seems doubtful that the same approach can prove Conjecture 2.8.1 in full strength. So, can the asymptotic multiplier ideals approach prove any of the statements in Problem 2.8.2, or at least weaker versions? We will see an example suggesting that the answer to this question is negative. The strategy of [13] is as follows. Let I be an ideal with bight.I / D e. Consider the graded system of ideals I ./ D fI .p/ gp2N , the sequence of symbolic powers of I . For each positive real number t there is an ideal associated to this graded system, called the t th asymptotic multiplier ideal of I ./ , and denoted J.t  I ./ /. These ideals enjoy a number of remarkable properties; see [13] (where they were introduced) or [35]. In particular it is shown in [13] that the following containments hold: I .re/  J.re  I ./ /  J.e  I ./ /r  I r : Here the first and third inclusions follow more or less directly from the definition of multiplier ideals, while the second inclusion follows from the subadditivity theorem of Demailly–Ein–Lazarsfeld [11], [35]. The weak version of 2.8.1 follows from the fact that if J.`  I ./ / D .1/, which holds for sufficiently small `, then I .re`/  J.re  I ./ /. (Conjecture 2.8.1 would follow if ` D e  1.) Note that the second and third containments remain the same; only the first is made tighter. To give a positive answer to Problem 2.8.2 by this approach would require an improvement at the far right-hand side of the above containments. An easy example

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suggests this may not be possible. Let R D CŒx; y; z and let I D .xy; xz; yz/, the ideal defining the union of the three coordinate axes (in particular e D bight.I / D 2). In general multiplier ideals are difficult to compute, and few examples are known; the situation is worse for asymptotic multiplier ideals, where even fewer examples have been computed. However, because this ideal I is a monomial ideal, it is possible to compute the asymptotic multiplier ideals appearing in the above containments. This is carried out in [53] (following [39]) with the result that, for each r, J.2  I ./ /r D I r : That is, it is not possible to improve the third containment while leaving the first and second the same. It is still possible that an improvement might be attainable for some ideals, just not this particular one. However at this point the asymptotic multiplier ideals approach does not seem promising.

Appendix Logarithmic differentials and the Miyaoka–Yau inequality A.1 Basics. Here we provide a very quick overview of logarithmic differentials, in particular, no proofs are given. For more complete discussions the reader should consult [14] or [30], for example. Let X be a smooth projective variety of dimension n over the complex numbers. A reduced divisor D D D1 C    C Dr is called a simple normal crossing divisor if all of its irreducible components are smooth, and every point of the support of D has an open neighborhood U such that D restricted to U looks like an intersection of coordinate hyperplanes. In particular, if D is prime, then it is forced to be smooth. With .X; D/ as above, let x 2 D, and D1 ; : : : ; Ds be the irreducible components of D containing the point x. By definition there exists an open neighborhood x 2 U  X, and a local coordinate system x1 ; : : : ; xn at x on U such that Di \ U D V .xi / for all 1 6 i 6 s. We call such a coordinate system a logarithmic coordinate system at x along D. In what follows, D denotes a simple normal crossing divisor. At first we look at the following heuristic picture. We will look at various interesting subbundles inside the tangent bundle TX of X . In order to give some geometric intuition, we will often switch to the language of differential geometry. With notation as above, TX ˝ D=X  TX is a sub-vector bundle, that can be identified with vector fields on X fixing D pointwise. On the other hand, it is natural to consider the subbundle corresponding to vector fields

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that stabilize the divisor D. This latter is denoted by TX . log D/, and is again a sub-vector bundle of TX . Summarizing, one has a sequence of inclusions of vector bundles TX ˝ D=X  TX . log D/  TX : With the help of a little homological algebra one obtains a dual sequence of inclusions .TX /  .TX . log D//  .TX ˝ D=X / : Observe that 1 ; .TX / ' X

1 .TX ˝ D=X / ' X ˝ OX .D/:

In fact, we can write 1 1 1  X .log D/  X ˝ OX .D/; X 1 .log D/, and the latter is called the sheaf of logarithmic where .TX . log D// ' X differentials with poles along D. The formal definition is as follows.

Definition A.1.1. With notation as above, the sheaf of logarithmic differentials with 1 1 poles along D, X .log D/, is the OX -submodule of X ˝OX OX .D/ determined uniquely by the following properties: 1 1 1. X .log D/jXD ' XD . 1 2. Let U  X be an open subset intersecting D. For an element f 2 .X ˝ OX .D//.U /, 1 .log D/.U / if and only if f 2 X

fx D

s X iD1

gi

n X dxi C hi dxi xi iDsC1

for every (closed) point x 2 D \ U , and every logarithmic coordinate system x1 ; : : : ; xn at x along D, where gi ; hi 2 OX;x . Next, we set

p 1 X .log D/ WD ^p X .log D/:

Remark A.1.2. For a smooth projective variety X of dimension n and a simple normal crossing divisor D on X we have the duality .j .log D// D nj .log D/ ˝ OX .KX  D/: In particular, for a surface X , 1 1 .log C // ' X .log C / ˝ OX .KX  C /; .X

hence also 1 1 .log C // ' Symm X .log C / ˝ OX .m.KX C C //: .Symm X

Here C  X denotes a simple normal crossing curve on the surface X .

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One of the fundamental tools for computing with logarithmic differentials is the collection of short exact sequences p p p1 0 ! X ! X .log D/ ! D !0

for all p > 1, in particular, 1 1 0 ! X ! X .log D/ ! OD ! 0:

(4)

From now on we assume that X is a smooth projective surface and C  X is an 1 irreducible curve. We will compute the Chern classes of the bundle F WD X .log C /. Since rank F D 2, we only care about c1 and c2 . From the Whitney sum formula [17, Theorem 3.2 (e)] and the exact sequence (4) we obtain 1 1 c t .X .log C // D c t .X /  c t .OC /;

and from the short exact sequence 0 ! OX .C / ! OX ! OC ! 0 it follows that c t .OC / D

1 D c t .OX .C //: c t .OX .C //

Hence 1 1 .log C //t C c2 .X .log C //t 2 1 C c1 .X 1 1 2 /t C c2 .X /t /  .1 C c1 .OX .C //t /; D .1 C c1 .X

which gives 1 1 c1 .X .log C // D c1 .OX .C // C c1 .X / D KX C C; 1 1 1 c2 .X .log C // D c2 .X / C c1 .X /  c1 .OX .C // D c2 .X / C .KX  C /:

When it comes to computing various expressions in terms of Chern classes, we can 1 identify c1 .OX .C // with C and c1 .X / with KX . 1 A.2 The Miyaoka–Yau inequality for X .log C /. The purpose of this section is to establish the logarithmic version of the Miyaoka–Yau inequality originally proved by Sakai [44, Theorem 7.6]. Sakai’s argument, as will our exposition, follows closely that of Miyaoka in [38]. Since it is our goal to provide a clear picture, we will present a reasonably self-contained proof. The main result is presented in Theorem A.2.8. In the course of this section X denotes an arbitrary smooth complex projective surface unless otherwise stated. We always work over the complex numbers. Our starting point is the following fundamental result of Bogomolov (see [54, Proposition 2.2] for a nice and detailed proof).

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1 be a line bundle, then L cannot be big. Proposition A.2.1. Let L  X

This result has been generalized to log differential forms (see [14, Corollary 6.9]). Theorem A.2.2 (Bogomolov–Sommese Vanishing). Let X be a smooth complex projective variety, L a line bundle, C a normal crossing divisor on X . Then   a .log C / ˝ L1 D 0 H 0 X; X for all a < .X; L/. A fundamental geometric consequence of Bogomolov–Sommese Vanishing is that sub-line bundles of the sheaf of log differentials cannot have many sections. More precisely, we have the following statement, which has already made an appearance as Corollary 3.3.8. We repeat the proof for the reader’s convenience. Corollary A.2.3. Let X be a smooth projective surface, C a normal crossing divisor 1 .log C / contains no big line bundles. on X. Then X 1 Proof. Note that L is big if and only if .X; D 2. An inclusion  L/  L ,! X .log C / 1 0 1 gives rise to a non-trivial section of H X; X .log C / ˝ L , which vanishes according to Theorem A.2.2.

As a consequence we obtain numerical criteria for line bundles contained in 1 X .log C /, with C a normal crossing divisor on X . 1 .log C / be a line bundle, OX .P / a nef line bundle Corollary A.2.4. Let OX .D/  X on X. Then either .P  D/ 6 0 or .D 2 / 6 0. In particular, if D is effective, then .D 2 / 6 0.

Proof. Assuming .P D/ > 0 we will show .D 2 / 6 0. The linear system jKX mDj D ¿ for large m > 0 because its intersection with the nef divisor P becomes negative for large m: .KX  mD/  P D KX  P  mP  D < 0 for m  0: By Serre duality,     H 2 X; OX .mD/ D H 0 X; OX .KX  mD/ D 0; which implies via Riemann Roch that for all m  0 we have         m > h0 X; OX .mD/ > h0 X; OX .mD/  h1 X; OX .mD/ D .X; OX .mD// D

.D 2 / 2 m C O.m/: 2

The left-most inequality comes from the fact that D is not big by Corollary A.2.3. It follows that .D 2 / must be non-positive.

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Connecting up with the Hodge Index Theorem gives one of the main technical ingredients of the logarithmic Miyaoka–Yau inequality (see also [38, Proposition 1]). Note that the original statement requires det F to be semi-ample but it is enough for the argument to assume det F is nef. This proposition gives a nice criterion for vanishing of the group of global sections for certain vector bundles. 1 Proposition A.2.5. Let F  X .log C / be a locally free sheaf of rank 2 with nef determinant, D an arbitrary divisor on X . If   H 0 X; F ˝ OX .D/ ¤ 0

then .det F  D/ 6 max f0; c2 .F /g : Proof. Consider the projective bundle  W V WD P .F / ! X associated to F . Denoting by H the tautological line bundle on V ,     H 0 V; OV .H    D/ ' H 0 X; F ˝ OX .D/ ¤ 0 by assumption. Let W 2 jH    Dj, and write it as W D W0 C   D 0 where W0  H    .D C D 0 / is a prime divisor and D 0 a suitable effective (possibly trivial) divisor on X. The line bundle det F is nef, therefore .det F  D 0 / > 0: Setting D 00 WD D C D 0 , we note that .det F  D/ 6 .det F  D 00 /; hence it suffices to prove the proposition for D 00 in place of D. An application of the Hodge Index Theorem in the guise of Lemma A.2.6 gives that .det F  D 00 / 6 c2 .F / C .D 002 /: (5)   The divisor W0 gives rise to a non-trivial section s 2 H 0 X; F ˝ OX .D 00 / , which 1 in turn embeds OX .D 00 / into F  X .log C /. Applying Corollary A.2.4 with the nef line bundle det F results in .det F  D 00 / 6 0

or

.D 002 / 6 0;

which, coupled with the inequality (5), finishes the proof. Lemma A.2.6. With notation as above, let W0 2 jH   D 00 j be an irreducible divisor on V . Then .det F  D/ 6 c2 .F / C .D 002 /:

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Proof. This is [38, Lemma 8]. The following theorem, which is [38, Theorem 3], generalizes Proposition A.2.5 to symmetric powers. Its proof is based on the so called ‘branched covering trick’ [1, Theorem I.18.2] and Remark A.1.2. 1 .log C / be a locally free sheaf of rank 2 with nef deterTheorem A.2.7. Let F  X minant bundle. If   H 0 X; Symm F ˝ OX .D/ ¤ 0

for some positive integer m, then .det F  D/ 6 max f0; mc2 .F /g :  Proof. Let s 2 H X; Symm F ˝ OX .D/ be a non-trivial section. By the aforementioned branched covering trick there exists a covering f W Y ! X such that   f  .s/ 2 H 0 Y; Symm f  F ˝ OY .f  D/  0

can be written as

f  .s/ D s1 : : : sm   with si 2 H 0 Y; f  F ˝ OY .Di / . Also note that .det f  F /˝m ' f  .det F /˝m is nef, and that we have canonical injections 1 f  F  f  X .log C /  1Y .log f 1 .C //:

The second containment comes from [30, Proposition 11.2] because f 1 .C / is a normal crossing divisor. Proposition A.2.5 applied to the divisors Di then gives ˚  .det f  F  Di / 6 max c2 .f  F /; 0 P for all 1 6 i 6 m. Summing them up and noting that f  D D i Di we obtain 







.det f F  f D/ D det f F 

m X

 ˚  Di 6 max mc2 .f  F /; 0 :

iD1

If d is the degree of the covering f , then .det f  F  f  D/ D d  .det F  D/ and

c2 .f  F / D d  c2 .F /;

which completes the proof. Theorem A.2.8 (Logarithmic Miyaoka–Yau inequality). Let X be a smooth complex projective surface, C a semi-stable curve on X such that KX C C is big. Then 1 1 c1 .X .log C //2 6 3c2 .X .log C //;

in other words,

.KX C C /2 6 3.e.X /  e.C //:

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1 1 Remark A.2.9. We recall that det X .log C / ' OX .KX C C / and c2 .X .log C // D e.X  C /, where e denotes the topological Euler characteristic (see [44, Proposition 7.1]).

Remark A.2.10. Let us quickly recall the definition of a semi-stable curve used here (following [44]). A reduced 1-cycle C  X is called semi-stable if 1. C is a normal crossing divisor (i.e., has only ordinary nodes as singularities), 2. each smooth rational component of C intersects the other components of C in at least two points. It is important to point out that an irreducible semi-stable curve can have ordinary nodes as singularities. If C were defined to be simple normal crossing, instead of just normal crossing, then C irreducible would imply C smooth. Remark A.2.11. With the notation of Theorem A.2.8, we will consider the following version of minimality (which could reasonably be called log-minimal). A smooth rational curve E  X is called C -exceptional, if .E 2 / D 1;

.C  E/ 6 1:

As outlined on [44, pp. 1–2], one can get rid of C -exceptional curves by blowing them down. Proof of Theorem A.2.8. According to Sakai’s hint [44, Last sentence in the proof of 1 . Theorem 7.6] we will follow Miyaoka’s original proof for X 1 Our plan is to apply Theorem A.2.7 to F D X .log C / itself. Exploiting Remark A.2.11, we may and will assume without loss of generality that X contains no C -exceptional curves. As proven in [44, Theorem 5.8]2 , KX C C is semi-ample pro1 .log C / is vided it is big, hence KX C C is nef. Consequently, the determinant of X nef. We still have to fight for the existence of a non-trivial section of a suitable vector bundle. 1 1 If c1 .X .log C //2 6 2c2 .X .log C //, then we are done. Hence we may assume 1 1 c1 .X .log C //2 > 2c2 .X .log C //;

and set

1 c2 .X .log C // 1 < : 1 2 c1 .X .log C //2 Here we used that ˛ > 0, as shown in [44, Section 7]. Fix a sufficiently small number ı 2 Q>0 . Observe that for sufficiently divisible m > 0 we have

˛ WD

1 .log C /  .m.˛ C ı/.KX C C /// D m.˛ C ı/..KX C C /2 / .det X 1 D m..˛ C ı/.c1 .X .log C //2 / 1 > m  c2 .X .log C //: 2 Note that it is implicitly assumed in the proof of this result that X is log-minimal, this is why we had to get rid of the C -exceptional curves first.

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By Theorem A.2.7, this means that   1 .log C / ˝ OX .m.˛ C ı/.KX C C // D 0 H 0 X; Symm X for all large m such that m.˛ C ı/ 2 Z. We next compute H 2 of the same vector bundle, with the help of Serre duality:   1 .log C / ˝ OX .m.˛ C ı/.KX C C // H 2 X; Symm X   1 D H 0 X; OX .KX / ˝ .Symm X .log C / ˝ OX .m.˛ C ı/.KX C C ///   1 .log C // ˝ OX .KX C m.˛ C ı/.KX C C // D H 0 X; .Symm X 1 D H 0 .X; .Symm X .log C / ˝ OX .m.KX C C /// ˝ OX .KX C m.˛ C ı/.KX C C ///   1 D H 0 X; Symm X .log C / ˝ OX .m.1  ˛  ı/.KX C C / C KX / ;

where we used Remark A.1.2 for the third equality. Next we apply Theorem A.2.7 to the 1 .log C / and the divisor D D m.1˛ı/.KX CC /KX . rank 2 vector bundle F D X 1 As noted above, the determinant of F D X .log C / is nef and .det F  .m.1  ˛  ı/.KX C C /  KX // D ..KX C C /  .m.1  ˛  ı/.KX C C /  KX // D m.1  ˛  ı/..KX C C /2 /  ..KX C C /  KX / > m  c2 .F / for all m sufficiently large for which m.1  ˛  ı/ 2 Z. Here we have used the observation that 1  ˛  ı > 0; which follows from having ˛ <

1 2

and having ı very small. In fact, we have

1˛ı >˛Cı and therefore Theorem A.2.7 implies that   1 .log C / ˝ OX .m.1  ˛  ı/.KX C C / C KX / D 0: H 0 X; Symm X According to our computations using Serre duality, this is equivalent to   1 .log C / ˝ OX .m.˛ C ı/.KX C C // D 0: H 2 X; Symm X Consequently, 1 .log C / ˝ OX .m.˛ C ı/.KX C C /// 6 0: .X; Symm X

On the other hand, the dictionary between the projective bundle 1 .log C // ! X  W V WD P .X

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and X (see [38], for example) gives 1 .X; Symm X .log C / ˝ OX .m.˛ C ı/.KX C C /// D .V; OV .m.H  .˛ C ı/  .KX C C ////;

where the right hand side grows with m asymptotically as 1 .H  .˛ C ı/  .KX C C ///3 m3 6 by the asymptotic Riemann–Roch Theorem. As a result, we obtain .H  .˛ C ı/  .KX C C ///3 6 0: By letting ı ! 0, we arrive at .H  ˛  .KX C C //3 6 0: To evaluate this, we recall the various intersection numbers on V (see [38, Lemma 5]): .H 3 / D c1 .F /2  c2 .F /; .H 2    D/ D .det F  D/; .H    D    D 0 / D .D  D 0 /: Therefore, .H  ˛  .KX C C //3 D .H 3 /  3.H 2  ˛.  .KX C C /// C 3.H  .˛  .KX C C //2 /  ..˛  .KX C C //3 / 1 1 D .c1 .X .log C //2  c2 .X .log C /// 1  3˛.c1 .X .log C //  .KX C C //

C 3˛ 2 ..KX C C /2 / C 0: 1 1 1 Since c1 .X .log C // D OX .KX C C / and c2 .X .log C // D ˛c1 .X .log C //2 (this latter by the definition of ˛), after collecting terms we get 1 0 > .H  ˛  .KX C C //3 D .1  4˛ C 3˛ 2 /c1 .X .log C //2 1 D .1  ˛/.1  3˛/c1 .X .log C //2 : 1 .log C // > 0 (since KX C C is big and nef), we conclude As 0 6 ˛ < 12 and c1 .X that 3˛ > 1, which is what we wanted to show.

Acknowledgement. We would like to thank the Mathematisches Forschungsinstitut Oberwolfach for providing a perfect venue for our workshop. We also thank Lawrence Ein, Laurent Evain, Jun-Muk Hwang and Roberto Muñoz, whose contributions to these notes through their active participation during the workshop we gratefully acknowledge, and Nick Shepherd-Barron for his comments regarding the bounded negativity problem. Finally, we are grateful to Stefan Kebekus for suggesting the use of subversion for helping to make possible the preparation of an article with so many coauthors and for his mathematical comments and support for the project.

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Thomas Bauer, Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein-Straße, 35032 Marburg, Germany E-mail: [email protected] Cristiano Bocci, Dipartimento di Scienze Matematiche e Informatiche “R. Magari”, Università di Siena, Pian dei Mantellini 44, 53100 Siena, Italy E-mail: [email protected] Susan Cooper, Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, U.S.A. E-mail: [email protected] Sandra Di Rocco, Department of Mathematics, Royal Institute of Technology (KTH), 10044 Stockholm, Sweden E-mail: [email protected] Marcin Dumnicki, Institute of Mathematics, Jagiellonian University, ul. Lojasiewicza 6, 30-348 Kraków, Poland E-mail: [email protected] Brian Harbourne, Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, U.S.A. E-mail: [email protected] Kelly Jabbusch, Department of Mathematics, Royal Institute of Technology (KTH), 10044 Stockholm, Sweden E-mail: [email protected] Andreas Leopold Knutsen, Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, 5008 Bergen, Norway E-mail: [email protected] Alex Küronya, Budapest University of Technology and Economics, Mathematical Institute, Department of Algebra, Pf. 91, 1521 Budapest, Hungary; current address: Albert-LudwigsUniversität Freiburg, Mathematisches Institut, Eckerstraße 1, 79104 Freiburg, Germany E-mail: [email protected] Rick Miranda, Department of Mathematics, Colorado State University, Fort Collins, CO 80523, U.S.A. E-mail: [email protected] Joaquim Roé, Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain E-mail: [email protected] Hal Schenck, Mathematics Department, University Illinois, Urbana, IL 61801, U.S.A. E-mail: [email protected] Tomasz Szemberg, Instytut Matematyki UP, 30-084 Kraków, Poland; current address: Albert-Ludwigs-Universität Freiburg, Mathematisches Institut, Eckerstraße 1, 79104 Freiburg, Germany E-mail: [email protected]

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Zach Teitler, Department of Mathematics, Boise State University, 1910 University Drive, Boise, ID 83725-1555, U.S.A. E-mail: [email protected]

Moduli of map germs, Thom polynomials and the Green–Griffiths conjecture Gergely Bérczi

1 Introduction Let Jk .n; m/ denote the complex vector space of k-jets of map germs from C n to C m reg mapping the origin to the origin. The open dense subset Jk .n; m/ consists of jets reg with regular linear part. Jk .1; 1/ is a group under composition of jets, and it acts via reparametrisation on Jk .1; n/. The dimension of the complex vector space Jk .1; 1/ is k, and with a natural choice reg of basis Jk .1; 1/ can be identified with the following linear subgroup of GL.k/: 9 80 1 ˛1 ˛2 ˛3 ::: ˛k > ˆ > ˆ > ˆ 2 C > ˆ 0 ˛ 2˛ ˛ : : : 2˛ ˛ C    = ˆ > ˆ A @0 0 0 :::  > ˆ > ˆ ; : k    ::: ˛1 where the polynomial in the .i; j / entry is X pi;j .˛/ N D

˛a1 ˛a2 : : : ˛ai :

a1 Ca2 CCai Dj

This paper is an exploration of this subgroup of GLk and the non-reductive quotient J k .1; n/=Jk .1; 1/, which is roughly speaking the moduli of k-jets of curves in C n . Principles and ideas of classical reductive geometric invariant theory of Mumford do not apply in this situation, for more details about the background see [13], [5]. We illustrate the importance of this moduli space for two classical problems. The first problem goes back to René Thom and his study of degeneracy loci of holomorphic maps between manifolds. Consider a holomorphic map f W N ! M between two complex manifolds, of dimensions n  m. For a singularity class O  Jk .n; m/ we can define the set ZO Œf  D fp 2 N I fp 2 Og; that is, the set of points where the germ fp belongs to O. Then, under some additional technical assumptions, ZO Œf  is an analytic subvariety of N . The computation of the Poincaré dual class ˛O Œf  2 H  .N; Z/ of this subvariety is one of the fundamental problems of global singularity theory. It turns out that these classes – the Thom polynomials of singularities – are certain equivariant intersection numbers on the moduli reg space Jk .1; n/=Jk .1; 1/.

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The second problem is an old conjecture of Green and Griffiths about holomorphic curves in smooth projective varieties. Their conjecture, from 1979, says that any projective variety X of general type contains a proper subvariety Y ¨ X such that any entire holomorphic curve f W C ! X sits in Y , that is, f .C/  Y . The strategy of Green, Griffiths, Demailly and Siu, and the recent work of Diverio, Merker and Rousseau [12] leads us to prove the positivity of an intersection number on the Demailly reg bundle, whose fibers are canonically isomorphic to Jk .1; n/=Jk .1; 1/. This survey paper is an extended version of my IMPANGA lectures given in the Banach Center, Warsaw in January 2011. I would like to thank to Piotr Pragacz for the warm welcome there. Most results presented here have already been published in the papers [3], [4], [5]. The only exception is the formula for the Euler characteristic of Demailly jet bundles in §8 Appendix and the relation to the curvilinear Hilbert scheme in the last section.

2 Equivariant cohomology It is well known that any group action on a topological space carries topological information about the space. Let G be a topological group. A principal G-bundle is a map E ! B, which is locally a projection U  G ! U . One of the main fundamental principles in topology is to find universal objects such that all objects in a given category can be “pulled-back” from this. Here a universal principal G-bundle is a bundle  W EG ! BG such that every principal G-bundle E ! B is a pull-back via a map B ! BG, which is unique up to homotopy. EG is contractible. In fact, if P is a contractible space with a free G-action then P ! P =G is a universal principle G-bundle. Theorem 1. EG exists for all topological group G, and unique up to equivariant homotopy. Example 1. BC  D P 1 .C/, and C 1 ! P 1 .C/ is a universal principle C  bundle. Similarly, BGLn D Hom.C n ; C 1 /=GLn D Gr.n; 1/; and EGL.n/ D Hom.C n ; C 1 / ! Gr.n; 1/ is the universal principle GL.n/-bundle. From this we can construct EGLn GLn C n ! BGLn ; which is a universal vector bundle, namely any vector bundle of rank n can be pulled back from this. The next step is to define equivariant cohomology. Let X be a G-space, i.e. a topological space with a G-action. If the action is free, then G-equivariant cohomology is the ordinary cohomology of the quotient H  .X=G/. For non-free actions the quotient X=G is not well-behaved and H  .X=G/ does not carry enough information. We

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need to “resolve” the action by replacing X with X  EG. This has a free (diagonal) G-action, and define HG .X / D H  .EG G X / Example 2. HG .pt/ D H  .BG/ D CŒhW , where h D Lie T is the Cartan algebra  .pt/ D S W D CŒx1 ; : : : ; xn Sn , the acted on by the Weil group W . For example HGL n algebra of symmetric polynomials. Properties of equivariant cohomology (1) f W X ! Y G-map induces H.f / W HG .Y / ! HG .X /. (2) h W G ! H homomorphism, then EH can serve as EG and we have a projection EH G X ! EH H X which induces H.h/ W HH .X / ! HG .X /. (3) HG .pt/ D H  .BG/ D CŒhW , and HG .X / is a HG .pt/-module. For example  HGL .pt/ D S W D CŒx1 ; : : : ; xn Sn . n Proposition 1 (Induction, Restriction). Let X be a G-space. • Restriction: If H  G then X naturally is a H -space, and there is an induced map HG .pt/ ! HH .pt/, HH .X / D HH .pt/ ˝HG .pt/ HG .X /: • Induction: If G  K then K G X is naturally a K-space, and there is an induced map HK .pt/ ! HG .pt/, HK .K G X / D HG .X /; but as a HK .pt/-module. Example 3. Let G D GLn . We have a left-right action of the upper Borel B  GLn on G B G. We compute HBB .G B G/ in the following steps:  .B/ D H  .B/ D S; HBB

so by induction

 .G B B/ D S 2 S W -mod-S; HGB

therefore by restriction  .G/ D S ˝S W S 2 S -mod-S HBB

and by induction again  .G B G/ D S ˝S W S 2 S W -mod-S HGB

and by restriction  .G B G/ D S ˝S W S ˝S W S 2 S -mod-S: HBB

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2.1 The equivariant de Rham model. Let G be a Lie group with Lie algebra g. For a smooth G-manifold M we can define equivariant differential forms, for more details see [7]. The equivariant differential forms are differential form valued polynomial functions on g: G .M / D f˛ W g ! .M /I ˛.gX / D g˛.X / for g 2 G; X 2 gg D .CŒg ˝ .M //G ; where .g  ˛/.X / D g  .˛.g 1  X //. Here CŒg denotes the algebra of complex values polynomial functions on g. We define an equivariant exterior differential dG on CŒg ˝ .M / by the formula .dG ˛/.X / D .d  .XM //˛.X /; where .XM / denotes the contraction by the vector field XM . This increases the degree of an equivariant form by 1 if the Z-grading is given on CŒg ˝ .M / by deg.P ˝ ˛/ D 2 deg.P / C deg.˛/ for P 2 CŒg; ˛ 2 .M /. The homotopy formula .X /d C d .X / D L.X / implies that dG2 .˛/.X / D L.X /˛.X / D 0 for any ˛ 2 CŒg ˝ .M /, and therefore .dG ; G .M // is a complex. Definition 1. The equivariant cohomology of the G-manifold M is the cohomology of the complex .dG ; G .M //: HG .M / D HdG : Note that ˛ 2 G .M / is equivariantly closed if ˛.X / D ˛.X /0 C    C ˛.X /n such that .XM /˛.X /i D d˛.X /i2 : Here ˛.X /i 2 i .M / is the degree-i part of ˛.X / 2 .M /. In other words, ˛i W g ! i .m/ is a polynomial function. The functoriality properties of equivariant cohomology now come for free: (1) If H ! G is a homomorphism of Lie groups then the restriction map CŒg ! CŒh induces a homomorphism of differential graded algebras G .M / ! H .M / and finally a homomorphism HG .M / ! HH .M /. (2) If  W N ! M is a map of G-manifolds which intertwines the actions of G then pull-back by  induces a homomorphism of differential graded algebras   W G .M / ! G .N / and a homomorphism HG .M / ! HG .N /.

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3 Equivariant localization 3.1 Integrating equivariant forms. If G is a Lie group and M is a G-manifold, we can integrate equivariant forms obtaining a map Z W G .M / ! CŒgG M

by the formula

Z M

 Z ˛ .X / D

M

Z ˛.X / D

M

˛Œn .X /:

That is, if ˛ is an equivariant differential form, then we integrate the top degree part of it, and the result is a polynomial function on g. This is well defined: if ˛ is equivariantly R exact, i.e. ˛ D dG ˇ for some ˇ 2 G .M /, then ˛.X R /n D dˇ.X /n and therefore M ˛.X / D 0. Thus if ˛ is equivariantly closed then M ˛ only depends on the equivariant cohomology class represented by ˛. It can be shown (see Proposition 7.10 in [7]) that if G is a compact Lie group, and M0 .X/ is the zero locus of the vector field R XM , then the form ˛.X /n is exact outside M0 .X/. This suggests that the integral M ˛.X / only depends on the restriction ˛.X /jM0 .X/ . Here we state the localization theorem in the special case when XM has isolated zeros. Theorem 2 (Atiyah–Bott, Berline–Vergne). Let G D T be a complex torus, M a T -manifold, ˛ 2 T .M /. Then Z X ˛0 .p/ ˛ D .2/l : Euler T .Tp M / M T p2M

In other words:

Z M

˛.X / D .2/l

X ˛.X /0 .p/ Q ; i i T

p2M

where i are the weights of the Lie action X W  2 Tp M ! ŒXM .p/;  2 Tp M: Most often we apply localization to compute certain intersection numbers on M . My favorite example illustrating the strength of the localization method is the following. 3.2 How many lines intersect two given lines and go through a point in P 3 ? We think of points, lines and planes in P 3 as 1-, 2-, and 3-dimensional subspaces in C 4 . For R 2 Grass.3; C 4 /, L 2 Grass.1; C 4 / define C2 .R/ D fV 2 Grass.2; 4/ W V  Rg;

C1 .L/ D fV 2 Grass.2; 4/ W L  V g:

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Standard Schubert calculus says that C1 .L/ (resp. C2 .R/) represents the cohomology class c1 . / (resp. c2 . /) where is the tautological rank 2 bundle over Grass.2; 4/. So the answer is Z C1 .L1 / \ C1 .L2 / \ C2 .R/ D c1 . /2 c2 . /: Grass.2;4/

Apply equivariant localization. The sufficient data are the following. • The diagonal torus T 4  GL.4/ acts on C 4 with weights 1 ; 2 ; 3 ; 4 2 t  HT .pt/.   • The induced action on Grass.2; 4/ has 42 fixed points, the coordinate subspaces indexed by .i; j /. 2 , where fs; t g D • The tangent space of Grass.2; 4/ at .i; j / is .C 2 /i;j ˝ Cs;t 2 f1; 2; 3; 4g n fi; j g, and Ci;j 2 Grass.2; 4/ is the subspace spanned by the i; j basis. Therefore, the weights on T.i;j / Grass are s  i ; s  j with s ¤ i; j .

• The weights of are identified with the Chern roots, so ci . / is represented by the i th elementary symmetric polynomial in the weights of . ABBV localization then gives Z X c1 . /2 c2 . / D Gr.2;4/

 2S4 =S2



. 1 C 2 /2 1 2 D 2: . 3  1 /. 4  1 /. 3  2 /. 4  2 /

(1)   On the right-hand side we sum over all 42 fixed points by taking appropriate permutation of the indices. It is not clear at first glance, why this rational expression is an integer. But it turns out that the sum is independent of i ’s and it is 2. 3.3 Iterated residues. We saw in the previous example that the ABBV localization results in a sum of rational expressions, but adding these together is not an obvious task. There is a short and elegant way to do this by identifying the summands as iterated residues of a certain meromorphic differential form on C d for some d , and then by applying the residue theorem saying that the sum of the residues at finite points is equal to minus the residue at infinity. The set-up is the following. • z1 ; : : : ; zd – coordinates on C d . • !1 ; : : : ; !N – affine linear forms on C d ; !i D ai0 C ai1 z1 C    C aid zd . • h.z/ a function h.z1 : : : zd /, and d z D dz1 ^    ^ dzd a holomorphic d -form.

Moduli of map germs, Thom polynomials and the Green–Griffiths conjecture

Definition 2. We define the iterated residue of h.z/ d z def Res : : : Res QN D z1 D1 zd D1 iD1 !i



1 2 i

h.z/ d z QN i D1 !i

at infinity as follows:

d Z

Z jz1 jDR1

147

:::

jzd jDRd

h.z/ d z ; QN iD1 !i

(2)

where 1  R1      Rd . The torus fjzm j D Rm I m D 1 : : : d g is oriented in such a way that Resz1 D1 : : : Reszd D1 d z=.z1 : : : zd / D .1/d . In practice, the iterated residue 2 may be computed using the following algorithm: for each i , use the expansion 1

X .a0 C ai1 z1 C    C aiq.i/1 zq.i/1 /j 1 D .1/j i ; !i .aq.i/ zq.i/ /j C1 j D0

(3)

i

where q.i/ is the largest value of m for which aim ¤ 0, then multiply the product of these expressions with .1/d h.z1 : : : zd /, and then take the coefficient of z11 : : : zd1 in the resulting Laurent series. Example 4. (1) use

1 z1 .z1 z2 /

(2) 4z12 z22

D

1 has two different Laurent expansions, z1 .z1 z2 / i 1 P1 1 i z1 iD0 .1/ z i C1 to get Res1 z1 z2 D 1. 2

1 ReszD1 .z1 z2 /.2z 1 z2 /

D coeff .z1 z2 /1 z12 .1 C 2

z1 z2

C

but on jz1 j  jz2 j we z12 z22

C    /.1 C

2z1 z2

C

C / D 3 Let us turn back to our toy example presented in §3.2. Define the differential form .z2  z1 /2 .z1 C z2 /2 z1 z2 d z ! D Q4 : Q4 iD1 . i  z1 / iD1 . i  z2 /

This is a meromorphic form in z2 on P 1 with poles at z2 D i , 1  i  4 and z2 D 1. The poles at i are non-degenerate and therefore applying the residue theorem we get Res ! D

z2 D1

4 X

. i  z1 /2 . i C z1 /2 i z1 dz1  Q4 Q j D1 . j  z1 / j ¤i . j  i / iD1 „ ƒ‚ … z2 Di

D

4 X iD1

. i  z1 /. i C z1 /2 i z1 dz1 Q Q : j ¤i . j  z1 / j ¤i . j  i /

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Doing the same again with z1 we get Res Res ! D

z1 D1 z2 D1

4 X X iD1 j ¤i

D

4 X X iD1 j ¤i

Q

. i  j /. i C j /2 i j Q k¤i;j . k  j / j ¤i . j  i /

. i C j /2 i j Q Q D k¤i;j . k  j / k¤i;j . k  i /

Z

c1 . /2 c2 . /: Gr.2;4/

On the other hand, using the above algorithm by expanding the rational form ! we get Res Res ! D 2:

z1 D1 z2 D1

We give another example, the so called Giembelli–Thom–Porteous formula in Section 4.2. 3.4 Localization on partial flag manifolds. The following proposition is a farreaching generalization of the idea presented in the previous section, and it provides a meromorphic differential form whose residue at infinity gives back the localization formula for a large class of forms. Let Flagd .n/ D fV1      Vd  C n W dim.Vi / D ig denote the full flags of d -dimensional subspaces of C n . The maximal torus T  GL.n/ acts on Flagd .n/, and the fixed points are parametrized by coordinate flags corresponding to certain permutations  2 .Flagd .n//T . The Chern classes of the tautological rank-d bundle over Flagd .n/ are elementary symmetric polynomials in the weight of T on C n , and the intersection numbers of can be computed as iterated residues according to Proposition 2 ([3]). Let Q.z/ D Q.z1 ; : : : ; zd / be a polynomial on C d of degree dim.Flagd .n//. Then Q X Q. 1     d / 1m
4 Singularities of maps The first problem we address goes back to the 1950s and the work of René Thom. For more details of the history and background of the problem see [1], [3]. The usual set-up for studying singularities of map germs is the following. Set up. We fix integers k  n  m.

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• Let A be a nilpotent algebra, dim A=C D k. We will take Ak D zCŒz=z kC1 . • Define Jk .n; m/ D fp D .p1 ; : : : ; pm / 2 Poly.C n ; C m / W deg pi  k; pi .0/ D 0g: This is the vector space of k-jets of map germs f W .C n ; 0/ ! .C m ; 0/. • Let †A D fp 2 Jk .n; m/ W CŒx1 ; : : : ; xn =hp1 ; : : : ; pm i D Ag be the set of map-germs with local algebra isomorphic to A. reg

• The germs Jk .n; n/ with non-degenerate linear part form a group, and reg reg Jk .n; n/  Jk .m; m/ naturally acts on Jk .n; m/ with .A; B/p D BpA1 : These are the polynomial reparametrizations of map germs. The central problem of global singularity theory is the computing the (co)homology classes of singularity loci of holomorphic maps between complex manifolds. Given a holomorphic map f W N n ! M m define ˚  Z.f / D p 2 N W fyp 2 †A ; where fyp is the germ of f at p 2 N . It was already known by Thom, which is now called the Thom principle, that for generic map f , Z.f / represents a cohomology cycle and there is a well-defined polynomial MDAn!m 2 CŒx1 ; : : : ; xn ; y1 ; : : : ; ym Sn Sm such that ŒZ.f / D MDA .T N; f  .TM // 2 H  .N; C/: Here MDA stands for multidegree, for explanation see the next section. Furthermore, in [18] Haefliger and Kosinski proves that if Qk c.f  .TM // .1 C m q/ D QmD1 c.q/ D c0 C c1 q C c2 q C    D n c.T N / iD1 .1 C i q/ 2

is the Chern classes of the difference bundle then MDA .T N; f  .TM // D TpAk!n .c1 ; c2 ; : : :/: That is, MDA is a polynomial in these difference Chern classes, and TpA is called the Thom polynomial of the algebra A.

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4.1 Multidegrees. The polynomial MDA stands for multidegree, which is also called equivariant Hilbert polynomial or equivariant Poincaré dual in the literature. This is defined for any G-invariant subvarieties of a complex G-vector spaces (i.e. G-representations, where G is a Lie algebra) as follows. Set up (1) V D C N complex vector space, with a G-action. (2) †  V is a G-invariant closed subvariety. (3) HG .V / D HG .pt/ is the G-equivariant cohomology ring of V . Recall that  .pt/ D CŒx1 ; : : : ; xd Sd . HGL.d / We give two definitions of a polynomial mdegŒ†; V  2 HGcodim.†2V / .pt/; called the multidegree of †: one topological and one algebraic definition. Vergne’s integral definition – topology. If †  V is a subvariety then EG G †  EG G V represents a homology cycle, and the multidegree is the ordinary Poincaré dual of the Borel construction EG G †: mdegŒ†; V  D PD.EG G †  EG G V /: By definition mdegŒ†; V  2 H  .EG G V / D HG .pt/ is a polynomial. Theorem 3 ([38]). There is an equivariant Thom class ThomG .V / 2 HGdim V .V / such that for any G-invariant subvariety †  V , Z mdegŒ†; V  D ThomG .V /: †

We give another, more algebraic definition of the multidegree, which also provides an algorithm to compute these polynomials. Sturmfels’ axiomatic definition Theorem 4 ([27]). Let †  V be a G-invariant subset of the G-representation V . Then mdegŒ†; V  is characterized by the following axioms: : S Additivity. If † D †i is the set of maximal irreducible components of †, then mdegŒ†; V  D

c X iD1

mult.†i /  mdegŒ†i ; W :

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Degeneration. The multidegree is constant under flat deformation of †. Normalization. For T -invariant linear subspaces of V , mdegŒ†; V  is defined to be equal to the product of weights in the normal direction. The recipe to compute the multidegree (although this recipe often ends up with difficult commutative algebra computations) is to choose a proper flat deformation of † into the union of coordinate spaces, that is, to deform its ideal into a monomial ideal. For example, choosing a monomial order on the coordinate ring, the initial ideal is monomial, and then normalization and additive properties of the multidegree give you the result. Example 5. .C  /3 acts on C 4 with weights 1 ; : : : ; 4 . Let 1 C 2 D 3 C 4 , and † D Spec.CŒy1 ; y2 ; y3 ; y4 =.y1 y2  y3 y4 //: Define the flat deformation † t D Spec.CŒy1 ; y2 ; y3 ; y4 =.y1 y2  ty3 y4 //: For t D 0, †0 D fy1 y2 D 0g, so normalization and additivity says mdegŒ†; C 4  D 1 C 2 D 3 C 4 : Now we can state Thom’s principle more precisely: Theorem 5 (Thom). Let †A  Jk .n; m/ denote the set of germs with local algebra isomorphic to A. This is a GLn  GLm -invariant subvariety of Jk .n; m/, and MDAn!m D mdegGLn GLm Œ†A ; Jk .n; m/: 4.2 Degeneracy loci of sections via localization. Here is another illustrating example for transformation of localization formulas into iterated residues. Given a rank-n vector bundle on a manifold M , and n generic sections 1 ; : : : ; n , it is an old question in topology to determine the cohomology class dual to the locus where the sections are linearly dependent. This class is the Thom polynomial TpA with A D t CŒt =t 2 , and we have †A D †1 D fA 2 Hom.n; m/I dim ker A D 1g D fA 2 Hom.n; m/I 9ŠŒv 2 P n1 Av D 0g: The goal is to compute mdegŒ†1 ; Hom.n; m/. We have the fibration  W †1 ! P n1 sending a linear map to its kernel. This is equivariant with respect to the GLn GLm action, and GLm acts fiberwise whereas GLn acts on P n1 . According to Vergne’s definition, we want to integrate the equivariant Thom class over †1 . The idea is to integrate first over the base P n1 and then along the fibers, and to apply ABBV localization on P n1 .

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We have n fixed points on P n1 corresponding to the coordinate axes. Let 1 ; : : : ; n denote the weights of T n  GLn on C n . The weights of Tpi P n1 are fs i I s ¤ ig, and the fiber at pi is the set of matrices A with all entries in the i th column Q vanishing. The normalization axiom says that the multidegree of the fiber at pi is jmD1 . j  i /, so Z Thom.C  /nCm .Hom.C n ; C m // mdegŒ†1 ; Hom.n; k/ D †1

Z D

P n1

Z Thom.C  /nCm D fiber

n X iD1

Qm

j D1 . j

 i /

s¤i .s

 i /

Q

:

Consider the rational differential form Qm j D1 . j  z/  Qn dz: iD1 .i  z/ The residues of this form at finite poles: fz D i I i D 1; : : : ; ng exactly recover the terms of the above sum. Applying the residue theorem, and change of variables z D 1=q, we get Qm dq j D1 .1 C q j / n m D cmnC1 ; mdegŒ†1 ; Hom.C ; C / D resqD0 Qn mnC2 q .1 C q / i iD1 where cmnC1 is the .mnC1/-th Chern class of the difference bundle f  .TM /T N . n!m This gives us the Thom polynomial Tp tCŒt=t 2 D cmnC1 . Note that it depends only on m  n.

5 Computing multidegrees of singularities Recall the following notation from the previous section. • Jk .n; m/ D f.p1 ; : : : ; pm / 2 Poly.C n ; C m / W deg pi  k; pi .0/ D 0g is the set of k-jets of map germs. ˚  • †k D p 2 Jk .n; m/ W CŒx1 ; : : : ; xn =hp1 ; : : : ; pm i Š zCŒz=z kC1 is the set of germs with Ak -singularity. reg

reg

• D D Jk .n; n/  Jk .m; m/ naturally acts on Jk .n; m/ with .A; B/p D BpA1 . Note that GLn  GLm  D. The goal now is to compute D mdegGLn GLm Œ†k ; Jk .n; m/; Tpn!m k the Thom polynomial of Morin singularities. The following theorem has first appeared in the work of Porteous and Gaffney, see [16].

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Theorem 6 (The test curve model of Morin singularities).  : ˚ reg †k .n; m/ D ‰ 2 Jk .n; m/ W 9 2 Jk .1; n/ such that ‰ ı  D 0 in Jk .1; m/ : : Here D denotes birational equality, that is, their Zariski closures are equal. reg

Note that if ' 2 Jk .1; 1/ D Gk , then ‰ ı  D 0 H) ‰ ı . ı '/ D 0: It can be shown that for ‰ 2 Jk .n; m/ whose linear part has corank 1, reg

‰ ı 1 D ‰ ı 2 D 0 () 9˛ 2 Jk .1; 1/ such that 1 D 2 ı ˛: Therefore: : Proposition 3. The Zariski open subset †0k D f‰ 2 †k W dim ker ‰ D 1g D †k fibers reg with linear fibres over Jk .1; n/=Gk . What are these fibers, and why are they linear? If  D v1 t C v2 t 2 C    C reg vd t d 2 Jk .1; n/ with vi 2 C n and v1 ¤ 0 and ‰.v/ D Av C Bv 2 C    with A 2 Hom.C n ; C k /, B 2 Hom.Sym2 .C n /; C k /, etc., then ‰ ı  D 0 is equivalent with the following k equations: A.v1 / D 0; A.v2 / C B.v1 ; v1 / D 0; A.v3 / C 2B.v1 ; v2 / C C.v1 ; v1 ; v1 / D 0; :: :

(5)

For fixed  D .1 ; : : : ; k / these are linear equations determining the fiber. According to Proposition 3,  : [˚ reg †k .n; m/ D Sol W  2 Jk .1; n/ ; where

Sol D Ann. / ˝ C m  Jk .n; m/

is the annihilator tensored by C k . To linearize the action of Gk let us make the following identifications. • Identify Jk .1; n/ with Hom.C k ; C n / by putting the coordinates  D .v1 ; : : : ; vk / into the columns of a matrix. Lk i n • Identify Jk .n; 1/ with Symk C n D iD1 Sym C , and then Jk .n; m/ D k n m Sym C ˝ C .

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Then Gk acts group: 80 ˛1 ˆ ˆ ˆB ˆ
on Jk .1; n/ by multiplication on the right by the following matrix ˛2 ˛12 0 0 

˛3 2˛1 ˛2 ˛13 0 

9 1 ::: ˛k > > > > : : : 2˛1 ˛k1 C : : :C = C  2 : : : 3˛1 ˛k2 C : : :C W ˛ 2 C ; ˛ 2 C I i C 1 > > A :::  > > ; ::: ˛1k

where the polynomial in the .i; j / entry is X pi;j .˛/ N D

(6)

˛a1 ˛a2 : : : ˛ai :

a1 Ca2 CCai Dj

This group is the central object of our study in this paper. It is a non-reductive linear subgroup of GLk , and therefore Mumford’s geometric invariant theory does not help reg us in handling the quotient Jk .1; n/=Gk . The following construction, which was the starting point of a general construction in [6] first appeared in [3]. Define the map  W Hom.C k ; C n / ! Hom.C k ; Symk C n /; 

.v1 ; : : : ; vk / D v1 ; v2 C v12 ; : : : ;

P a1 Ca2 CCai Dk

(7) 

va1 va2 : : : vai ;

where in the j th coordinate we sum over all ordered partitions of j into positive integers. Note that these correspond to the monomials in j th column of the matrix Gk . For more details see [6]. Theorem 7 ([3]). Let Hom0 .C k ; C n / D f.v1 ; : : : ; vk 2 Hom.C k ; C n / W v1 ¤ 0g D reg Jk .1; n/. Then  (defined in (7)) descends to an injective map on the orbits Grass W Hom0 .C k ; C n /=Gk ,! Grass.k; Symk C n /; and therefore also descends to Flag W Hom0 .C k ; C n /=Gk ,! Flagk .Symk C n /: Composing with the Plücker embedding we get Proj D Pluck ı Grass W Hom0 .C k ; C n /=Gk ,! P .^k .Symk C n //: Note that  is GLn -equivariant with respect to the multiplication on the left on Hom0 .C k ; C n /=Gk and the induced action on Grass.n; Symk C n / coming from the standard action on C n . This embedding allows us to give a geometric description of some generators in the invariant ring reg CŒJk .1; n/Uk  CŒf 0 ; : : : ; f .k/ ;

Moduli of map germs, Thom polynomials and the Green–Griffiths conjecture

155

where Uk  Gk is the maximal unipotent subgroup. Namely, the coordinate ring of the image is a subring of the invariant ring: CŒim./  CŒJk .1; n/Uk : reg

CŒJk .1; n/Uk has been long studied in relation with the Green–Griffiths conjecture. In his seminal paper [10], Demailly suggested a strategy to the Green– Griffiths conjecture through the investigation of the invariant jet differentials. These are sections of a bundle, whose fibers are canonically isomorphic to the invariant ring reg CŒJk .1; n/Uk . It is a major unsolved problem to prove the finite generation of this ring, and to find the generators. The main obstacle is that Gk is a non-reductive group, and therefore classical Geometric Invariant Theory ([28]) does not apply. For an introduction on non-reductive group actions see [13]. reg Following Demailly’s notation, let .f 0 ; f 00 ; : : : ; f .k/ / 2 Jk .1; n/ denote the k-jet of a germ f and fi.j / denote the i th coordinate of the j th derivative, 1  i  n; 1  j  k. This is a simple rescaling, namely vi D f .i/ = i Š. reg

Theorem 8 ([5]). Let  dim.Symk .n//  I D i1 ;:::;ik .f / W .i1 ; : : : ; ik / 2 C CŒf 0 ; : : : ; f .k/  k 



be the ideal generated by the k k minors of .f 0 ; : : : ; f .k/ / 2 Hom.C k ; Symk C n /. Then reg I  CŒJk .1; n/Uk Example 6. n D 2; k D 4. In this case J4 .1; 2/ D f.f10 ; f20 ; f100 ; f200 ; f1000 ; f2000 ; f10000 ; f20000 / 2 .C 2 /4 I .f10 ; f20 / ¤ .0; 0/g; reg

and fixing a basis fe1 ; e2 g of C 2 and fe1 ; e2 ; e12 ; e1 e2 ; e22 ; e13 ; : : : ; e1 e24 ; e24 g of Sym4 C 2 the map  W J4 .1; 2/ ! Hom.C 4 ; Sym4 C 2 / sends .f10 ; f20 ; f100 ; f200 ; f1000 ; f2000 ; f10000 ; f20000 / to a 4  15 matrix, whose first five columns (corresponding to Sym2 C 2 ) are 0 B B @

f10 1 00 2Š f1 1 000 3Š f1 1 0000 4Š f1

f20 1 00 2Š f2 1 000 3Š f2 1 0000 4Š f2

0 .f10 /2 f10 f100 1 2 0 000 00 2 3Š f1 f1 C 2Š2Š .f1 /

0 f10 f20 .f10 f200 C f100 f20 / 1 2 0 000 000 0 00 00 3Š .f1 f2 C f1 f2 / C 2Š f1 f2

1 0 C .f20 /2 C; A f20 f200 1 2 0 000 00 2 3Š f2 f2 C 2Š2Š .f2 /

and the next four columns (corresponding to Sym3 C 2 ) are 0 B @

0 0 .f10 /3 3 0 2 00 2Š ..f1 / f1 /

0 0 0 2 0 .f1 / f2 3 0 2 00 0 0 00 2Š ..f1 / f2 C 2f1 f2 f1 /

0 0 0 f1 .f20 /2 3 0 2 00 0 0 00 2Š ..f2 / f1 C 2f2 f1 f2 /

1 0 0 C A; .f20 /3 3 0 2 00 2Š ..f2 / f2 /

156

G. Bérczi

and the remaining five columns (corresponding to Sym3 C 3 ) are 0 1 0 0 0 0 0 0 0 0 0 C B 0 : @ 0 0 0 0 0 A 0 4 0 3 0 0 2 0 2 0 0 3 0 4 .f1 / .f1 / f2 .f1 / .f2 / f1 .f2 / .f2 / 4 Then the weight 1 C 2 C 3 C 4 D 10 piece CŒJ4 .1; 2/U 10 of the invariant algebra U4 CŒJ4 .1; 2/ is generated by the 4  4 minors of this 4  15 matrix.

5.1 The computation: double localization + vanishing theorem. According to Proposition 3 and Theorem 7 we have the following picture, also called the snowmanmodel after the figure in Section §6 in [3]: †k .n; m/

 Jk .n; m/

 reg Jk .1; n/=Gk

 Flagk .Symk C n /

(8)

 reg Jk .1; n/=Bk D Flagk .C n /:

Here Bk  GLk is the upper Borel subgroup. Now we apply ABBV localization Rto compute mdegŒ†k .n; m/; Jk .n; m/. According to Vergne, we have to compute n †k .n;m/ Thom.Jk .n; m/, and we do this in two steps: first we localize on Flagk .C / and use Proposition 2 to turn the localization formula into an iterated residue. Then we integrate along the fibers. The fibers are canonically isomorphic to Bk =Gk and in the second step we apply ABBV localization on the image .fiber/  Grass.k; Symk C n /. Surprisingly – for some unclear geometric reason – in this second localization all fixed points but a distinguished one contributes 0 to the sum, and a lengthy computation leads us in [3] to Theorem 9 ([3]). We have Q Tpmn k

D ReszD1

  k  zj / Qk .z1 : : : zk / Y 1 zlmn dzl ; c  .z C z  z / z j l l iCj lk i

i<j .zi

Q

(9)

lD1

where • we integrate on the cycle jz1 j > jz2 j >    > jzk j, which determines the Laurent expansion, • c.q/ D 1 C c1 q C c2 q 2 C    , • Qk .z1 ; : : : ; zk / is the multidegree of a Borel orbit in .C k / ˝ Sym2 .C k /, for details see [3], and Q1 D Q2 D Q3 D 1;

Q4 D 2z1 C z2  z4 :

Moduli of map germs, Thom polynomials and the Green–Griffiths conjecture

157

The polynomial Qk is known up to k  6, but with enough computer capacity – in principle – it can be computed for any k. But no general formula is known at the moment. We give a concise – and not complete – summary of the history of Thom polynomial computations. For a more detailed overview see [3], [20]. • Multidegrees of singularities have been studied for nearly 60 years now. We call these Thom polynomials in the honour of René Thom and his pioneering work in the 1950s. He proved the existence of these polynomials ([37]). He studied real manifolds and singularities of differentiable maps between them. Later Damon in [9] studied complex contact singularities. D cmnC1 . The • The case k D 1 is the classical formula of Porteous: Tpn!m 1 was k D 2 case was computed by Ronga in [32]. An explicit formula for Tpn!k 3 proposed in [2] and P. Pragacz has given a proof [29]. He also studied Thom polynomials in [30], [22], the latter written with A. Lascoux. Finally, using his method of restriction equations, Rimányi [31] was able to treat the n D k case, and computed Tpn!n for k  8 (cf. [16] for the case d D 4). k • More recently, Kazarian ([21]) has worked out a framework for computing Thom polynomials of contact singularities in general. He suggests studying certain non-commutative associative algebras to get a polynomial QA and an iterated residue formula similar to (9) for any local algebra A. Unfortunately, the explicit computation of QA is difficult, his description does not give more information for Morin singularities, where Qk is unknown for k > 6. The structure of Thom polynomials of contact singularities was also studied in [14], [15]. Finally, let us mention a conjecture of R. Rimányi about the positivity of these Thom polynomials. Conjecture 1 (Rimányi, 1998). Tpmn 2 NŒc1 ; : : : ; ck.mnC1/ ; k i.e. the coefficients of the Thom polynomials are non-negative. This would follow from the more general conjecture that Q i<j .zi  zj / Qk .z1 : : : zk / Q > 0; iCj lk .zi C zj  zl / that is, the coefficients of the Thom series are non-negative.

6 The Green–Griffiths conjecture First we list some results related to hyperbolic varieties and the Green–Griffiths conjecture. This is a selection of classical results and it is far from being complete.

158

G. Bérczi

6.1 Hyperbolic varieties. Let X be a complex manifold, n D dimC .X /. X is said to be hyperbolic • in the sense of Brody, if there are no non-constant entire holomorphic curves f W C ! X; • in the sense of Kobayashi, if the Kobayashi–Royden pseudo-metric on TX is non-degenerate. This pseudo-metric is defined as follows. The infinitesimal Kobayashi–Royden metric is kX ./ D inff > 0 W 9f W  ! X; f .0/ D x; f 0 .0/ D g for x 2 X ,  2 TX;x . The Kobayashi pseudo-distance d.x; y/ is the geodesic pseudo-distance obtained by integrating the Kobayashi–Royden infinitesimal metric. X is hyperbolic in the sense of Kobayashi if d.x; y/ > 0 for x ¤ y. The following theorem of Kobayashi tells that positivity of the cotangent bundle implies hyperbolicity. Theorem 10 (Kobayashi, ’75). Let X be a smooth projective variety with ample cotangent bundle. Then X is hyperbolic. Conversely, Conjecture 2. IfVa compact manifold X is hyperbolic, then it should be of general type, i.e.L KX D n T  X should be big. (That is, X has maximal Kodaira dimension, 0 i i.e. dim 1 iD0 H .X; K / D dim X .) Conjecture 3 (Green–Griffiths, ’79). Let X be a projective variety of general type. Then there exists an algebraic variety Y   X such that for all non-constant holomorphic f W C ! X one has f .C/  Y . Diophantine properties Theorem 11 (Faltings, ’83). A curve of genus greater than 1 has only finitely many rational points. Theorem 12 (Moriwaki, ’95). Let K be a number field (finitely generated over Q), and X a smooth projective variety. If T  X is ample and globally generated then X.K/ is finite. Conjecture 4 (S. Lang). (1) If a projective variety X is hyperbolic, then it is mordellic, i.e. X.K/ is finite for any K finitely generated over Q. S (2) Let Exc.X / D ff .C/I f W C ! X g. Then X n Exc.X / is mordellic. Highlights in the history of the Green–Griffiths conjecture. Here is a short (incomplete) list of results related to the Green–Griffiths conjecture, which first appeared in [17].

Moduli of map germs, Thom polynomials and the Green–Griffiths conjecture

159

• In [24] McQuillen gives a positive answer to the conjecture for general surfaces if the second Segre class c12  c2 > 0 is positive. • In the seminal paper [10] Demailly – using ideas of Green, Griffiths and Bloch – works out a strategy for projective hypersurfaces. • In [33], [34] Siu gives a positive answer for hypersurfaces of high degree, without an effective lower bound for the degree. • In [12] Diverio, Merker and Rousseau give an effective lower bound, proving 5 that for a generic projective hypersurface of dimension n and degree > 2n the Green–Griffiths conjecture holds. • Recently, Merker ([26]) has proved the existence of global jet differentials of high order for projective hypersurfaces in the optimal degree. Demailly in [11] has proved the existence of global jet differentials (of possibly high order) for compact manifolds in general. 6.2 A promising strategy (Green, Griffiths, Demailly, Siu, Diverio, Merker, Rousseau). The main idea of this strategy is to find differential equations which must be satisfied by (the jet of) any entire holomorphic curve in X , and then to find enough independent equations such that their solution set is a proper subvariety of X . For more details on the history of this approach see [12], [10]. Let f W C ! X; t 7! f .t/ D .f1 .t /; f2 .t/; : : : ; fn .t //; be a curve written in some local holomorphic coordinates .z1 ; : : : ; zn / on X . Let Jk X be the k-jet bundle over X of holomorphic curves, whose fiber at x 2 X is .Jk X /x D ffOŒk I f W .C; 0/ ! .X; x/g ! X sending fŒk to f .0/. This fibre is canonically isomorphic to Jk .1; n/. reg The group of reparametrizations Gk D Jk .1; 1/ acts fiberwise on Jk X . The fibres of Jk X can be identified with Jk .1; n/, and the action is linearised as in (6) before. Note that Gk D C  Ë Ud is a C  -extension of its maximal unipotent subgroup, and for  2 C  , .  f /.t/ D f .  t /; so   .f 0 ; f 00 ; : : : ; f .k/ / D .f 0 ; 2 f 00 ; : : : ; k f .k/ /: Algebraic differential operators correspond to polynomial functions on Jk X , and we call these polynomial functions jet differentials. They have the form X Q.f 0 ; f 00 ; : : : ; f .k/ / D a˛1 ;˛2 ;:::˛k .f .t //.f 0 .t /˛1 f 00 .t /˛2    f .k/ .t /˛k /; ˛i 2N n

where a˛1 ;˛2 ;:::˛k .z/ are holomorphic coefficients on X and t ! z D f .t / is a curve.

160

G. Bérczi

Q is homogeneous of weighted degree m under the C  action if and only if Q.f 0 ; 2 f 00 ; : : : ; k f .k/ / D m Q.f 0 ; f 00 ; : : : ; f .k/ /: Definition 3. We recall the following notation. GG denote the sheaf of algebraic differential operators of (Green–Griffiths ’78) Let Ek;m order k and weighted degree m.

(Demailly, ’95) The bundle of invariant jet differentials of order k and weighted degree GG m is the subbundle Ek;m  Ek;m , whose elements are invariant under arbitrary changes of parametrization, i.e. for  2 Gk , Q..f ı /0 ; .f ı /00 ; : : : ; .f ı /.k/ / D  0 .0/m Q.f 0 ; f 00 ; : : : ; f .k/ /: We want to apply the general principle that for a G-space X the ring of invariant functions on X can be identified with polynomial functions on the quotient X=G. Roughly speaking we want L L GG U U m .Ek;m /x D m .Ek;m /x D O..Jk X /x / D O.Jk .1; n/=U/: Applying Theorem 7 fibrewise we get Proposition 4. (1) The quotient Jk X=Gk has the structure of a locally trivial bundle over X, and there is a holomorphic embedding  P W Jk X=Gk ,! P .^k .TX ˚ Sym2 .TX / ˚    ˚ Symk .TX //: The fibrewise closure of the image Xk D im P is a relative compactification of Jk .TX /=Gk over X . (2) We have .k / OXk .m/ D O.Ek;m.kC1/ /; 2

where k W P .^k .TX ˚ Sym2 .TX / ˚    ˚ Symk .TX /// ! X is the projection. The strategy to solve the Green–Griffiths conjecture is based on the following Theorem 13 (Fundamental vanishing theorem, Green–Griffiths ’78, Demailly ’95, Siu ’96). Let P 2 H 0 .X; Ek;m ˝ O.A// be a global algebraic differential operator whose coefficients vanish on some ample divisor A. Then for any f W C ! X , P .fŒk .C//  0. (Note that fŒk .C/  Jk X .) Corollary 1. (1) Let  be a non-zero element of H 0 .Xk ; OXk .m/ ˝   O.A// ' H 0 .X; Ek;m.kC1/ ˝ O.A//: 2

Then fŒk .C/  Z , where Z  Xd is the zero divisor of  . T (2) If fj g is a basis of global sections then the image f .C/ lies in Y D k . ZPj /, hence the Green–Griffiths conjecture holds if there are enough independent differential T equations so that Y D k . .ZPj //   X:

161

Moduli of map germs, Thom polynomials and the Green–Griffiths conjecture

It is crucial to control in a more precise way the order of vanishing of these differential operators along the ample divisor. Thus, we need here a slightly different theorem. Theorem 14 ([12]). Assume that n D k, and there exist a ı D ı.n/ > 0 and D D D.n; ı/ such that H 0 .Xn ; OXn .m/ ˝   KXım / ' H 0 .X; En;m.nC1/ TX ˝ KXım / ¤ 0 2

whenever deg.X / > D.n; ı/ provided that m > mD;ı;n is large enough. Then the Green–Griffiths conjecture holds for 

deg.X / max D.n; ı/;

 n2 C 2n CnC2 : ı

The goal is therefore to find a global section of OXn .m/˝  KXım keeping D.n; ı/ small. Following [12], we use the algebraic Morse inequalities of Demailly/Trapani. Theorem 15 ([36]). Let L ! X be a holomorphic line bundle given as L D F ˝ G 1 with F , G nef bundles. Then for any non-negative integer q we have q X

qj

.1/

j

˝m

h .X; L

j D0

! q mn X qj n F nj  G j C o.mn /: ˝ E/  r .1/ j nŠ j D0

Applying this with q D 1 we get F n  nF n1 G > 0 H) H 0 .L˝m / ¤ 0

for m 0:

(10)

In [4] we prove that F and G are nef bundles in the following equality: ı .nC1 2 /

OX .1/ ˝   KX „ n ƒ‚

…

L

ı .nC1/ D .OXn .1/ ˝   OX .2n2 // ˝ .  OX .2n2 / ˝   KX 2 /1 : „ ƒ‚ … „ ƒ‚ … F

G

We introduce the notations h D c1 .OX .1//I

c1 .KX / D c1 .X / D .d  n  2/hI

OXn .1/ D det ;

where ! Xn is the tautological n-bundle. Now dim.Xn / D n2 , and according to (10) we want to prove the positivity of the following intersection number on Xn : Z 2 .c1 .det / C 2n2   h/n Xn ! nC1 2 2  n2 1 2   n .c1 .det / C 2n  h/ .2n  h C ı .d  n  2/h/: 2

162

G. Bérczi

We apply localization using the double fibration model (8) on the fibers of Xn . We need a stronger version of the vanishing property of the iterated residue to guarantee that only one fixed point’s contribution is non-zero. After going through these technical difficulties in [4] we arrive at Residue formula for the Demailly intersection number Q

Z I D

X

i<j .zi

ReszD1 Q 

 zj / Qd .z1 : : : zn /R.z; h; d; ı/  zl /.z1 : : : zn /n

1iCj ln .zi C zj Y n n  Y

1C

lD1

dh zl

1

lD1

h2 h C 2   zl zl

nC2

where R.z; h; d; ı/ D .z1      zn C 2n2 h/n

2

n2 1

 n2 .z1      zn C 2n2 h/

! n C 1 .2n2 h C ı .d  n  2/h/: 2

Analysis of the formula 1 , z1 :::zn

• The iterated residue is the coefficient of

and has the form hn p.d; n; ı/.

• Integration on X is the substitution hn D d , so the result is dp.d; n; ı/. • p.n; d; ı/ D an .n; ı/d n C    C a0 .n; ı/ is a degree-n polynomial in d D deg.X/, with polynomial coefficients in n, ı. • The leading coefficient is an .n; ı/ D 1  n2 where

! ! nC1 ı ‚.n/; 2

Y

Q.z/

2

.zi  zj /.z1 C    C zn /n

i<j

‚.n/ D constant term of

Y

.zi C zj  zl /.z1 : : : zn /n

iCj ln

Z

Note that ‚.n/ D

Xn

c1 . /n

is positive, as is an ample bundle. Therefore

2

:

Moduli of map germs, Thom polynomials and the Green–Griffiths conjecture

Corollary 2. For ı < number is positive.

2 n3 .nC1/

163

the leading coefficient of the Demailly intersection

The background and experimental evidences of the following conjecture is explained in [4]. It says that quotients of “neighbouring” coefficients of the Thom polynomial are polynomials. Conjecture 5. Define

Q m
Tpk .z1 ; : : : ; zk / D Then

coeff coeff

i

i

i

z11 :::zkk

i C1

z11 zl l

i

zmm

Tpk 1

ik Tpk

< k2:

zk

Some further computations in [4] leads us to Theorem 16 ([4]). Conjecture 5 and Conjecture 1 for Thom polynomials of An singularities imply the Green–Griffiths conjecture for d D deg.X/ > n6 .

7 Appendix The given iterated residue formula is suitable to compute other intersection numbers as well. The Euler characteristic of the Demailly bundle is defined as .X; Ek;m TX / D

n X

.1/i dim H i .X; Ek;m TX /:

iD0

It is well known (see [19]) that .X; Ek;m / D

Z X

ŒCh.Ek;m /  Td.TX /n ;

1 .c12 C c2 / C    where Ch.OXn .1// is the Chern character and Td.TX / D 1 C 12 c1 C 12 is the Todd class.

Theorem 17 (Iterated residue formula for the Euler-characteristics). .X;  OXn .m// Q Z i<j .zi  zj / Qn .z1 : : : zn /Ch.OXn .m//Td.TX / Q ReszD1 D n X 1iCj ln .zi C zj  zl /.z1 : : : zn / nC2  n  n  Y h2 dh Y h  C 2   1C 1 zl zl zl lD1

lD1

where Ch.OXn .1// D e m.z1 CCzn / ;

1 1 Td.TX / D 1 C c1 C .c12 C c2 / C    : 2 12

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G. Bérczi

8 Curviliear Hilbert schemes The goal of this section is to give a general framework for our localization argureg ments. If Gk D Jk .1; 1/ denotes the group of k-jets of reparametrization germs of reg C and Jk .1; n/ the k-jets of germs of curves f W .C; 0/ ! .C n ; 0/, then the quotient reg reg Jk .1; n/=Jk .1; 1/ plays an important role in our applications, namely: reg

reg

(1) †k fibers over Jk .1; n/=Jk .1; 1/ with linear fibers. The Thom polynomials of Morin singularities are certain equivariant intersection numbers on †k . reg

reg

(2) Jk .1; n/=Jk .1; 1/ is isomorphic to the fibers of the Demailly jet bundle Ek over a smooth manifold of dimension n. The positivity of the Demailly intersection number implies the Green–Griffiths conjecture. In both applications we are going to compute certain (equivariant) intersection reg reg numbers on the quotient Jk .1; n/=Jk .1; 1/, thereby using equivariant localization on reg reg  Grass .Jk .1; n/=Jk .1; 1//. Let H0 .k; n// be the punctual Hilbert scheme of k points on C n , that is, the set of zero dimensional subschemes of C n of length k supported at the origin. There is an important subset of H0 .k; n//, namely the punctual curvilinear Hilbert scheme, defined as follows. Definition 4. The punctual curvilinear Hilbert scheme is the closure of the set of ideals C.k; n/ D fI  CŒx1 ; : : : ; xn  W CŒx1 ; : : : ; xn =I ' t CŒt =t kC1 g; that is, C H .k; n/ D C .k; n/: If m D .x1 ; : : : ; xn /  OC n ;0 denotes the maximal ideal at the origin, then Symk C n WD m=mkC1 D

k L iD1

Symi C n

is the set of function-germs of degree  n, and the punctual Hilbert scheme naturally sits in the Grassmannian H0 .k; n/  Grass.k; Symk C n /: Looking at our embedding  Grass it is not hard to check (see [6]) that Proposition 5. We have reg

reg

C H .k; n/ D  Grass .Jk .1; n/=Jk .1; 1// :

Moduli of map germs, Thom polynomials and the Green–Griffiths conjecture

165

This roughly means that C H .k; n/ can be described as a certain compactification of a non-reductive quotient. When n D 2 we further know that the punctual curvilinear component C H .k; n/ is dense in H0 .k; n/, and therefore Corollary 3. We have reg

reg

H0 .k; 2/ D  Grass .Jk .1; 2/=Jk .1; 1// : We have developed localization methods to compute intersection numbers on the punctual curvilinear Hilbert scheme C H .k; n/ for k  n. A more detailed study of non-reductive quotients allows us to improve this technique, the details with more applications will be published later.

References [1] V. I. Arnold, V. V. Goryunov, O. V. Lyashko, and V. A. Vasilliev, Singularity theory I. Reprint of Dynamical systems VI, Encyclopaedia Math. Sci. 6, Springer-Verlag, Berlin 1998. [2] G. Bérczi, L. M. Fehér, and R. Rimányi, Expressions for resultants coming from the global theory of singularities. In Topics in algebraic and noncommutative geometry, Contemp. Math. 324, Amer. Math. Soc., Providence, RI, 2003, 63–69. [3] G. Bérczi and A. Szenes, Thom polynomials of Morin singularities. Ann. of Math. 175 (2012), 567–629. [4] G. Bérczi, Thom polynomials and the Green-Griffiths conjecture. Preprint, arXiv:1011.4710v1 [math.AG]. [5] G. Bérczi, and F. C. Kirwan, A geometric construction for invariant jet differentials. In Algebra and geometry: In memory of C. C. Hsiung, Surv. Differ. Geom. 17, International Press, Somerville, Mass., 2011 79–127. [6] G. Bérczi, and F. Kirwan, A Grassmannian model for non-reductive quotients. In preparation. [7] N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators. Grundlehren Math. Wiss. 298, Springer-Verlag, Berlin 1992. [8] N. Berline, and M. Vergne, Zéros d’un champ de vecteurs et classes characteristiques équivariantes. Duke Math. J. 50 (1973), no. 2, 539–549. [9] J. Damon, Thom polynomials for contact class singularities. Ph.D. Thesis, Harvard University, 1972. [10] J.-P. Demailly, Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials. In Algebraic geometry–Santa Cruz 1995, Proc. Sympos. Pure Math. 62, Part 2, Amer. Math. Soc., Providence, RI, 1997, 285–360. [11] J.-P. Demailly, Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture. Pure Appl. Math. Q. 7 (2011), 1165–1208. [12] S. Diverio, J. Merker, and E. Rousseau, Effective algebraic degeneracy. Invent. Math. 180 (2010), 161–223.

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[13] B. Doran and F. Kirwan, Towards non-reductive geometric invariant theory. Pure Appl. Math. Q. 3 (2007), 61–105. [14] L. M. Fehér and R. Rimányi, Thom polynomial computing strategies. A survey. In Singularity theory and its applications, Adv. Stud. Pure Math. 43, Math. Soc. Japan, Tokyo 2006, 45–53. [15] L. M. Fehér and R. Rimányi, On the structure of Thom polynomials of singularities. Bull. London Math. Soc. 39 (2007), 541–549. [16] T. Gaffney, The Thom polynomial of †1111 . In Singularities, Part 1, Proc. Sympos. Pure Math. 40, Amer. Math. Soc., Providence, RI, 1983, 399–408. [17] M. Green and P. Griffiths, Two applications of algebraic geometry to entire holomorphic mappings. In The Chern Symposium 1979, Springer-Verlag, New York 1980, 41–74. [18] A. Haefliger and A. Kosinski, Un théorème de Thom sur les singularités des applications diffèrentiables. Sèminaire Henri Cartan, 9 Exposè 8, 1956/57. [19] F. Hirzebruch, Topological methods in algebraic geometry. Grundlehren Math. Wiss. 131, Springer-Verlag, Berlin 1966. [20] M. Kazarian, Thom polynomials. In Singularity theory and its applications, Adv. Stud. Pure Math. 43, Math. Soc. Japan, Tokyo 2006, 85–135. [21] M. Kazarian, unpublished. [22] L. Lascoux and P. Pragacz, Thom polynomials and Schur functions: the singularities A3 . Publ. Res. Inst. Math. Sci. 46 (2010), no. 1, 183–200. [23] M. McQuillan, Diophantine approximation and foliations. Inst. Hautes Études Sci. Publ. Math. 87 (1998), 121–174. [24] M. McQuillan, Holomorphic curves on hyperplane sections of 3-folds. Geom. Funct. Anal. 9 (1999), 370–392. [25] J. Merker, Applications of computational invariant theory to Kobayashi hyperbolicity and to Green-Griffiths algebraic degeneracy. J. Symbolic Comput. 45 (2010), no. 10, 986–1074. [26] J. Merker, Complex projective hypersurfaces of general type: toward a conjecture of Green and Griffiths. Preprint, arXiv:1005.0405v1 [math.AG]. [27] E. Miller and B. Sturmfels, Combinatorial commutative algebra. Grad. Texts in Math. 227, Springer-Verlag, New York 2005. [28] D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory. 3rd edition, Ergeb. Math. Grenzgeb. 34, Springer-Verlag, Berlin 1994. [29] P. Pragacz, Thom polynomials and Schur-functions I. Preprint, arXiv:math.AG/0509234. [30] P. Pragacz, Thom polynomials and Schur-functions: towards the singularities Ai . In Real and complex singularities, Contemp. Math. 459 Amer. Math. Soc., Providence, RI, 2008, 165–178. [31] R. Rimányi, Thom polynomials, symmetries and incidences of singularities. Invent. Math. 143 (2001), no. 3, 499–521. [32] F. Ronga, Le calcul des classes duales aux singularités de Boardman d’ordre 2. C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A582–A584.

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[33] Y.-T. Siu, Some recent transcendental techniques in algebraic and complex geometry. In Proceedings of the International Congress of Mathematicians (Beijing, 2002),Vol. I, Higher Education Press, Beijing 2002, , 439–448. [34] Y.-T. Siu, Hyperbolicity in complex geometry. In The legacy of Niels Henrik Abel, SpringerVerlag, Berlin 2004, 543–566. [35] Y.-T. Siu and S.-K. Yeung, Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane. Invent. Math. 124 (1996), 573–618. [36] S. Trapani, Numerical criteria for the positivity of the difference of ample divisors. Math. Z. 219 (1995), no. 3, 387–401. [37] R. Thom, Les singularités des applications différentiables. Ann. Inst. Fourier 6 (1955/56), 43–87. [38] M. Vergne, Polynomes de Joseph et représentation de Springer. Ann. Sci. École Norm. Sup. (4) 23 (1990), no. 4, 543–562. Gergely Bérczi, Mathematical Institute, University of Oxford, 24–29 St Giles, OX1 3LB Oxford, UK E-mail: [email protected]

The Minimal Model Program revisited Paolo Cascini and Vladimir Lazi´c

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . 2 Zariski decomposition on surfaces . . . . . . . . 3 Diophantine approximation and a lifting theorem 4 Finite generation . . . . . . . . . . . . . . . . . 5 Examples . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction The purpose of Mori theory is to give a meaningful birational classification of a large class of algebraic varieties. This means several things: that varieties have mild singularities (in the sense explained below), that we do some surgery operations on varieties which do not make singularities worse, and that objects which are end-products of surgery have some favourable properties. Smooth projective varieties belong to this class, and even though the end-product of doing Mori surgery on a smooth variety might not be – and usually is not – smooth, it is nevertheless good enough for all practical purposes. One of the main properties of these surgeries is that they do not change the canonical ring. Recall that given a smooth projective variety X , the canonical ring is M R.X; KX / D H 0 .X; OX .mKX //: m0

More generally, for any Q-divisors D1 ; : : : ; Dk on X , we can consider the divisorial ring M    P R.X I D1 ; : : : ; Dk / D H 0 X; OX b mi Di c : .m1 ;:::;mk /2N k

Rings of this type were extensively studied in Zariski’s seminal paper [Zar62]. In particular, the following is implicit in [Zar62]: Question 1.1. Let X be a smooth projective variety. Is the canonical ring R.X; KX / finitely generated?  The first author was partially supported by an EPSRC grant. We would like to thank the referee for many useful comments.

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The answer is trivially affirmative for curves, and in the appendix of [Zar62], Mumford confirmed it when X is a surface. Since it is invariant under operations of Mori theory, this question largely motivated conjectures of Mori theory, which postulate that Mori program should terminate with a variety with KX semiample. This was confirmed in the case of threefolds in the 1980s by Mori et al. Very recently, the question was answered affirmatively in any dimension in [BCHM10] by extending many of the results of Mori theory from threefolds to higher dimensions. An analytic proof in the case of varieties of general type was announced in [Siu06]. One of the main inputs of Mori theory is that instead of considering varieties, we should consider pairs .X; /, where X is a normal projective variety and   0 is a Q-divisor on X such that the adjoint divisor KX C  is Q-Cartier. In order to get a good theory one has to impose a further technical condition on KX C, namely how its ramification formula behaves under birational maps. For our purposes, the condition means that X is a smooth variety, the support of  has simple normal crossings, and the coefficients of  are strictly smaller than 1, that is bc D 0. Therefore, we might pose the following: Conjecture 1.2. Let X be a smooth projective variety. Let B1 ; : : : ; Bk be Q-divisors P on X such that bBi c D 0 for all i , and such that the support of kiD1 Bi has simple normal crossings. Denote Di D KX C Bi for every i . Then the ring R.XI D1 ; : : : ; Dk / is finitely generated. Divisorial rings of this form are called adjoint rings. This conjecture obviously generalises the affirmative answer to Question 1.1, by choosing k D 1 and B1 D 0. Here we survey a different approach to the finite generation problem, recently obtained in [CL10a], which avoids all the standard difficult operations of Mori theory. In that paper, we give a new proof of the following result, only using the Kawamata– Viehweg vanishing and induction on the dimension. Theorem A. Let X be a smooth projective variety. Let B1 ; : : : ; Bk be Q-divisors on X P such that bBi c D 0 for all i, and such that the support of kiD1 Bi has simple normal crossings. Let A be an ample Q-divisor on X , and denote Di D KX C A C Bi for every i . Then the ring R.XI D1 ; : : : ; Dk / is finitely generated. Theorem A, in fact, also gives the affirmative answer to Question 1.1 – details can be found in [CL10a]. This theorem was originally proved in [BCHM10] as a consequence of Mori theory, and later in [Laz09] by induction on the dimension and without Mori theory; for an easy introduction to the latter circle of ideas, see [Cor10]. In [CL10a], we build on and significantly simplify arguments from [Laz09]. The aim of this survey is to show that, in spite of its length and modulo some standard technical difficulties, our paper [CL10a] is based on simple ideas. We hope here to make these ideas more transparent and the general flow of the proof more accessible.

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Moreover, by the results of [CL10b], Theorem A implies all currently known results of Mori theory, so in effect this approach inverts the conventional order of the program on its head, where earlier finite generation came at the end of the process, and not at the beginning. Further, it is shown that Conjecture 1.2 implies the most of outstanding conjectures of the theory, in particular the Abundance conjecture. Finally, one might ask whether the rings R.X I D1 ; : : : ; Dk / are finitely generated when the divisors Di are not adjoint, or whether the assumption on singularities can be relaxed. However, Example 5.1 shows that this ring might not be finitely generated in similar situations. Notation and conventions We work with algebraic varieties defined over C. We denote by RC and QC the sets of non-negative real and rational numbers, and by Cx the topological closure of a set C  RN . Let X be a smooth projective variety and R 2 fZ; Q; Rg. We denote by DivR .X / the group of R-divisors on X , andPR and  denote Pthe R-linear and numerical equivalence of R-divisors. If A D ai Ci and B D bi Ci are two R-divisors on X, then bAc is the round-down of A, dAe is the round-up of A, and X A^B D minfai ; bi gCi : Further, if S is a prime divisor on X , mult S A is the order of vanishing of A at the generic point of S . In this paper, a log pair .X; / consist of a smooth variety X and an R-divisor   0. A projective birational morphism f W Y ! X is a log resolution of the pair .X; / if Y is smooth, Exc f is a divisor and the support of f1  C Exc f has simple normal crossings. A log pair .X; / with bc D 0 has canonical singularities if for every log resolution f W Y ! X , we have KY C f1  D f  .KX C / C E for some f -exceptional divisor E  0. If X is a smooth projective variety and D is an integral divisor, Bs jDj denotes the base locus of D, and Fix jDj and Mob.D/ denote the fixed and mobile parts of D. Hence jDj D j Mob.D/j C Fix jDj, and the base locus of j Mob.D/j contains no divisors. More generally, if V is any linear system on X , Fix.V / is the fixed divisor of V . If S is a prime divisor on X such that S ª Fix jDj, then jDjS denotes the image of the linear system jDj under restriction to S . If D is an R-divisor on X , we denote by B.D/ the intersection of the sets Supp D 0 for all D 0  0 such that D 0 R D, and we call B.D/ the stable base locus of D; we set B.D/ D X if no such D 0 exists. Definition 1.3. Let X P be a smooth variety, and let S; S1 ; : : : ; Sp be distinct prime P divisors such that S C piD1 Si has simple normal crossings. Let V D piD1 RSi 

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DivR .X/, and let A be a Q-divisor on X . We define P L.V / D fB D bi Si 2 V j 0  bi  1 for all i g; EA .V / D fB 2 L.V / j there exists 0  D R KX C A C Bg; BAS .V / D fB 2 L.V / j S ª B.KX C S C A C B/g: Now, let X be a smooth projective variety, let V  DivR .X / be a finite dimensional vector space, and let C  V be a rational polyhedral cone, i.e. a convex cone spanned by finitely many rational vectors. Then the divisorial ring associated to C is M R.X; C / D H 0 .X; OX .D//: D2C \DivZ .X/

Note that C \ DivZ .X / is a finitely generated monoid by Gordan’s lemma, and if all elements in it are multiples of adjoint divisors, the corresponding ring is an adjoint ring. This generalises divisorial and adjoint rings introduced earlier. If S is a prime divisor on X , the restriction of R.X; C/ to S is defined as M   resS R.X; C / D resS H 0 X; OX .D/ ; D2C \DivZ .X/

 0



where resS H X; OX .D/ is the image of the restriction map H 0 .X; OX .D// ! H 0 .S; OS .D//: Similarly, we denote by resS R.X I D1 ; : : : ; Dk / the restricted divisorial ring for Qdivisors D1 ; : : : ; Dk in X . We finish this section with some basic definitions from convex geometry. Let C  RN be a convex set. A subset F  C is a face of C if F is convex, and whenever u C v 2 F for u; v 2 C , then u; v 2 F . Note that C is itself a face of C . We say that x 2 C is an extreme point of C if fxg is a face of C . It is a well known fact that any compact convex set C  RN is the convex hull of its extreme points. A polytope in RN is a compact set which is the intersection of finitely many half spaces; equivalently, it is the convex hull of finitely many points in RN . A polytope is rational if it is an intersection of rational half spaces; equivalently, it is the convex hull of finitely many rational points in RN .

2 Zariski decomposition on surfaces Zariski decomposition, introduced by Zariski in [Zar62] in the case of surfaces, has proven to be a powerful tool in birational geometry. In this section, after reviewing some basic notions about the Zariski decomposition and its generalization in higher dimension, we show that the finite generation of the canonical ring on surfaces is an easy

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consequence of this decomposition, combined with the Kawamata–Viehweg vanishing theorem. It is worth noting that while the vanishing theorem holds in any dimension, and it is the most important ingredient for the proof of the lifting theorem described in the next section, the Zariski decomposition a priori only holds on surfaces, see [Laz04, §2.3.E]. Definition 2.1. Let X be a smooth projective variety and let D be a pseudo-effective Q-divisor. Then D admits a Zariski decomposition if there exist a nef Q-divisor P (positive part) and a Q-divisor N  0 (negative part) such that (1) D D P C N , (2) for any positive integer m, the natural homomorphism H 0 .X; OX .bmP c// ! H 0 .X; OX .bmDc// is an isomorphism. In particular, we have R.X; D/ ' R.X; P /. In the case of surfaces, Zariski showed Theorem 2.2. Let X be a smooth projective surface and let PD be a pseudo-effective Q-divisor. Then there exist unique Q-divisors P and N D riD1 i Ni  0 such that: (1) D D P C N , (2) P is nef, (3) P Ni D 0 for every i , (4) the intersection matrix .Ni Nj /i;j is negative definite. In particular, this defines a Zariski decomposition of D. Unfortunately, in higher dimensions it is easy to find examples of divisors which do not admit a Zariski decomposition, even after taking their pull-back to a suitable modification, see [Nak04]. This birational version of the decomposition is usually called Cutkosky–Kawamata–Moriwaki decomposition. P However, if D is a big Q-divisor on a smooth surface X and if N D i Ni is the negative part of D as in Theorem 2.2, it can be shown that i D lim sup m!1

1 mult Ni jmDj; m

see [Laz04, Corollary 2.3.25]. This motivates the following definition of the positive and negative parts of a pseudo-effective divisor over a smooth projective variety. These were introduced by Nakayama in [Nak04] (see also [Bou04] for an analytic construction).

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Definition 2.3. Let X be a smooth projective variety, let A be an ample R-divisor, and let  be a prime divisor. If D 2 DivR .X / is a big divisor, define 0 .D/ D inffmult D 0 j 0  D 0 R Dg: If D 2 DivR .X / is pseudo-effective, set  .D/ D lim 0 .D C "A/: "!0

Then the negative part of D is defined as P N .D/ D   .D/ ; where the sum runs over all prime divisors  on X . The positive part of D is given by P .D/ D D  N .D/. Then one easily shows that 0 .D/ D  .D/ when D is big. Note that, unlike in the case of surfaces, P .D/ is in general not nef. The main reason for this is that on surfaces, every mobile divisor is nef since curves and divisors coincide. Nevertheless, this decomposition turned out to be very useful in the recent advances of the Minimal Model Program, e.g. see [BCHM10]. The following result summarises its main properties, see [Nak04]. Lemma 2.4. Let X be a smooth projective variety, and let D be a pseudo-effective R-divisor. Then N .D/ depends only on the numerical class of D. Moreover, if  is a prime divisor, then the function  is homogeneous of degree one, convex and lower semicontinuous on the cone of pseudo-effective divisors on X , and continuous on the cone of big divisors. We now describe the role of Zariski decompositions in the study of the canonical ring of a smooth projective variety. By the basepoint free theorem [KM98, Theorem 3.3], it follows immediately that if X is a smooth minimal projective variety of general type, i.e. if KX is big and nef, then KX is semi-ample, and thus the canonical ring R.X; KX / is finitely generated. In [Kaw85], Kawamata generalized this result by showing that if X is a smooth projective variety of general type, the existence of a Zariski decomposition for KX implies the finite generation of the canonical ring. Using the same techniques, we next show how Theorem 2.2 implies the finite generation of the canonical ring on surfaces. We first recall the following important result of Zariski [Zar62] on the structure of base loci of linear systems. Theorem 2.5. Let X be a projective variety, and let D be a Cartier divisor on X such that the base locus of D is a finite set. Then D is semiample.

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Corollary 2.6. Let X be a smooth projective surface and let D be a Q-divisor on X such that .X; D/  0. Then, for every sufficiently divisible positive integer k, there exist a semiample Q-divisor Mk whose coefficients are at most 1=k, and a divisor Fk  0 such that Supp Fk  B.D/ and D Q Mk C Fk . Furthermore, if the support of D has simple normal crossings, then the support of Mk C Fk has simple normal crossings. Proof. Let m be a positive integer such that B.D/ D Bs jmDj. If we write mD D M C F; where M is the mobile part of jmDj and F is its fixed part, then M is semiample by Theorem 2.5 and Supp F  B.D/. Thus, for every sufficiently divisible positive integer k, the linear system jkM j is basepoint free. Bertini’s theorem implies that a general section Nk of jkM j is reduced and irreducible. Thus, it is enough to define Mk D Nk =km and Fk D F=m. The last claim follows from the construction. Our goal is to show that for any smooth projective surface X and for any divisor  on X such that the support of  has simple normal crossings and bc D 0, the ring R.X; KX C/ is finitely generated. As in the higher dimensional case, see [BCHM10] and [CL10a], by the existence of log resolutions and by a result of Fujino and Mori [FM00], we may and will assume, without loss of generality, that the divisor  can be written as  D A C B, where A is an ample Q-divisor and B is a Q-divisor such that the support of B has simple normal crossings and bBc D 0. First we need an easy result about adjoint divisors on curves. Lemma 2.7. Let X be a smooth projective curve of genus g  1, let D be an integral divisor, and let ‚ be a Q-divisor such that such that deg ‚ > 0 and D Q KX C ‚. Then H 0 .X; OX .D// ¤ 0. Proof. By Riemann–Roch theorem we have h0 .X; OX .D//  deg D  g C 1 D 2g  2 C deg ‚  g C 1 > g  1  0; which proves the lemma. We recall the following particular version of the Kawamata–Viehweg vanishing theorem which we need in this paper. P Theorem 2.8. Let X be a smooth variety and let B D bj Bj be a Q-divisor whose support has simple normal crossings and such that 0  bj  1 for each j . Let A be an ample Q-divisor and assume D is an integral divisor such that D Q KX C B C A. Then H i .X; OX .D// D 0 for every i > 0. Finally, we have:

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Theorem 2.9. Let X be a smooth projective surface, and let B be a Q-divisor such that the support of B has simple normal crossings and bBc D 0. Let A be an ample Q-divisor and let  D A C B. Then the ring R.X; KX C / is finitely generated. Proof. We may assume that .X; KX C /  0, since otherwise R.X; KX C / is trivial. Let KX C D P CN be the Zariski decomposition of KX C, whose existence is guaranteed by Theorem 2.2. Since R.X; KX C / ' R.X; P /, we may assume that P is not semiample, and in particular, B.P / contains a curve by Theorem 2.5. Furthermore, we can assume that P  0. Replacing X by a log resolution, we may assume that the support of B C P C N has simple normal crossings. For every positive integer k, let P Q Mk C Fk D Pk be the decomposition as in Corollary 2.6. Note that bB  N c  0, and let k D supft  0 j bB C tPk  N c  0g: Then k > 0, and if †k is the sum of all the prime divisors in B C k Pk  N with coefficient equal to 1, then the support of †k is contained in the support of Pk . Furthermore, by choosing k large enough, we can assume that the support of †k is contained in B.P /, and hence without loss of generality, we may assume that P D Pk . Denote † D †k and  D k . Let X RD .mult T dN  B  P e/ T; T

where the sum is over all prime divisors T such that multT .N  B  P / > 0, and denote B 0 D B C P  N C R. Then R is an integral divisor, the coefficients of B 0 lie in the interval .0; 1, and we have 0  R  dN e;

and bB 0 c D †:

Fix a prime divisor S in Supp †  B.P /. Let m >  C 1 be a sufficiently large positive integer such that mA, mB, mP and mN are integral divisors and B.P / D Bs jmP j. Let A0 D A C .m  1  /P , and note that A0 is ample since P is nef and m  1   > 0. Then mP C R Q KX C A C B  N C .m  1/P C R D KX C A0 C B 0 ; and the Kawamata–Viehweg vanishing theorem implies H 1 .X; OX .mP CRS // D 0. We claim that H 0 .S; OS .mP C R// ¤ 0: We first show that the claim implies the theorem. The long cohomology sequence associated to the exact sequence 0 ! OX .mP C R  S/ ! OX .mP C R/ ! OS .mP C R/ ! 0

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yields that the map H 0 .X; OX .mP CR// ! H 0 .S; OS .mP CR// ¤ 0 is surjective, and therefore S ª Bs jmP C Rj. Since 0  R  dN e  dmN e D mN , the composition of injective maps H 0 .X; OX .mP // ! H 0 .X; OX .mP C R// ! H 0 .X; OX .mP C mN // is an isomorphism by the property of Zariski decompositions, hence so is the first map. But this implies Bs jmP C Rj D Bs jmP j [ Supp R, and thus S ª Bs jmP j D B.P /, a contradiction. Finally, to prove the claim, let g be the genus of S. Note that S ª Supp R by construction, and therefore deg.mP C R/jS  0 since P is nef. In particular, this implies the claim for g D 0. If g  1, note that .mP C R/jS Q KS C A0jS C .B 0  S /jS and deg.A0jS C .B 0  S/jS / > 0, hence the claim follows by Lemma 2.7.

3 Diophantine approximation and a lifting theorem Both Diophantine approximation and the Kawamata–Viehweg vanishing theorem play crucial roles in recent developments in Mori theory. Diophantine approximation was first introduced in birational geometry, as an essential tool, by Shokurov in [Sho03], in order to give a conceptual proof of the existence of certain surgery operations called flips. The aim of this section is to sketch how these two tools can be naturally combined to obtain many of the results in [CL10a]. Diophantine approximation We first recall Diophantine approximation, see for instance [BCHM10, Lemma 3.7.7]. Lemma 3.1. Let k k be a norm on RN and let x 2 RN . Fix a positive integer k and a positive real number ". Then there are finitely many points xi 2 RN and positive integers ki divisible by k, such that ki xi =k are integral, kx  xi k < "=ki , and x is a convex linear combination of xi . The previous lemma implies the following characterization of rational polytopes in RN (similar techniques were used to prove [CL10a, Theorem 4.4]). Proposition 3.2. Let k k be a norm on RN and let P  RN be a bounded convex subset. Then P is a rational polytope if and only if there is a constant " > 0 and a positive integer k with the following property: for every rational v 2 RN , if there exist w 2 P and a positive integer l such that lv is integral and kv  wk < "= lk, then v 2 P .

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Proof. Since any two norms on RN are equivalent, we can assume that k k is the standard Euclidean norm. Let h ; i be the standard scalar product. Suppose that P is a rational polytope. Then there exist finitely many ci 2 Z and N i 2 Z such that w 2 P if and only if h i ; wi  ci for every i . Pick " > 0 such that k i k < 1=" for all i and let k be any positive integer. Assume that for v 2 RN there exist w 2 P and a positive integer l such that lv is integral and kv  wk < "= lk. Then by the Cauchy–Schwarz inequality we have ci  h

i ; vi

 h i ; wi  h i ; vi D h i ; w  vi  k i kkw  vk < k i k"= lk < 1= lk

for any i. But lk.ci  h i ; vi/ D lkci  h i ; klvi is an integer, and so h i ; vi  ci . Therefore v 2 P . Assume now that there exists " and k as in the statement of the proposition, and let w 2 Px . By Lemma 3.1, there exist points wi 2 RN and positive integers mi divisible by k, such that w is a convex linear combination of wi and kw  wi k < "=mi

and mi wi =k is integral

for every i . In particular, there exists w 0 2 P such that kw 0  wi k < "=mi . By the assumption, it follows that wi 2 P for all i, and in particular w 2 P . Therefore, P is a closed set, and moreover, every extreme point of P is rational. If P is not a rational polytope then there exist infinitely many extreme points vi of P , with i 2 N. Since P is compact, by passing to a subsequence we obtain that there exist v1 2 P such that v1 D lim vi : i!1

0 By Lemma 3.1, there exists a positive integer m divisible by k and v1 2 RN such that 0 0 0 mv1 =k is integral and kv1  v1 k < "=m. By assumption, it follows that v1 2 P. Pick j 0 so that 0 0 kvj  v1 k  kvj  v1 k C kv1  v1 k<

" : m

Therefore, there exists a positive integer m0 0 divisible by k such that .mCm0 /vj =k is integral, and such that if we define v0 D

m C m0 m 0 0 0 vj  0 v1 2 v1 C RC .vj  v1 /; 0 m m

then m0 v0 =k is integral and kv0  vj k D

m " 0 kvj  v1 k < 0: m0 m

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By assumption, this implies that v0 2 P , and since vj D follows that vj is not an extreme point of P , a contradiction. Thus, P is a rational polytope.

m0 v mCm0 0

C

m v0 , mCm0 1

it

Lifting property Let X be a smooth projective variety, let S be a prime divisor, let A be an ample Q-divisor, and let B  0 be a Q-divisor such that S ª Supp B, bBc D 0, and Supp.S C B/ has simple normal crossings. Let C  0 be a Q-divisor on S such that C  BjS , and let m be a positive integer such that mA, mB and mC are integral. We say that .B; C / satisfies the lifting property Lm if the image of the restriction morphism     H 0 X; OX .m.KX C S C A C B// ! H 0 S; OS .m.KS C AjS C BjS //   contains H 0 S; OS .m.KS C AjS C C //  , where  is a global section of OS .m.BjS  C // vanishing along m.BjS  C /. The following theorem is [CL10a, Theorem 3.4], and it is a slight generalization of the lifting theorem by Hacon–Mc Kernan [HM10], which is itself a generalization of results by Siu [Siu98], [Siu02] and Kawamata [Kaw99]. Similar results were also obtained in [Tak06], [P˘au07] and [EP12]. Theorem 3.3. Let X be a smooth projective variety, let S be a prime divisor, let A be an ample Q-divisor, and let B  0 be a Q-divisor such that S ª Supp B, bBc D 0, and Supp.S C B/ has simple normal crossings. Let C  0 be a Q-divisor on S such that .S; C / is canonical, and let m be a positive integer such that mA, mB and mC are integral. Assume that there exists a positive integer q 0 such that qA is very ample, 1 A/j and S 6 Bs jq m.KX C S C A C B C m C  BjS  BjS ^

1 Fix jq m.KX C S C A C B C qm

1 A/jS : m

Then .B; C / satisfies Lm . We omit the proof of the theorem, but we emphasise that it is a direct consequence of the Kawamata–Viehweg vanishing plus some elementary arithmetic. Before we show how the lifting theorem and Diophantine approximation are related to each other, it is useful to spend a few words on the assumptions of the theorem. It is well known that, in general, the lifting theorem does not hold if we just take C D BjS . A simple counterexample is given by considering the blow-up of P 2 at one point, see [Hac05], [CKL11] for more details. On the other hand, the condition that .S; C / is canonical is a bit more subtle and it might look artificial, but Example 5.2 shows that it is essential. Clearly if X is a surface, this condition is guaranteed by the fact that X is smooth, Supp.S C B/ has simple normal crossings and bBc D 0.

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We now proceed with Theorem 3.4. Let X be a smooth P projective variety, and let S; S1 ; : : : ; Sp be distinct prime divisors such that S C piD1 Si has simple normal crossings. Let V  DivR .X / be the subspace spanned by S1 ; : : : ; Sp , and let W  DivR .S / be the subspace spanned by all the components of SijS . Let A be an ample Q-divisor on X . Let Q0 be the set of pairs of rational divisors .B; C / 2 BAS .V /  EAjS .W / such that C  B and .B; C / satisfies Lm for infinitely many m; and let Q be the intersection of the closure of Q0 and the convex hull of Q0 . Then Q is a finite union of rational polytopes. Note that, since the aim is to provide a proof of Theorem A by induction on the dimension of X, in this context we are assuming that Theorem A holds in dimension dim X  1. In particular, we may assume that EAjS .W / is a rational polytope, see [CL10a, Theorem 5.5]. We now explain briefly how Proposition 3.2 can be applied to get Theorem 3.4. First note that the property Lm for the pair .B; C / immediately implies that Fix jm.KS C .A C C /jS /j C m.BjS  C /  Fix jm.KX C S C A C B/jS : Thus, after doing some simple algebra of divisors, the lifting result in Theorem 3.3 can be rephrased by saying that if the property Lm holds for a pair .B 0 ; C 0 / sufficiently “close” to .B; C /, then also the pair .B; C / will satisfy the property Lm . Here the distance between .B; C / and .B 0 ; C 0 / is bounded in terms of a positive integer q such that qB and qC are integral. Thus, we can apply Proposition 3.2 to get the desired result. Theorem 3.4 implies two crucial results which are related to Theorem A. Theorem 3.5. Under the assumption of Theorem 3.4, the set BAS .V / is a rational polytope and, for any B1 ; : : : ; Bk 2 L.V /, the ring resS R.X I KX C S C A C B1 ; : : : ; KX C S C A C Bk / is finitely generated. To prove the first statement, it is sufficient to show that BAS .V / is the image of the set defined in Theorem 3.4 through the first projection, and the result follows immediately by the convexity of BAS .V /. Note that, in particular, BAS .V / is compact, which is one of the main ingredients in our proofs of several results in [CL10a]. The second statement is more delicate. Theorem 3.4 implies that the restricted algebra is spanned by a finite union of adjoint rings on S . Thus, the result follows by induction on the dimension.

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4 Finite generation In this section we present the proof of a special case of the finite generation theorem which already contains almost all fundamental problems of the general case, and it is particularly easy to picture what is going on. We prove the following: Theorem 4.1. Let X be a smooth projective variety, and let S1 and S2 be distinct prime divisors such that S1 C S2 has simple normal crossings. Let B D b1 S1 C b2 S2 be a Q-divisor such that 0  b1 ; b2 < 1, and let A be an ample Q-divisor. Assume that KX C A C B Q D for some D 2 QC S1 C QC S2 . Then the ring R.X; KX C A C B/ is finitely generated. It will become clear from the scheme of the proof that it is necessary to work with higher rank algebras even in this simple situation. The proof will mostly be “by picture”, and for that reason we restrict ourselves to the case of two components. The proof in the general case follows the same line of thought, the only difference is that it is more difficult to visualise. The following result will be used often without explicit mention in this section; the proof can be found in [ADHL10]. L Lemma 4.2. Let   Zr be a finitely generated monoid and let R D s2 Rs be an L graded algebra. Let  0   be a finitely generated submonoid and let R0 D s2 0 Rs . (1) If R is finitely generated over R0 , then R0 is finitely generated over R0 . (2) If R0 is Noetherian, R0 is a Veronese subring of finite index of R, and R0 is finitely generated over R0 , then R is finitely generated over R0 . Sketch of the proof of Theorem 4.1. Let V D RS1 CRS2  DivR .X / be the subspace spanned by S1 and S2 , let B  V be the rectangle with vertices D, D C .1  b1 /S1 , D C .1  b2 /S2 and D C .1  b1 /S1 C .1  b2 /S2 , and denote C D RC B. For i 2 f1; 2g, consider the segments Bi D ŒD C .1  bi /Si ; D C .1  b1 /S1 C .1  b2 /S2   V and the cones Ci D RC Bi . It is clear from the picture (see Figure 1) that C D C1 [C2 , and that there exists M > 0 such that the “width” of the cones Ci in the half-plane fxS1 C yS2 j x C y  M g is bigger than 1. More precisely, (1) if xS1 C yS2 2 Ci for i 2 f1; 2g and for some x; y 2 N with x C y  M , then xS1 C yS2  Si 2 C . We further claim that: (2) for i 2 f1; 2g, the ring resSi R.X; Ci / is finitely generated.

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y

x+y =M C2 B2

C1

B1

S2

D x

S1 Figure 1

To show (2), without loss of generality we assume that i D 1. Let fD1 ; : : : ; D` g be a set of generators of C1 \ Div.X /. Then for every j D 1; : : : ; `, the line through 0 and Dj intersects the segment B1 , and therefore, there exist rational numbers 0  tj  1  b2 and kj > 0 such that Dj D kj .D C .1  b1 /S1 C tj S2 /: Since there is the natural projection resS1 R.X I D1 ; : : : ; D` / ! resS1 R.X; C1 /; it suffices to show that the first ring is finitely generated, and hence that the ring  R1 D resS1 R X I D C .1  b1 /S1 C t1 S2 ; : : : ;  : : : ; D C .1  b1 /S1 C t` S2 is finitely generated. But this follows from Theorem 3.5, as D C .1  b1 /S1 C tj S2 Q KX C A C S1 C .b2 C tj /S2 : Note that, in order to prove the theorem, it is enough to show that R.X; C / is finitely generated. Let i 2 H 0 .X; OX .Si // be sections such that div i D Si , and let R  R.XI S1 ; S2 / be the ring spanned by R.X; C /, 1 and 2 . Then it suffices to show that R is finitely generated. By (2), for i 2 f1; 2g there are finite sets Hi of sections in the rings R.X; Ci / such that resSi R.X; Ci / are generated by the sets f jSi j 2 Hi g, and denote H D f1 ; 2 g [ H1 [ H2 . Let M be the intersection of the cone C with the half-plane fxS1 C yS2 j x C y  M g. Possibly by enlarging H , we may assume that the elements of H generate H 0 .X; G/ for every integral G 2 M.

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We claim that H generates the whole ring R. Indeed, let 2 R. By definition of P   R, we may write D i 1 i 2 i i , where i 2 H 0 .X; OX .Gi // for some integral Gi 2 C and some i ; i 2 N. Thus, it is enough to show that i are generated by the elements in H , and therefore we may assume from the start that 2 H 0 .X; G/ for some integral G 2 C . If 2 M, we conclude by the definition of H . Otherwise, assume that G 2 C1 , the case G 2 C2 being analogous. Then there are 1 ; : : : ; z 2 H and a polynomial ' 2 CŒX1 ; : : : ; Xz  such that jS1 D '. 1jS1 ; : : : ; zjS1 /, so the exact sequence  1

0 ! H 0 .X; OX .G  S1 // ! H 0 .X; OX .G// ! H 0 .S1 ; OS1 .G// gives

 '. 1 ; : : : ; z / D 1 0

for some 0 2 H 0 .X; OX .G  S1 //. Note that G  S1 2 C by (1) above, and we continue with 0 instead of . This “zig-zag” process terminates after finitely many steps, once we reach M (see Figure 2). We are done. y

x+y =M

C2

G C1 M

S2

x

S1

Figure 2

Remark 4.3. In the case of surfaces, the ring R.X; C / which appears in the proof above, can be seen as the ring associated to the positive parts P .D/, for any divisor D 2 C . By [ELMC 06], the finite generation of this ring implies that the function P .D/ is piecewise linear on C and P .D/ is semiample for any Q-divisor D 2 C . This implies finiteness of ample models on the rational polytope B, see [BCHM10].

5 Examples In this final section, we give examples which show that the results presented above are optimal.

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Example 5.1. This example is similar to [Laz04, Example 2.3.3]. Here we show that in Theorem A, even for curves, the assumption of ampleness for the divisor A cannot be replaced by nefness. Let X be an elliptic curve and let A be a non-torsion integral divisor on X of degree 0. Let B1 D 0 and B2  0 be a non-zero Q-divisor such that bB2 c D 0. Note that A is nef and .X; Bi / is canonical, for i 2 f1; 2g. We want to show that the ring R D R.XI KX C A C B1 ; KX C A C B2 / is not finitely generated. To that end, let k be a positive integer such that kB2 is integral. We have that M Rm1 ;m2 ; RD .m1 ;m2 /2N 2

where Rm1 ;m2 D H 0 .X; OX .b.m1 C m2 /A C m2 B2 c//. If R were finitely generated, then the set  D RC f.m1 ; m2 / 2 N 2 j Rm1 ;m2 ¤ 0g would be a rational polyhedral cone, and in particular a closed subset of R2 . However, we have Rm1 ;0 D 0 for all m1 > 0, and Rm1 ;k ¤ 0 for every k > 0 by Riemann–Roch, since .m1 C k/A C kB2 has positive degree. Therefore  D R2C nf.r; 0/ j r > 0g, hence R is not finitely generated. Example 5.2. In this example, we show that in Theorem 3.3, the assumption that .S; C / is canonical cannot be replaced by the weaker assumption that only bC c D 0. Below, we are allowed to take C D B. The construction is similar to Mukai’s flop, see [Tot09] and [Deb01, 1.36]. Let E D OP 1 ˚ OP 1 .1/˚3 and let X D P .E/ with the projection map W P .E/ ! 1 P . Thus, X is a smooth projective 4-fold. Let S ' P 3 be a fibre of and denote  D c1 .OX .1//. Then  is basepoint free by [Laz04, Lemma 2.3.2], and KX D  .KP 1 C det E/  4 D S  4: The linear system j  S j contains smooth divisors S1 ; S2 ; S3 corresponding to the quotients E ! OP 1 ˚ OP 1 .1/˚2 , and it is obvious that S C S1 C S2 C S3 has simple normal crossings. If we denote BD

8 .S1 C S2 C S3 / 9

and

AD

7 1  C S; 3 6

then A is ample and B Q 83 .  S/. Note that bBc D 0, and it is easy to check that .S; BjS / is not canonical. Setting  D S C A C B, we have 1 KX C  Q   S: 2 Let P be the curve in X corresponding to the trivial quotient of E. Then P D S1 \ S2 \ S3 , and we have the relations S P D 1 and S  D 0. In particular, for any sufficiently divisible positive integer m, we have P D Bs jm.  S /j and P  Bs jm.2  S/j  Bs jmj [ Bs jm.  S /j D P:

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Therefore, the base locus of the linear system j2m.KX C /j is the curve P , and hence Fix j2m.KX C /j D 0: We want to show that for m sufficiently divisible, the restriction map H 0 .X; OX .2m.KX C /// ! H 0 .S; OS .2m.KS C .A C B/jS /// is not surjective, thus demonstrating that the assumption of canonicity cannot be removed from Theorem 3.3. Assume the contrary. Then the sequence 0 ! H 0 .X; OX .2m.KX C /  S// ! H 0 .X; OX .2m.KX C /// ! H 0 .S; OS .2m.KS C .A C B/jS /// ! 0 is exact. After some calculations, we have h0 .X; OX .2m.KX C /  S// D h0 .P 1 ; S 2m E.m  1// ! 2m X j C2 D .j  m/ 2 j DmC1

and h0 .X; OX .2m.KX C /// D h0 .P 1 ; S 2m E.m// ! 2m X j C2 D .j  m C 1/: 2 j Dm

Furthermore, one sees that KS C .A C B/jS represents a hyperplane in S , and thus ! m C 3 : h0 .S; OS .m.KS C .A C B/jS /// D h0 .P 3 ; OP 3 .m// D m It is now straightforward to derive a contradiction.

References [ADHL10] I. Arzhantsev, U. Derenthal, J. Hausen, and A. Laface, Cox rings. Book in preparation, preprint (Chapter 1), arXiv:1003.4229v2 [math.AG]. [BCHM10] C. Birkar, P. Cascini, C. Hacon, and J. Mc Kernan, Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23 (2010), no. 2, 405–468. [Bou04]

S. Boucksom, Divisorial Zariski decompositions on compact complex manifolds. Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 1, 45–76.

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[CKL11]

A. Corti,A.-S. Kaloghiros, andV. Lazi´c, Introduction to the Minimal Model Program and the existence of flips. Bull. London Math. Soc. 43 (2011), 415–448.

[CL10a]

P. Cascini and V. Lazi´c, New outlook on the Minimal Model Program, I. Duke Math. J., to appear; preprint, arXiv:1009.3188 [math.AG].

[CL10b]

A. Corti and V. Lazi´c, New outlook on the Minimal Model Program, II. Preprint, arXiv:1005.0614 [math.AG].

[Cor10]

A. Corti, Finite generation of adjoint rings after Lazi´c: an introduction. In Classification of algebraic varieties, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich 2011, 197–220.

[Deb01]

O. Debarre, Higher-dimensional algebraic geometry. Universitext, Springer-Verlag, New York 2001.

[ELMC 06] L. Ein, R. Lazarsfeld, M. Musta¸ta˘ , M. Nakamaye, and M. Popa, Asymptotic invariants of base loci. Ann. Inst. Fourier (Grenoble) 56 (2006), no. 6, 1701–1734. [EP12]

L. Ein and M. Popa, Extension of sections via adjoint ideals. Math. Ann. 352 (2012), no. 2, 373–408.

[FM00]

O. Fujino and S. Mori, A canonical bundle formula. J. Differential Geom. 56 (2000), no. 1, 167–188.

[Hac05]

C. Hacon, Extension theorems and their applications. Lecture notes, Seattle 2005.

[HM10]

C. Hacon and J. Mc Kernan, Existence of minimal models for varieties of log general type II. J. Amer. Math. Soc. 23 (2010), no. 2, 469–490.

[Kaw85]

Y. Kawamata, The Zariski decomposition of log-canonical divisors. In Algebraic geometry, Bowdoin, 1985, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987, 425–433.

[Kaw99]

Y. Kawamata, On the extension problem of pluricanonical forms. In Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), Contemp. Math. 241, Amer. Math. Soc., Providence, RI, 1999, 193–207.

[KM98]

J. Kollár and S. Mori, Birational geometry of algebraic varieties. Cambridge Tracts Math. 134, Cambridge University Press, Cambridge 1998.

[Laz04]

R. Lazarsfeld, Positivity in algebraic geometry. I. Ergeb. Math. Grenzgeb. 48, Springer-Verlag, Berlin 2004.

[Laz09]

V. Lazi´c, Adjoint rings are finitely generated. Preprint, arXiv:0905.2707v3 [math.AG].

[Nak04]

N. Nakayama, Zariski-decomposition and abundance. MSJ Memoirs. 14, Mathematical Society of Japan, Tokyo 2004.

[P˘au07]

M. P˘aun, Siu’s invariance of plurigenera: a one-tower proof. J. Differential Geom. 76 (2007), no. 3, 485–493.

[Sho03]

V. V. Shokurov, Prelimiting flips. Proc. Steklov Inst. Math. 240 (2003), 82–219.

[Siu98]

Y.-T. Siu, Invariance of plurigenera. Invent. Math. 134 (1998), no. 3, 661–673.

[Siu02]

Y.-T. Siu, Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type. In Complex geometry (Göttingen, 2000), Springer-Verlag, Berlin 2002, 223–277.

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[Siu06]

Y.-T. Siu, A general non-vanishing theorem and an analytic proof of the finite generation of the canonical ring. Preprint, arXiv:math/0610740v1 [math.AG]

[Tak06]

S. Takayama, Pluricanonical systems on algebraic varieties of general type. Invent. Math. 165 (2006), no. 3, 551–587.

[Tot09]

B. Totaro, Jumping of the nef cone for Fano varieties. J. Algebraic Geom. 21 (2012), 375–396.

[Zar62]

O. Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface. Ann. of Math. (2) 76 (1962), 560–615.

Paolo Cascini, Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK E-mail: [email protected] Vladimir Lazi´c, Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany E-mail: [email protected]

Invariants of hypersurfaces and logarithmic differential forms Sławomir Cynk and Sławomir Rams

1 Introduction The main goal of this note is to give an overview of several methods to compute certain invariants of hypersurfaces in projective manifolds. The case of Hodge numbers of a smooth hypersurface in a complex projective space was studied in the classical paper of Griffiths ([26]). The first formulae for Hodge numbers of resolutions of singular varieties of dimension three were given by Clemens (the case of nodal double solids – see [6]) and Werner (nodal hypersurfaces in P 4 studied in [48]). Those seminal results admit various generalizations. We focus our interest on two kinds of problems closely related to the above-mentioned facts. First, we want to study hypersurfaces in (or double coverings of) more general projective manifolds. Unfortunately, in most cases under consideration one needs some Botttype vanishing assumptions in order to obtain applicable formulae. The other natural question we want to consider, is the behaviour of Hodge structure of a resolution if we allow certain higher singularities (i.e. other than ordinary double points) on the studied varieties. Then we are interested in invariants of a fixed resolution of singularities. Moreover, we consider only resolutions that are given by a sequence of blow-ups with smooth centers. The main tool we use in the paper are differential forms with logarithmic poles along a divisor, i.e. differential forms ! such that ! and d! have at most simple poles. In the case of a simple normal crossing divisor those forms are well behaved. They form a locally free sheaf that appears in exact sequences given by the Poincaré residue and the restriction map. For the convenience of the reader we collect basic information on logarithmic differential forms in Section 2. Section 3 is devoted to the study of behavior of differential forms under a blow-up with a smooth center. The next section contains an overview of basic properties of Hodge numbers. The first case where behaviour of Hodge numbers of a resolution becomes very subtle are threefolds. For three-dimensional varieties (resp. their resolutions) one has two Hodge numbers that are difficult to study/compute. The numbers in question can be related using the Euler characteristic. In Section 5 we discuss a method to study the difference between the Euler characteristic of a smooth model and the degree of the Fulton–Johnson class, i.e. the Milnor number. 

Research partially supported by MNiSW grant no. N N201 388834.

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Section 6 contains a discussion of infinitesimal deformations of double coverings of algebraic manifolds. The original motivation was that for a Calabi–Yau threefold one of the Hodge numbers (i.e. h1;2 of the manifold in question) equals the dimension of the Kuranishi space. In Section 7 we discuss the defect formulae by Clemens and Werner, and study Hodge numbers of nodal hypersurfaces and nodal double coverings. Last two sections contain an overview of the most general results on Hodge numbers of resolutions of hypersurfaces with A-D-E singularities.

2 Logarithmic differential forms and logarithmic vector fields j Let Y be a reduced divisor on a smooth algebraic manifold X , and let X .Y / stand for the sheaf of differential j -forms on X with at most simple poles along Y .

Definition 2.1 ([14]). A differential j -form with logarithmic poles along Y on an open subset V  X is a meromorphic j -form ! on V regular on V n Y and such that both ! and d! have at most simple poles along Y . Differential j -forms with logarithmic poles along Y form a sheaf denoted by j X .log Y /. For any open subset V  X we have j j j C1 .log Y // D f! 2 X .Y / W d! 2 X .Y /g: .V; X

A normal crossing divisor Y in X is a reduced divisor which is locally defined by an equation of the form f D f1 : : : fp , where f1 ; : : : ; fn are local coordinates for X, p  n. j .log Y / is a locally free sheaf. In this case If Y is a normal crossing divisor then X j a form ! 2 X .log Y / can be written locally in the following way: !D

X

fk1 :::kj ık1 ^    ^ ıkj ;

1k1 <
where ıi D

dfi fi

if i  j , and ıi D dfi if i > j . In particular, we have j X .log Y / D

^j

1 X .log Y /:

If Y is a smooth divisor, then we have the following exact sequences ([21, Proposition 2.3]): 1 1 ! X .log Y / ! OY ! 0; 0 ! X j j 0 ! X ! X .log Y / ! jY1 ! 0;

0 !

j X .log Y /.Y /

!

j X

!

jY

! 0:

(2.1)

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191

j The map X .log Y / ! jY1 in the exact sequence (2.1) is the Poincaré residue j X .log Y / 3 ! ^

df f

7! !jY 2 jY1 ;

where f D 0 is a local equation of Y in X . In particular, for dim X D 4, we get the exact sequence (see [40, p. 444]) 3 3 ! X .log Y / ! 2Y ! 0; 0 ! X

(2.2)

3 .log Y / (see [40, p. 445]): and the following resolution of the sheaf X 3 3 0 ! X .log Y / ! X .Y / ! KX .2Y /=KX .Y / ! 0:

(2.3)

A more detailed exposition of other properties of logarithmic forms can be found in [21]. 1 .log Y / is the sheaf The dual sheaf to the sheaf of differential one-forms X ‚X .log Y / of logarithmic vector fields along Y , that is defined by the following exact sequence (cf. [18, (2.1)]): 0 ! ‚X .log Y / ! ‚X ! NY jX ! 0:

(2.4)

The sheaf ‚X .log Y / is the kernel of the natural restriction map ‚X ! NY jX . Consequently, it is the subsheaf of the tangent bundle ‚X consisting of the vector fields that carry the ideal sheaf of Y into itself.

3 Blow-up In this section we study a fixed resolution of a singular variety by a sequence of blow-ups with smooth centers. Let C  X be a smooth subvariety of codimension k D .n C 1  d / and let  W Xz ! X the blow-up of X along C with the exceptional divisor E. We put N WD NC jY (resp. N _ ) to denote the normal bundle of C in Y (resp. its dual). Moreover, S l N stands for the l-th symmetric power of N . Our strategy in most proofs in next sections will be to consider separately the impact of each blow-up on the Hodge numbers of the studied varieties. In further sections we will sketch certain proofs and omit some computations. Below we collect basic technical facts that are necessary to work out the details (see [1, Theorem I.9.1] and [27, Exercise III.8.4]). Proposition 3.1. We have  OXz Š OX ; Ri  OXz D 0 for i > 0; OXz .E/ ˝ OE Š OE .1/;  .OE .l// Š S l N _ ;

for l  0;

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 .OE .l// D 0;

for l < 0;

i

R  .OE .l// D 0; R

k1

for i 6D 0; k  1;

 .OE .l// D 0;

for l > k;

Rk1  .OE .l// Š S lk N ˝

k ^

N ; for l  k:

Moreover the following “relative Euler sequence” 0 !

pE=C

! 



p ^

 N _ ˝ OE .p/ ! p1 ! 0 E=C

is exact. Proposition 3.2. For a non-negative integer m we have (1)  OXz .mE/ Š OX , (2) Rk1  OXz .mE/ Š

mk L j D0

S j .N / ˝

Vk

N,

(3) Ri  OXz .mE/ D 0, for i 6D 0; k  1.

4 Hodge numbers The Hodge number hp;q .Y / (0  p; q  dim Y ) of a compact complex manifold is defined as the dimension of the Dolbeault cohomology hp;q .Y / D dimC H p;q .Y /;

H p;q .Y / D H q .pY /:

Hodge numbers are usually collected in the Hodge diamond: h0;0 h0;1

h0;n

h1;0

h0;2 h1;1 h2;0 ..................................... … h1;n1 hn;0 hn1;1 ..................................... hn;n2 hn1;n1 hn2;n hn;n1 hn1;n hn;n

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Moreover, Hodge numbers of a projective manifold Y satisfy the following symmetries: hp;q D hq;p hp;q D hnp;nq

Hodge duality, Serre duality.

Furthermore, by the Hodge decomposition one has the equality k X

hi;ki D bk ;

iD0

where bk .Y / WD dimC H k .Y; C/ is the k-th Betti number. If Y is a smooth ample divisor in a projective manifold X , then by Lefschetz’s hyperplane section theorem we have isomorphisms H p;q .Y / Š H p;q .X /;

for p C q  dim Y  1:

Consequently, the Hodge diamond of an n-dimensional complete intersection Y in a projective space P n looks as follows: 1

h0;n

0 0 0 1 0 ........................... … h1;n1 hn;0 hn1;1 ........................... 0

1 0

0 0

1 Moreover, the Euler characteristic of Y can be computed using the Gauss–Bonnet formula. For a degree-d hypersurface in P nC1 one obtains ! n X n C 2 e.Y / D ; where d D dim Y: .1/k d kC1 nk kD0

We can also compute the geometric genus n;0

pg D h

! d 1 .Y / D : nC1

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As a consequence, we get the Hodge diamond of any (smooth) hypersurface of dimension less or equal 3: dim Y D 1 h00 D h11 D 1   h01 D h10 D d 1  2  h02 D h20 D d 1 3

dim Y D 2 h00 D h22 D 1 h01 D h10 D h03 D h30 D 0   h03 D h30 D d 1 4 h11 D h11 D

2d 3 6d 2 C7d 3

dim Y D 3 h00 D h11 D h22 D h33 D 1 h01 D h10 D h02 D h20 D h04 D h40 D h05 D h50 D 0   h03 D h30 D d 1 4 h12 D h21 D

11d 4 50d 3 C85d 2 70d C24 24

The Hodge decomposition of higher dimensional hypersurfaces in projective spaces was given by Griffiths ([26]). Theorem 4.1. If Y D fF D 0g is a smooth degree d hypersurface in P nC1 , then H0p;np .Y / Š .CŒX0 ; : : : ; XnC1 =JF /d.pC1/.nC2/ ; where H0p;q denotes the primitive cohomology, JF is the Jacobian ideal of Y generated by partial derivatives of F . Recall, that the primitive cohomology H0p;q is the kernel of the map H p;q .Y / ! H pC1;qC1 .Y / defined by multiplication with a class of a hyperplane. Consequently, in the above theorem hp;q .Y / D hp;q 0 .Y / hp;p .Y / D hp;p 0 .Y / C 1

unless n is even and p D q D n2 ; if n D 2p:

5 Euler characteristic of a smooth model of a singular hypersurface Let Y be a hypersurface of dimension n in a smooth algebraic manifold X . If Y is smooth then the topological Euler characteristic of Y can be computed using the

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adjunction formula as c.‚X / \ ŒY  D e.Y / D deg c.NY jX /

Z X n

.1/k ck .X /ŒY nk ;

X kD0

where ŒY  is its cohomology class. We are interested in the case when Y is singular. In this situation the topological Euler characteristic of the hypersurface in question is not determined by its cohomology class ŒY . The number Z X n c.‚X / \ ŒY  .1/k ck .X /ŒY nk D eQ .Y / D deg c.NY jX / X kD0

is the degree of the Fulton–Johnson class c FJ (see [23, Example 4.2.6] or [24]), while the Euler characteristic e.Y / is the degree of the Schwartz–MacPherson class c SM (see [31]). The difference (up to a sign convention) of these classes is called the Milnor class (see [39]). In the case of an isolated singularity, the degree of the Milnor class agrees with the classical Milnor number studied in [35]. It equals the codimension of the Jacobian ideal. In the case of higher dimensional singularities, it agrees with the generalization of the Milnor number studied by Parusi´nski in [37] (see also [38], [39]). In [9] we gave a method for computing the difference between the degree of the Fulton–Johnson class eQ .Y / and the Euler characteristic e.Yz / of a non-singular model Yz of Y (in the paper [9] we work in a more general setup of a complete intersection). We shall consider a non-singular model satisfying the following property: there is a sequence of blow-ups with smooth centers  W Xz ! X such that Yz  Xz is the strict transform of Y . If Y is a smooth manifold its Euler characteristic can be computed as an alternating sum of holomorphic Euler characteristics of sheaves of differential forms X .1/i .iY /: e.Y / D i

From the exact sequence 0 ! OX .Y / ! OX ! OY ! 0 and the additivity of the holomorphic Euler characteristic we get .OY / D .OX /  .OX .Y //: Similarly, from the exact sequences (2.1) tensored with powers of OX .Y / we get 1 1 .1Y / D .X /  .X .log Y /.Y //; 1 1 .X .log Y /.Y // D .X .Y // C .OY .Y //; .OY .Y // D .OX .Y //  .OX .2Y //;

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and consequently 1 1 .1Y / D .X /  .X .Y //  .OX .Y // C .OX .2Y //:

More generally, for any locally free sheaf F on X and any p D 0; : : : ; n we have .pY ˝ F / D

p X

  pq pq .1/q .X .qY / ˝ F /  .X ..q C 1/Y / ˝ F / :

qD0

As the above formulae make sense for any divisor Y and (by the Riemann–Roch theorem) depend only on the class of Y , they give the degree of the Fulton–Johnson class eQ .Y /. Our goal is to compute the difference .Qe.Yz /  eQ .Y //. In order to do this we have to study p ..qY ///; .pz ..q Yz ///  .X X

so it is enough to compute the numbers p ˝ L1 / Dp .L; m/ WD .pz ˝   L1 ˝ OXz .mE//  .X X

for an effective line bundle L on X and non-negative integers m, p. Exact formulae for Dp , p  2 are given in [9, Theorem 5]. Applying those formulae we obtain Theorem 5.1 ([9, Theorem 6]). Let Y be a surface in a smooth threefold X ,  W Xz ! X be a blow-up of a smooth irreducible variety C  Y . Denote by Yz the strict transform of Y and by m the multiplicity of Y at a generic point of C . Then 8 ˆ m3 C 2m2 if dim C D 0; ˆ < eQ .Yz /  eQ .Y / D .3m2  2m  1/Y C C .m3 C 1/c1 .N / if dim C D 1: ˆ ˆ : 2 C.m C m/c1 .C / Theorem 5.2 ([9, Theorem 7]). Let Y be a threefold in a smooth fourfold X ,  W Xz ! X be a blow-up of a smooth irreducible variety C  Y . Denote by Yz the strict transform of Y and by m the multiplicity of Y at a generic point of C . Then 8 4 if dim C D 0; m  3m3 C 2m2 C 2m; ˆ ˆ ˆ ˆ ˆ ˆ 3 2 4 3 ˆ if dim C D 1 ˆ.m C 2m /c1 .C / C .m C m ˆ ˆ 3 2 2 ˆ ˆ < Cm  m/c1 .N / C .4m  6m C 2/Y C;  eQ .Yz /  eQ .Y / D  if dim C D 2 m4 C m3 C 2m2 c2 .N /; ˆ ˆ  2   2  2 ˆ ˆ ˆ C m  m c2 .C / C 6m  3m  1 Y C ˆ ˆ     ˆ ˆ 3 2 ˆ C 2m Yc .N / C 3m C 2m C 1 Yc1 .C / C 4m 1 ˆ ˆ  4  3   2 : 2 C m  m c1 .N / C m  m c1 .C /c1 .N /:

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6 Deformations of double coverings An infinitesimal deformation of X is a scheme X 0 flat over the ring of dual numbers D D CŒt=Œt 2  and such that X 0 ˝D C Š X . If the variety X is smooth, then the space of infinitesimal deformations is isomorphic to the cohomology group H 1 ‚X of the tangent bundle ‚X . Let  W X ! Y be a double cover of a smooth algebraic variety branched along a smooth divisor D. The cover  is not determined by D itself, we have also to fix a line bundle L on Y s.t.  OX Š OY ˚ L1 . This L satisfies L˝2 Š OY .D/. Since the map  is finite, we have H i .‚X / Š H i . ‚X /. From [21, Lemma 3.16] we get  ‚X Š ‚Y ˝ L1 ˚ ‚Y .log D/ and so H 1 ‚X Š H 1 .‚Y .log D// ˚ H 1 .‚Y ˝ L1 /: Consequently, we obtain the following proposition describing the deformations of a double covering with smooth branched divisor. Proposition 6.1 ([13, Proposition 2.2]). We have (a) H 1 .‚Y .log D// Š CoKer.H 0 ‚Y ! H 0 NDjY /˚Ker.H 1 ‚Y ! H 1 NDjY /; (b) H 1 .‚Y .log D// is isomorphic to the space TX1!Y of infinitesimal deformations of X which are double covers of deformations of Y ; 1 of infinitesimal (c) CoKer.H 0 ‚Y ! H 0 NDjY / is isomorphic to the space TX=Y deformations of X which are double covers of Y .

Corollary 6.2 ([13], Corollary 2.3). (a) Every deformation of X is a double cover of a deformation of Y iff H 1 .‚Y ˝ L1 / D 0. (b) Every deformation of X is a double cover of Y iff H 1 .‚Y ˝ L1 / D 0 and the map H 1 ‚Y ! H 1 NDjY is injective (e.g. Y is rigid). The situation becomes more complicated when we allow singularities of the branch divisor D. Then the double cover X is singular. We shall consider a resolution of singularities of X obtained by a special embedded resolution of D.P zC z For any birational morphism  W Yz ! Y we have   D D D j nj Ej (where D is the strict transform of D, Ej are the  -exceptional divisors and nj  0). Therefore, the divisor X X j nj k zC D D D Ej D   D  2 Ej 2 j

26 jnj

is reduced and even. In fact, it is the only reduced and even divisor satisfying z  D     D: D Let

 Q Xz ! Yz

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P n be the double cover branched along D  defined by   L ˝ OY . j b 2j cEj /. We can  find a birational morphism Xz ! X that fits into the following commutative diagram: 

Xz

/X 

 Q

 Yz



 / Y.

It follows from the Hironaka desingularization theorem that we can find a sequence of blow-ups with smooth centers  W Yz ! Y such that D  is a smooth divisor. Obviously, such a sequence gives a resolution of singularities of the double cover. Assume that  W Yz ! Y is a sequence  D n1 ı    ı 0 of blow-ups i W YiC1 ! Yi of smooth subvarieties Ci  Yi such that D  is smooth, Y0 D Y , Yn D Yz . Let mi  be the integer such that DiC1 D i Di  mi Ei , where Ei  YiC1 is the exceptional divisor of i . Theorem 6.3 ([13], Theorem 4.1). H 1 .‚Yz .log D  // is isomorphic to the space of simultaneous deformations of D  Y which have simultaneous resolution i.e. which can be lifted to deformations of Ci  Di  Yi in such a way that the multiplicity of the deformation of Di along the deformation of Ci is at least mi . Definition 6.4. We call an infinitesimal deformation of D in Y equisingular if it satisfies the assertion of the above theorem. Theorem 6.3 is particularly useful when we have an explicit description of infinitesimal deformations of Y , for instance when Y is rigid. Corollary 6.5 ([13], Corollary 4.3). If the variety Y is rigid, then the space of equisingular deformations of D in Y is isomorphic to H 1 .‚Yz .log D  //. We can compute H 1 .‚Yz .log D  // explicitly in local coordinates. Let .Ci / stand m for the ideal sheaf of Ci in Yi , and let zi i be (for a nonnegative integer mi ) the pushforward .i1 ı    ı 0 / ..Ci /mi / of the mi -th power of .Ci / to Y . Denote by Ji the image of the homomorphism ‚Yi ˝ ODi ! NDi jYi , and by Jzi its pushforward .i1 ı    ı 0 / .Ji / to Y . Let J stand for the image of the map H 0 .‚Y / ! H 0 NDjY induced by the exact sequence (2.4). Theorem 6.6 ([13, Theorem 4.6]). Under the above assumptions we have 1



H .‚Yz .log D // Š

n1 \

 ı  m H 0 zi i ˝ NDjY C Jzi J:

iD0

In the most interesting case Y D P N this formula can be written in a form more suitable for computations with a computer algebra system. Indeed, define the equisingular

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199

ideal of D in P N (w.r.t. ) as Ieq .D/ D

n1 \

 z i /.mi / C JFi ; I.C

iD0

where Czi is the image of Ci in P n , and JFi is the homogeneous ideal associated to Ji . Theorem 6.7 ([13, Theorem 4.7]). The space of equisingular deformations of D is isomorphic to the space of degree-d forms in the quotient of the equisingular ideal modulo the Jacobian ideal   H 1 .‚Yz .log D  // Š Ieq .D/=JF d : The remaining part of the deformations space H 1 .‚Y ˝ L1 / coming from the deformations that fail to be a double cover is much more difficult to understand. However, in many situations it is easy to compute its dimension. There is a special case where the formula is particularly simple. Proposition 6.8 ([13, Proposition 5.1]). If KY D L1 and  W Yz ! Y is a sequence of blow-ups satisfying the condition 12 D  C KYz D   . 12 D C KY /, then we have the equality X h1 .‚Yz ˝ Lz1 / D h1 .‚Y ˝ L1 / C h0 .KCi /: codim Ci D2

If dim Y D 3, then the above assumptions correspond to a construction of a Calabi– Yau threefold. In this case the second summand coincides with the sum of genera of the blown-up (double and triple) curves. P If we specialize further to the case Y D P 3 and D D 8iD1 Di , where D1 , : : :, D8 are eight planes satisfying the conditions • no six intersect, • no four contain a common line, the above construction gives a Calabi–Yau threefold called a double octic ([10], [34]). For a generic choice of the eight planes the singularities of the octic surface are given by 28 double lines with threefold intersections at 56 triple points. In order to obtain a Calabi–Yau smooth model it suffices to blow-up only the double lines, the triple points do not require any special treatment. At each triple point exactly one of the intersecting planes is blown-up (the one that does not contain the double line through that point which was blown-up first). Consequently the resulting Calabi–Yau threefold is a double covering of the projective space P 3 blown-up 28 times at a line branched along a disjoint sum of 8 planes blown-up 56 times at a point and so its Euler characteristic equals 2.4 C 28  2/  .8  3 C 56/ D 40:

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For special arrangements we have to take into account the number and types of singularities of D ([10, Theorem 2.1]). In this special case the equisingular ideals becomes Ieq D

 multC D IC C JF ;

\ C

the intersection being taken over all multiple curves and points of the arrangement D, and  @F @F JF WD ;:::; @z3 @z0 is the Jacobian ideal of D.

7 Clemens’ and Werner’s defect formulae The first examples of singular hypersurfaces are the ones that have ordinary double points (nodes) as the only singularities. We shall call such hypersurfaces nodal. A threedimensional node admits two kinds of resolutions: the big one (blow-up of the singular point) and small resolutions (blow-ups along analytic smooth surfaces through the node in question). A small resolution replaces a singular point with a smooth rational curve. However, as we blow-up along a local analytic submanifold, the resulting manifold may fail to be projective (the delicate problem of an existence of a projective small resolution of a nodal variety is treated in details in Chapter III of Werner’s thesis [48]). There are two different small resolutions of a node; each of which corresponds to a ruling of the projectivised tangent cone. A big resolution may be obtained from a small resolution by blowing-up the exceptional rational curve. In particular, the Hodge numbers of a small resolution are uniquely determined. A double covering of a smooth algebraic variety branched along a nodal hypersurface is also nodal. In the seminal paper [6] Betti numbers of so-called double solids; i.e. double coverings X of P 3 branched along a nodal surface D of an even degree d , are studied. Clemens proves that certain Betti numbers of a double solid depend not only on the number of nodes but on their position as well. The latter is encoded in the so-called defect. Clemens defines the defect of a nodal double solid as the difference between the second and the fourth Betti number of the singular threefold. Let S WD sing.D/ be the set of nodes. Moreover, let us put V to denote the vector space of degree-. 32 d  4/ homogeneous polynomials on P 3 , and by VS the subspace of V that consists of polynomials vanishing at S . Then (see [6]) one has the equality ı D dim VS  .dim V  /;

where  D #S;

i.e. the defect equals the number of dependent conditions that vanishing on S imposes on the homogeneous polynomials of degree . 32 d  4/.

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The following formula for the Hodge numbers of the big resolution Xz of the nodal double solid X is given in [6]: h1;1 .Xz / D 1 C  C ı;

! ! 3d=2  1 d=2 4   C ı: h1;2 .Xz / D 3 3 Clemens uses topological arguments in his proof. Using basic properties of logarithmic differential forms we can give a simple algebraic proof of the following direct generalization. Theorem 7.1. Let Y be a smooth projective three-dimensional variety, and let D  Y be an ample nodal hypersurface. Moreover, assume that D is even, i.e. there exists a line bundle L on Y such that D D L˝2 in Pic.Y /, and the following equality holds: H 2 .1Y ˝ L1 / D 0: If X is the double covering of Y branched along D and defined by L, and  W Xz ! X is the big resolution, then one has the formula h1;1 .Xz / D h1;1 .Y / C  C ı; where

ı D h0 .L˝3 ˝ KY ˝ S /  .h0 .L˝3 ˝ KY /  /;

and S stands for the ideal of S . Proof. Let  W Xz ! Yz be the double covering of Yz branched along the strict transform z of D. By [21, Lemma 3.16 (d)] we have D z ˝ Lz1 ;  1Xz D 1Yz ˚ 1Yz .log D/ where Lz D   L ˝ OXz .E/, E is the sum of exceptional divisors of  . Since the map  is finite it suffices to prove z ˝ Lz1 / D ı: h1 .1Yz .log D/ By the first exact sequence of (2.1) we have z ˝ Lz1 ! O z ˝ Lz1 ! 0: 0 ! 1Yz ˝ Lz1 ! 1Yz .log D/ D Using Proposition 3.1 we show that  .1z ˝ Lz1 / D 1Y ˝ L1 and Ri  .1z ˝ Y Y Lz1 / D 0 for i  1. By the (degenerate case of) Leray spectral sequence and Nakano vanishing H i .1z ˝ Lz1 / D H i .1Y ˝ L1 / D 0, for i D 1; 2 and so it remains to Y show the equality ı D h1 .ODz ˝ Lz1 /: (7.1)

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Since Lz1 D   L1 ˝ OXz .E/ from Proposition 3.1 and projection formula we get  Lz1 D L1

and

Ri  Lz1 D 0 for i > 0:

So by the Leray spectral sequence H i .Lz1 / D H i .L1 / D 0 for i D 1; 2. Using the cohomology exact sequence associated to 0 ! Lz3 ! Lz1 ! ODz ˝ Lz1 ! 0 we get

H 1 .ODz ˝ Lz1 / Š H 2 .Lz3 /:

Again, using Proposition 3.1 and projection formula we get  Lz3 D L3 ;

R1  Lz3 D 0;

R2  Lz3 D L3 ˝ OS (a sky-scraper sheaf):

Now, the Leray spectral sequence implies h2 Lz3 D   .h3 .L3 /  h3 .Lz3 //: By Serre duality h3 .L3 / D h0 .L˝3 ˝ KY / while h3 .Lz3 / D h0 .Lz˝3 ˝KYz / D h0 .  .L˝3 ˝KY /˝OYz .E// D h0 .L˝3 ˝KY ˝S / and the theorem follows. Nodal hypersurfaces were studied by J. Werner in his PhD thesis ([48]). Using some topological arguments he was able to deduce an analogue of Clemens’ defect formula for nodal hypersurfaces in the projective space P 4 .C/ (see Example 8.1). Using a similar line of arguments as in the proof of Theorem 7.1 one can obtain the following generalization of Werner’s result. We have the following formulae for the Hodge numbers of the big resolution of a nodal threefold hypersurface. Theorem 7.2 ([13, Theorem 2]). Let Y be a nodal hypersurface in a smooth projective four-dimensional manifold satisfying the conditions A1 the line bundle M WD OX .Y / is ample, 1 D 0, A2 H 2 X 1 ˝ M 1 / D 0. A3 H 3 .X

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203

Then 1 h1 1Yy D h1 X C ı;

h2 1Yy D h0 .OX .2Y C KX // C h3 OX  h0 .L0 ˝ KX / 1 1  h4 .X ˝ L1  h3 X 0 /   C ı;

where  is the number of nodes and ı is a non-negative integer called defect equal to the number of dependent equations that vanishing at nodes of S imposes on the global sections of the line bundle .M ˝2 ˝ KX / on X. One can show, that if Yz (resp. Yy ) is the big (resp. a small resolution) of a nodal threefold Y with  nodes, then their Hodge numbers are related by the equalities h1;1 .Yz / D h1;1 .Yy / C ; h1;2 .Yz / D h1;2 .Yy /: In [48] Werner gave also a sufficient and necessary condition for projectivity of a given small resolution. A necessary condition for a nodal threefold hypersurfaces to admit a projective small resolution is the existence of a Weil but not Q-Cartier divisor. Recall that a variety X is called Q-factorial if every Weil divisor on X is Q-Cartier. An easy observation is that a nodal hypersurface in the projective space P 4 (or a nodal double covering of the projective space P 3 ) is Q-factorial iff its defect is zero. The simplest example of a nodal hypersurface of degree d in P 4 is given by a degree-d polynomial of the form pk qd k C rl sd l ; where pk , qd k , rl , sd l are generic homogeneous polynomials of degrees k, d  k, l, d  l respectively (1  k, l  d  1). One easily verifies that F D 0 is degree-d nodal variety with exactly k.d  k/l.d  l/ nodes given by pk D qd k D rl D sd l D 0. As the Hilbert function of the singular locus of X equals .1  t k /.1  t l /.1  t d k /.1  t d l / ; .1  t /5 a simple computation shows that the defect of Y is always one. The above construction is a special case of the following Theorem 7.3 ([29, Theorem 2.1]). Let D  P N be a smooth surface that is a schemetheoretic base locus of a linear system of hypersurfaces of degree d . Then the generic complete intersection Y of N  3 hypersurfaces of degree d containing D is a nodal threefold. The above examples cover the case of complete intersection surfaces in P 4 . In the case when D is a plane, we obtain a degree-d non-factorial hypersurface with .d  1/2

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nodes. Cheltsov ([5]) proved that the above number of nodes is minimal possible: every degree-d nodal hypersurface in P 4 with at most .d  1/2  1 singular points is factorial. For a general case we can use the last formula from Theorem 5.2 to compute the number of nodes and obtain  D c2 .N / C Y 2 D  Yc1 .N /: If we consider degree-d surfaces, where d  6, that are considered in [36], we get ten non Q-factorial quintics with the following number of nodes: deg.D/

KD H



P P1  P1 D1;3

1 2

3 4

16 24

3

3

24

F1

3 4 4

5 4 6

34 36 46

5 5

5 3

50 40

6 6

0 2

36 46

2

D2;2 Veronese PC .E/ Castelnuovo D2;3 Bordiga

For a more detailed account of the above construction see [12]. The question which number of nodes can be realized on a hypersuface of degree d is open. Even the maximal number n .d / of nodes on a degree-d hypersurface in P n remains unknown. The best known upper bound is Varchenko’s spectral bound n .d /  Ar n .D/; where Ar n .D/ is the Arnold number: n o n

˘ P ki D nd Ar n .D/ WD # .k0 ; : : : ; kn / W ki 2 f1; : : : ; d  1g; C 1 : 2 iD0

This bound is sharp in the case of a cubic. For surfaces in P 3 the exact values are known for d  6: 3 .3/ D 4;

3 .4/ D 16;

3 .5/ D 31;

3 .6/ D 65:

The upper bound for the octic surface is 3 .8/  174, whereas the best known example has 168 nodes and was constructed by Endrass ([20]). In higher dimensions much less

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is known. For a quintic hypersurface in P 4 there is the upper bound 4 .5/  135; and the best known example is due to van Straten ([47]). It is a quintic with 130 nodes. The pairs of integers ; ı that can be realized as a number of nodes and the defect of a nodal quintic threefold or a nodal double octic were studied by Borcea ([3]). He uses deformation arguments to prove the following result. Theorem 7.4 ([3]). Let X be a nodal quintic threefold (resp. a double solid ramified along a nodal octic surface) with nodes and defect ı. Then for all but ı integers d 2 f0; 1; : : : ; g, there exists a nodal quintic (resp. nodal double octic) with d nodes. The defect of the Endrass double octic is 19, so all integers smaller than 169 with at most 19 exceptions are realized as the number of nodes of an octic surface. In his thesis Werner computed defect for some examples getting the pairs .108; 0/, .123; 3/, .136; 7/ and .144; 9/, so all integers up to 108 can be realized, there are at most three gaps up to 123 etc. The defect of van Straten’s example is 29 so there are at most 29 gaps in the region below 130. There exist examples with  D 50; ı D 1 (degeneracy locus of a generic .5  5/ matrix of linear forms in x0 ; : : : ; x4 ) and  D 100; ı D 3 (dependency locus of two generic sections in the Horrocks–Mumford bundle). Consequently there is at most 1 gap up to 50 and at most 3 gaps up to 100.

8 The case of A-D-E singularities In this section we deal with hypersurfaces with certain higher singularities. Let Y be a hypersurface in a smooth four-dimensional projective variety X . Moreover, we assume that sing.Y / consists of A-D-E points. In general, A-D-E points can be defined in various ways (see [19]). The following definitions/characterizations of this class are of use for us: According to [19, Characterization C 9], a point P 2 sing.Y / is A-D-E iff we can choose (analytic) coordinates x1;P ; : : : ; x4;P centered at P such that the germ of Y at P is given by the semiquasihomogenous equation 2 C F .x1;P ; x2;P ; x3;P ; x4;P / D 0; n.x1;P ; x2;P ; x3;P / C x4;P

(8.1)

where n.x1;P ; x2;P ; x3;P / is the normal form of the equation of a two-dimensional AD-E singularity and F .x1;P ; x2;P ; x3;P ; x4;P / is a polynomial of order strictly greater than 1 with respect to the weights wn .x1;P /, wn .x2;P /, wn .x3;P /, wn .x4;P / given in

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the table below: n.x1 ; x2 ; x3 /

(wn .x1 /; : : : ; wn .x4 //

Am , m  1

x1mC1 C x22 C x32

1 . mC1 ; 12 ; 12 ; 12 /

Dm , m  4

x1  .x22 C x1m2 / C x32

1 m2 . m1 ; 2.m1/ ; 12 ; 12 /

E6

x14 C x23 C x32

. 14 ; 13 ; 12 ; 12 /

E7

x13  x2 C x23 C x32

. 29 ; 13 ; 12 ; 12 /

E8

x15 C x23 C x32

. 15 ; 13 ; 12 ; 12 /

(8.2)

In particular the singularities of Y are absolutely isolated, i.e. we have the big resolution  W Yz ! Y of the threefold Y ([19, p. 137]) obtained as the composition  D n ı    ı 1 W Yz ! Y DW Yz 0 ;

(8.3)

where Yz WD Yz n is smooth and j W Yz j ! Yz j 1 , for j D 1; : : : ; n, is the blow-up with the center sing.Yz j 1 / ¤ ;, that consists of isolated points. By direct computation, the singularities of Yz j are double points for each j  n  1. The number of singular points (different from P ) which are infinitely near P is as follows: Am , m  1

Dm , m  4

E6

E7

E8

dm=2e  1

2  bm=2c  1

3

6

7

(8.4)

By [46, Theorem 1] the above property of the big resolution characterizes A-D-E singularities: if P 2 Y is an absolutely isolated double point (i.e. P is an isolated double point and all singularities infinitely near P are isolated double points), then it is an A-D-E singularity. Let Xz 0 WD X and let Xz j stand for the fourfold obtained from Xz j 1 by blowing composition of the it up along sing.Yz j 1 /, j D 1; : : : ; n. We put Xz WD Xz n . TheP blow-ups in question is denoted by  W Xz ! X . Moreover, let l kl El WD KX=X z P  z and let  Y D Y C l ml El , where El are (reduced) components of the exceptional locus of  . In order to generalize the notion of defect one defines the following sheaf ([11, Definition 2.1]): IY WD  .OXz ..kl  2ml /El // (8.5) Then the defect of the hypersurface Y is defined as the integer ıY D h0 .KX .2Y / ˝ IY /  .h0 .KX .2Y //  Y /;

(8.6)

where Y stands for the number of singularities and infinitely near singularities of Y . The motivation for the choice of the sheaf IY in the above definition becomes clear, if

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one applies the projection formula to the map  and the sheaf X KXz C 2Yz    .KX C 2Y / C .kl  2ml /El ; to obtain the equality h0 .KXz .2Yz // D h0 .KX .2Y / ˝ IY /. One can follow a more direct approach (see [43, 3]) and consider the space VY of sections H 2 H 0 .KX .2Y / ˝ sing.Y / / such that • if P 2 Y is an Am point, with m  1 then

@j H j @x1;P

.P / D 0 for j  dm=2e  1;

• for every Dm singularity of P 2 Y , where m  4, one has @j H @H .P / D j .P / D 0 for j  bm=2c  1; @x2;P @x1;P .P / D • if P is an Em point, where m D 6; 7; 8, then @x@H 2;P m  5;

@j H j @x1;P

.P / D 0 for j 

where x1;P ; : : : ; x4;P are analytic local coordinates centered at the point P such that the hypersurface Y is given near P by the semiquasihomogenous equation (8.1). By [43, Lemma 3.3] (see also [ibid., (4.2)]), the defect of Y can be expressed as ıY D dim.VY /  h0 .KX .2Y // C Y : In particular (for A-D-E singularities) we have kl  2ml and the sheaf IY is indeed a sheaf of ideals. Using the properties of logarithmic differential forms (see Section 2 and Section 3) one obtains the following formulae for Hodge numbers: Theorem 8.1 ([11, Theorem 2.4]). Let X be a smooth projective fourfold, and let Y  X be a hypersurface with A-D-E singularities. If 1 1 h2 .X / D h3 .X .Y // D h3 .OX .Y // D h2 .OX .Y // D 0 ;

then 1 1 h1;1 .Yz / D h1;1 .X / C ..X .Y //  h4 .X .Y ///  ..OX .2Y //

 h4 .OX .2Y ///  2h1 .OX .Y // C Y C ıY ; h1;2 .Yz / D h4;1 .X / C h0;2 .X / C h0 .KX .2Y //  h3;1 .X / 1  h4 .X .Y //  h0 .KX .Y //  Y C ıY ;

where ıY (resp. Y ) is the defect (resp. the number of singularities and infinitely near singularities) of Y .

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Let us comment on the proof of the above result. The reasoning consists of three steps: 3 1 (and its twists) instead of X (resp. Step 1. The Serre duality is applied to study X its twists).

Step 2. One uses the Leray spectral sequence to compare the cohomologies of various twisted sheaves of differentials on X and on the blow-up Xz . Step 3. The Hodge numbers of the big resolution Yz are computed via Poincaré residue on the blow-up Xz . The assumption on singularities of Y implies vanishing of certain higher direct image sheaves, which in particular yields the equality 3 .Y //: h1 .3Xz .Yz // D h1 .X

Once one knows that h1 .3z .Yz //, h2 .3z / vanish, one can see that the exact sequence X X of cohomology associated to the Poincaré residue (resp. to the resolution of the sheaf of logarithmic differentials – see (2.1)) breaks into shorter exact sequences. The advantage of replacing one-forms with three-forms becomes apparent when we recall the resolution (2.3) of the sheaf of logarithmic three-forms. Details of the proof can be found in [11]. It should be pointed out, however, that the above formulae cease to hold once we weaken the assumptions (see [11, Example 3.8]). In order to see how Theorem 8.1 works, let us consider the following example. Example 8.1. Let Y  P4 be a degree-d hypersurface with A-D-E singularities, where d  3. Recall that VY was defined (see the paragraph preceding 8.1) as the space of degree-.2d  5/ polynomials that vanish along sing.Y / and such that some of their partial derivatives vanish in every Am (resp. Dm , Em ) point of Y . Theorem 8.1 implies the equalities ! 2d  1 1;1 z h .Y / D 1 C 2  Y C dim.VY /  ; 4 ! d 1;2 z : h .Y / D dim.VY /  5  4 In particular, for a nodal hypersurface we regain Werner’s formula [48, Satz on p. 27]. In general, big (divisorial) resolutions perturb the canonical class: if one starts from a (singular) hypersurface with trivial canonical class (e.g. Calabi–Yau threefold), one obtains a smooth threefold without that property. That is why small resolutions are of interest. As we already mentioned in Section 7 small resolutions exist for threedimensional nodal hypersurfaces. One can show that analogous resolutions can be also constructed for certain higher three-dimensional singularities. If we suppose that there

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exists a proper holomorphic map O W Yy ! Y such that Yy is smooth , O jYy nO 1 .sing.Y // is an isomorphism onto the image and the exceptional set Ey WD O 1 .sing.Y // is a curve, then O (and sometimes Yy ) is called a small resolution of Y . By [22, Theorem 1.3] (see also [45]) it suffices to assume that Y has Gorenstein singularities to show that the exceptional set Ey consists of smooth rational curves meeting transversally. Let us put Ez to denote the exceptional divisor of the big resolution. Then, the following simple lemma can be applied to use Theorem 8.1 to compute Hodge numbers of small resolutions. z C/ D 0 and Yy is Proposition 8.2 ([43, Proposition 6.1]). If h1 .OY / D 0, h3 .E; Kähler, then z C/ : h2;2 .Yz / D h2;2 .Yy / C h4 .E;

9 Further generalizations A large class of ambient varieties X where the result from the previous section can be applied consists of toric varieties. Indeed one has Bott-type formulae (see e.g. [32]) for cohomology of various twisted sheaves of differentials, even if the considered ambient space is singular. Having that in mind we assume now that X is a four-dimensional normal complex variety, so the canonical (Weil) divisor KX is well defined (up to the linear equivalence). We have one-to-one correspondence between the linear equivalence classes of Weil divisors and isomorphism classes of rank-1 reflexive sheaves on X: D ! OX .D/: x 3 WD j 3 ; where j W reg.X / ! X stands for the inclusion, to denote We put  X reg.X/ the Zariski sheaf of germs of 3-forms. In order to obtain a direct generalization of Theorem 8.1 one has to assume that Y  X is a hypersurface with A-D-E singularities such that sing.X / \ Y D ;: (9.1) In particular, the above assumptions assure that the defect ıY of Y is well defined. One has the following theorem: Theorem 9.1 ([11, Theorem 3.2]). Let Y  X satisfy the assumptions of this section. If 3 3 xX xX h1 .OX .Y C KX // D h2 . / D h1 . .Y / D 0

and h2 .OX .Y // D h3 .OX .Y // D 0;

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then the following equalities hold: 3 3 3 xX xX xX / C .. .Y //  h0 . .Y /// C ..OX .Y C KX // h1;1 .Yz / D h3 . 3 xX /  ..OX .2Y C KX //  h0 .OX .Y C KX /// C h1 .OX /  h4 .

 h0 .OX .2Y C KX ///  h1 .OX .Y // C Y C ıY ; 3 3 xX xX / C h2 .OX / C h0 .OX .2Y C KX //  h1 . / h1;2 .Yz / D h0 . 0 0 x3  h .OX .Y C KX //  h .X .Y //  Y C ıY :

Here, we no longer assume X to be Cohen–Macaulay, so we cannot apply the Serre duality. The proof consists of steps 2, 3 of the proof of Theorem 8.1. Indeed, the assumption (9.1) enables us to work with logarithmic 3-forms as in the smooth case. Again, as in the previous section the first three vanishings are essential. The others are needed to control the Hodge numbers h1;0 .Yz /, h2;0 .Yz /. In particular, in toric case, many summands in Theorem 9.1 vanish and one arrives at the following result. Corollary 9.2 ([11, Corollary 3.5]). Let X be a complete toric fourfold, and let Y  X be a hypersurface with A-D-E singularities such that sing.X / \ Y D ;. If OX .Y / is ample, then 3 xX / C  Y C ıY ; h1;1 .Yz / D h3 . 1;2 z 0 3 xX h .Y / D h .OX .2Y C KX //  h0 .OX .Y C KX //  h0 . .Y //  Y C ıY :

It is an interesting exercise to see what generalizations of the classical Clemens formula for double solids one can derive from Corollary 9.2. Such generalizations can be found in [43, 5]. In view of recent progress concerning study of behaviour of reflexive differential forms on singular varieties (see e.g. [25]), one should ask what is natural set-up for generalization of Theorem 9.1. In particular, it seems natural to ask to what extent the assumption (9.1) can be weakened. In general, a residue map does not have to exist (see [25, 11.B]). However, [ibid., Thm. 11.7] suggests that a generalization (possibly with extra correction summands) can be obtained provided .X; Y / is a dlt pair.

References [1] W. Barth, K. Hulek, C. Peters, and C. Van de Ven, Compact complex surfaces. Ergeb. Math. Grenzgeb. 4, Springer-Verlag, Berlin 2004. [2] V. Batyrev and D. A. Cox, On the Hodge structure of projective hypersurfaces in toric varieties. Duke Math. J. 75 (1994), 293–338. [3] C. Borcea, Nodal quintic threefolds and nodal octic surfaces. Proc. Amer. Math. Soc. 109 (1990), no. 3, 627–635.

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[4] I. Cheltsov, On factoriality of nodal threefolds. J. Algebraic Geom. 14 (2005), 663–690. [5] I. Cheltsov, Factorial threefold hypersurfaces. J. Algebraic Geom. 19 (2010), no. 4, 781–791. [6] C. H. Clemens, Double solids. Adv. in Math. 47 (1983), 107–230. [7] S. Cynk, Cohomologies of a double covering of a non-singular algebraic 3-fold. Math. Z. 240 (2002), 731–743. [8] S. Cynk, Defect of a nodal hypersurface. Manuscripta Math. 104 (2001), 325–331. [9] S. Cynk, Euler characteristic of a complete intersection. In Complex and differential geometry (W. Ebeling, K. Hulek, K. Smoczyk, eds.), Springer Proc. Math. 8, Springer-Verlag, Berlin 2011, 99–114 [10] S. Cynk and C. Meyer, Geometry and arithmetic of certain double octic Calabi-Yau manifolds. Canad. Math. Bull. 48 (2005), no. 2, 180–194. [11] S. Cynk and S. Rams, Defect via logarithmic differential forms. Math. Nachr. 284 (2011), no. 17–18, 2148–2158. [12] S. Cynk and S. Rams, On the geometry of certain Calabi–Yau threefolds containing a smooth surface. In preparation. [13] S. Cynk and D. van Straten, Infinitesimal deformations of double covers of smooth algebraic varieties. Math. Nachr. 279 (2006), 716–726. [14] P. Deligne, Equations différentielles à points singulieres réguliers. Lecture Notes in Math. 163, Springer-Verlag, Berlin 1970. [15] V. Di Gennaro and D. Franco, Factoriality and Néron-Severi groups. Commun. Contemp. Math. 10 (2008), 745–764. [16] A. Dimca, Betti numbers of hypersurfaces and defects of linear systems. Duke Math. J. 60 (1990),285–294. [17] A. Dimca, Singularities and topology of hypersurfaces. Universitext, Springer-Verlag, New York 1992. [18] I. V. Dolgachev, Logarithmic sheaves attached to arrangements of hyperplanes. J. Math. Kyoto Univ. 47 (2007), no. 1, 35–64. [19] A. H. Durfee, Fifteen characterizations of rational double points and simple critical points. Enseign. Math. (2) 25 (1979), 132-163. [20] S. Endrass, A projective surface of degree eight with 168 nodes. J. Algebraic Geom. 6 (1997), no. 2, 325–334. [21] H. Esnault and E. Viehweg, Lectures on vanishing theorems. DMV Seminar 20, Birkhäuser, Basel 1992. [22] R. Friedman, Simultaneous resolution of threefold double points. Math. Ann. 274 (1986), 671–689. [23] W. Fulton, Intersection theory. Ergeb. Math. Grenzgeb. (3) 2, Springer-Verlag, Berlin 1984. [24] W. Fulton and K. Johnson, Canonical classes on singular varieties. Manuscripta Math. 32 (1980), no. 3-4, 381–389.

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[25] D. Greb, S. Kebekus, S. J. Kovacs, and T. Peternell, Differential forms on log canonical spaces. Publ. Math. Inst. Hautes Études Sci. 114 (2011), 87–169. [26] P. A. Griffiths, On the periods of certain rational integrals. I, II. Ann. of Math. (2) 90 (1969), 460–495; ibid. 90 (1969), 496–541. [27] R. Hartshorne, Algebraic geometry. Grad. Texts Math. 52, Springer-Verlag, NewYork 1977. [28] K. Hulek and R. Kloosterman, Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces. Ann. Inst. Fourier 61 (2011), no. 3, 1133–1179. [29] G. Kapustka, Primitive contractions of Calabi-Yau threefolds. II. J. London Math. Soc. (2) 79 (2009), no. 1, 259–271. [30] J. Kollár, S. Mori, Birational geometry of algebraic varieties. Cambridge Tracts in Math. 134, Cambridge University Press, Cambridge 1998. [31] R. MacPherson, Chern classes for singular algebraic varieties. Ann. Math. 100 (1974), 423–432. [32] A. Mavlyutov, Cohomology of rational forms and a vanishing theorem on toric varieties. J. Reine Angew. Math. 61 (2008), 45–58. [33] J. McCleary, User’s guide to spectral sequences. Cambridge Stud. Adv. Math. 58, Cambridge University Press, Cambridge 2001. [34] C. Meyer, Modular Calabi-Yau threefolds. Fields Inst. Monogr. 22, Amer. Math. Soc., Providence, RI, 2005. [35] J. Milnor, Singular points of complex hypersurfaces. Ann. of Math. Stud. 61, Princeton University Press, Princeton, NJ, 1968. [36] C. Okonek, Moduli reflexiver Garben und Flächen von kleinem Grad in P 4 . Math. Z. 184 (1983), no. 4, 549–572. [37] A. Parusi´nski, A generalization of Milnor number. Math. Ann. 281 (1988), 247–254. [38] A. Parusi´nski and P. Pragacz, A formula for the Euler characteristic of singular hypersurfaces. J. Algebraic Geom. 4 (1995), 337–351. [39] A. Parusi´nski and P. Pragacz, Characteristic classes of hypersurfaces and characteristic cycles. J. Algebraic Geom. 10 (2001), 63–79. [40] C. Peters and J. Steenbrink, Infinitesimal variations of Hodge structure and the generic Torelli problem for projective hypersurfaces. In Classification of algebraic and analytic manifolds (Katata, 1982), Progr. Math. 39, Birkhäuser, Boston, Mass., 1983, 399–463. [41] C. Peters and J. Steenbrink, Mixed Hodge structures. Ergeb. Math. Grenzgeb. (3) 52, Springer-Verlag, Berlin 2008. [42] V. V. Przhiyalkovskii, I. Cheltsov, and K. A. Shramov, Hyperelliptic and trigonal Fano threefolds. Izv. Ross. Akad. Nauk Ser. Mat. 69 (2005), 145–204; English transl. Izv. Math. 69 (2005), no. 2, 365–421. [43] S. Rams, Defect and Hodge numbers of hypersurfaces. Adv. Geom. 8 (2008), 257–288. [44] M. Reid, Canonical 3-folds. In Journées de géométrie algébrique d’Angers Journees de geometrie algebrique, Angers/France 1979, Sijthoff & Noordhoff, Alphen aan den Rijn; Germantown, Md., 1980 273–310.

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[45] M. Reid, Minimal models of canonical 3-folds. In Algebraic varieties and analytic varieties, Adv. Stud. Pure Math. 1 (1983), 131–180. [46] R. Treger, Rational singularities in dimension  2. In Algebraic geometry, Lecture Notes in Math. 732, Springer-Verlag, Berlin 1979, 592–604. [47] D. van Straten, A quintic hypersurface in P4 with 130 nodes. Topology 32 (1993), 857–864. [48] J. Werner, Kleine Auflösungen spezieller dreidimensionaler Varietäten. Bonner Math. Schriften 186 (1987). Sławomir Cynk, Institute of Mathematics, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków; and ´ Institute of Mathematics of the Polish Academy of Sciences, ul. Sniadeckich 8, 00-956 Warszawa, Poland E-mail: [email protected] Sławomir Rams, Institute of Mathematics, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland E-mail: [email protected]

Prym varieties and their moduli Gavril Farkas

1 Prym, Schottky and 19th century theta functions Prym varieties are principally polarized abelian varieties associated to étale double covers of curves. They establish a bridge between the geometry of curves and that of abelian varieties and as such, have been studied for over 100 years, initially from an analytic [Wi], [SJ], [HFR] and later from an algebraic [M] point of view. Several approaches to the Schottky problem are centered around Prym varieties, see [B1], [D2] and references therein. In 1909, in an attempt to characterize genus g theta functions coming from Riemann surfaces and thus solve what is nowadays called the Schottky problem, F. Schottky and H. Jung, following earlier work of Wirtinger, associated to certain two-valued Prym differentials on a Riemann surface C new theta constants which then they related to the classical theta constants, establishing what came to be known as the Schottky–Jung relations. The first rigorous proof of the Schottky–Jung relations has been given by H. Farkas [HF]. The very definition of these differentials forces one to consider the parameter space of unramified double covers of curves of genus g. The aim of these lectures is to discuss the birational geometry of the moduli space Rg of Prym varieties of dimension g  1. Prym varieties were named by Mumford after Friedrich Prym (1841–1915) in the very influential paper [M] in which, not only did Mumford bring to the forefront a largely forgotten part of complex function theory, but he developed an algebraic theory of Prym’s, firmly anchored in modern algebraic geometry. In particular, Mumford gave a simple algebraic proof of the Schottky–Jung relations. To many algebraic geometers Friedrich Prym is a little-known figure, mainly because most of his work concerns potential theory and theta functions rather than algebraic geometry. For this reason, I find it appropriate to begin this article by mentioning a few aspects from the life of this interesting transitional character in the history of the theory of complex functions. Friedrich Prym was born into one of the oldest business families in Germany, still active today in producing haberdashery articles. He began to study mathematics at the University of Berlin in 1859. After only two semesters, at the advice of Christoffel,  These notes are based on lectures delivered in July 2010 in Bedlewo at the IMPANGA Summer School on Algebraic Geometry and in January 2011 in Luminy at the annual meeting Géométrie Algébrique Complexe. I would like to thank Piotr Pragacz for encouragement and for asking me to write this paper in the first place, Maria Donten-Bury and Oskar Kedzierski for writing-up a preliminary version of the lectures, as well as Herbert Lange for pointing out to me the historical figure of F. Prym. This work was finalized during a visit at the Isaac Newton Institute in Cambridge.

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he moved for one year to Göttingen in order to hear Riemann’s lectures on complex function theory. The encounter with Riemann had a profound effect on Prym and influenced his research for the rest of his life. Having returned to Berlin, in 1863 Prym successfully defended his doctoral dissertation under the supervision of Kummer. The dissertation was praised by Kummer for his didactic qualities, and deals with theta functions on a Riemann surface of genus 2. After a brief intermezzo in the banking industry, Prym won professorships first in Zürich in 1865 and then in 1869 in Würzburg, at the time a very small university. In Prym’s first year in Würzburg, there was not a single student studying mathematics, in 1870 there were only three such students and in the year after that their number increased to four. Prym stayed in Würzburg four decades until his retirement in 1909, serving at times as Dean and Rector of the University. In 1872 he turned down a much more prestigious offer of a Chair at the University of Strasbourg, newly created after the Franco–German War of 1870–71. He did use though the offer from Strasbourg in order to improve the conditions for mathematical research in Würzburg. In particular, following more advanced German universities like Berlin or Göttingen, he created a Mathematisches Seminar with weekly talks. By the time of his retirement, the street in Würzburg on which his house stood was already called Prymstrasse. Friedrich Prym was a very rich but generous man. According to [Vol], he once claimed that while one might argue that he was a bad mathematician, nobody could ever claim that he was a bad businessman. In 1911, Prym published at his own expense in 1000 copies his Magnum Opus [PR]. The massive 550 page book, written jointly with his collaborator Georg Rost 1 explains the theory of Prym functions, and was distributed by Prym himself to a select set of people. In 1912, shortly before his death, Prym created the Friedrich Prym Stiftung for supporting young researchers in mathematics and endowed it with 20000 Marks, see again [Vol]. Unfortunately, the endowment was greatly devalued during the inflation of the 1920s. We now briefly discuss the role Prym played in German mathematics of the 19th century. Krazer [Kr] writes that after the death of both Riemann and Roch in 1866, it was left to Prym alone to continue explaining “Riemann’s science” (“...Prym allein die Aufgabe zufiel, die Riemannsche Lehre weiterzuführen”). For instance, the paper [P1] published during Prym’s years in Zürich, implements Riemann’s ideas in the context of hyperelliptic theta functions on curves of any genus, thus generalizing the results from Prym’s dissertation in the case g D 2. This work grew out of long conversations with Riemann that took place in 1865 in Pisa, where Riemann was unsuccessfully trying to regain his health. During the last decades of the 19th century, Riemann’s dissertation of 1851 and his 1857 masterpiece Theorie der Abelschen Funktionen, developed in staggering gener1 Georg Rost (1870–1958) was a student of Prym’s and became Professor in Würzburg in 1906. He was instrumental in helping Prym write [PR] and after Prym’s death was expected to write two subsequent volumes developing a theory of n-th order Prym functions. Very little came to fruition of this, partly because Rost’s interests turned to astronomy. Whatever Rost did write however, vanished in flames during the bombing of Würzburg in 1945, see [Vol].

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ality, without examples and with cryptic proofs, was still regarded with mistrust as a “book with seven seals” by many people, or even with outright hostility by Weierstrass and his school of complex function theory in Berlin 2 . In this context, it was important to have down to earth examples, where Riemann’s method was put to work. Prym’s papers on theta functions played precisely such a role. The following quote is revelatory for understanding Prym’s role as an interpreter of Riemann. Felix Klein [Kl] describes a conversation of his with Prym that took place in 1874, and concerns the question whether Riemann was familiar with the concept of abstract manifold or merely regarded Riemann surfaces as representations of multivalued complex functions. The discussion seems to have played a significant role towards crystalizing Klein’s view of Riemann surfaces as abstract objects: “Ich weiss nicht, ob ich je zu einer in sich abgeschlossenen Gesamtauffassung gekommen wäre, hätte mir nicht Herr Prym vor längeren Jahren eine Mitteilung gemacht, die immer wesentlicher für mich geworden ist, je länger ich über den Gegenstand nachgedacht habe. Er erzählte mir, dass die Riemannschen Flächen ursprünglich durchaus nicht notwendig mehrblättrige Flächen über der Ebene sind, dass man vielmehr auf beliebig gegebenen krummen Flächen ganz ebenso komplexe Funktionen des Ortes studieren kann, wie auf den Flächen über der Ebene” (“I do not know if I could have come to a self-contained conception [about Riemann surfaces], were it not for a discussion some years ago with Mr. Prym, which the more I thought about the subject, the more important it became to me. He told me that Riemann surfaces are not necessarily multi-sheeted covers of the plane, and one can just as well study complex functions on arbitrary curved surfaces as on surfaces over the plane”)3 . Prym varieties (or rather, theta functions corresponding to Prym varieties) were studied for the first time in Wirtinger’s monograph [Wi]. Among other things, Wirtinger observes that the theta functions of the Jacobian of an unramified double covering split into the theta functions of the Jacobian of the base curve, and new theta functions that depend on more moduli than the theta functions of the base curve. The first forceful important application of the Prym theta functions comes in 1909 in the important paper [SJ] of Schottky and his student Jung. Friedrich Schottky (1851–1935) received his doctorate in Berlin in 1875 under Weierstrass and Kummer. Compared to Prym, Schottky is clearly a more important and deeper mathematician. Apart from the formulating the Schottky Problem and his results on theta functions, he is also remembered today for his contributions to Fuchsian 2

Weierstrass’ attack centered on Riemann’s use of the Dirichlet Minimum Principle for solving boundary value problems. This was a central point in Riemann’s work on the mapping theorem, and in 1870 in front of the Royal Academy of Sciences in Berlin, Weierstrass gave a famous counterexample showing that the Dirichlet functional cannot always be minimized. Weierstrass’ criticism was ideological and damaging, insofar it managed to create the impression, which was to persist several decades until the concept of Hilbert space emerged, that some of Riemann’s methods are not rigorous. We refer to the beautiful book [La] for a thorough discussion. Note that Prym himself wrote a paper [P2] providing an example of a continuous function on the closed disc, harmonic on the interior and which contradicts the Dirichlet Principle. 3 It is amusing to note that after this quote appeared in 1882, Prym denied having any recollection of this conversation with Klein.

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groups, for the Schottky Groups, as well as for a generalization of Picard’s Big Theorem on analytic functions with an essential singularity. To illustrate Schottky’s character, we quote from two remarkable letters that Weierstrass wrote. To Sofja Kowalewskaja he writes [Bo]: “... [Schottky] is of a clumsy appearance, unprepossessing, a dreamer, but if I am not completely wrong, he possesses an important mathematical talent”. The following is from a letter to Hermann Schwarz [Bi]: “... [Schottky] is unsuited for practical life. Last Christmas he was arrested for failing to register for military service. After six weeks however he was discharged as being of no use whatsoever to the army. [While the army was looking for him] he was staying in some corner of the city, pondering about linear differential equations whose coefficients appear also in my theory of abelian integrals. So you see the true mathematical genius of times past, with other inclinations” (“... das richtige mathematische Genie vergangener Zeit mit anderen Neigungen”). Schottky was Professor in Zürich and Marburg, before returning to the University of Berlin in 1902 as the successor of Lazarus Fuchs4 . Schottky remained at the University of Berlin until his retirement in 1922. Due to his personality he could neither attract students nor play a leading role in the German mathematical life and his appointment can be regarded as a failure that accentuated Berlin’s mathematical decline in comparison with Göttingen. The paper [SJ] deals with the characterization of theta-constants # Œ ı  .; 0/ of period matrices  2 Hg in the Siegel upper-half space that correspond to Jacobians of algebraic curves of genus g. Schottky and Jung start with a characteristic of genus g  1, that is a pair ; ı 2 f0; 1gg1 and note that if one completes this characteristic to one of genus g by adding one column in two possible ways, the product of the two theta constants satisfies the following proportionality relation: #2

h  i h  .…; 0/  # ı ı

h  0 i .; 0/  # 0 ı

0 i .; 0/: 1

Here  2 Hg is the period matrix of a Jacobian of a genus g curve but the novelty is that … 2 Hg1 no longer corresponds to any Jacobian but rather to a Prym variety, constructed from an unramified double covering of the curve whose period matrix is  . This allows one to obtain theta relations for Jacobians in genus g starting with any theta relation in genus g  1 (for instance Riemann’s theta formula). Schottky himself carried out this approach for g D 4.

4 Fuchs’ chair was offered initially to Hilbert, but he declined preferring to remain in Göttingen after the university, in an effort to retain him, created a new Chair for his friend Hermann Minkowski. It was in this way that Schottky, as second on the list, was controversially hired at the insistence of Frobenius, and despite the protests of the Minister, who (correctly) thought that Schottky’s teaching was totally inadequate (“durchaus unbrauchbar”) and would have preferred that the position be offered to Felix Klein instead [Bo]. Schottky could never be asked to teach beginner’s courses, not even in the dramatic years of World War I.

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2 The moduli space of Prym varieties The main object of these lectures is the moduli space of unramified double covers of genus g curves, that is, the parameter space ˚  Rg D ŒC;  W C is a smooth curve of genus g;  2 Pic0 .C /  fOC g; ˝2 D OC : In its modern guise, that is, as a coarse moduli space representing a stack, this space appears for the first time in Beauville’s influential paper [B2]. The choice of the name corresponds to the French word revêtement. We begin by recalling basic facts about the algebraic theory of Prym varieties. As a general reference we recommend [ACGH], Appendix C, [BL], Chapter 12 and especially [M]. We fix an integer g  1 and denote by Hg WD f 2 Mg;g .C/ W  Dt ; Im  > 0g the Siegel upper half-space of period matrices for abelian varieties of dimension g, hence Ag WD Hg =Sp2g .Z/. The Riemann theta function with characteristics Œ ı  is defined as the holomorphic function # W Hg  C g ! C, where  h  i X        exp  i t m C 2  m C 2 C 2 i t m C 2 z C 2ı ; # .; z/ WD ı g m2Z

where  D .1 ; : : : ; g /, ı D .ı1 ; : : : ; ıg / 2 f0; 1gg . For any period matrix  2 Hg , the pair  n h 0 i o Cg A WD g ; ‚ WD # .; z/ D 0  0 Z C   Zg defines a principally polarized abelian variety, that is, ŒA ; ‚  2 Ag . To a smooth curve C of genus g, via the Abel–Jacobi isomorphism Pic0 .C / D

H 0 .C; KC /_ ; H1 .C; Z/

(1)

one associates a period matrix as follows. Let .˛1 ; : : : ; ˛g ; ˇ1 ; : : : ; ˇg / a symplectic 0 basis R by .!1g; : : : ; !g / the basis of H .C; KC / characterized R of H1 .C; Z/, and denote by ˛i !j D ıij . Then  WD ˇi !j i;j D1 2 Hg is a period matrix associated to C and     # 00 .; 0/ is the theta constant associated to C . The theta function # 00 .; z/ is the (up to scalar multiplication) unique section of the bundle OA .‚ /. The theta functions with characteristic # Œ ı  .; z/ are the unique sections of the 22g symmetric line bundles on A algebraically equivalent to OA .‚ /. A very clear modern discussion of basics of theta functions can be found in [BL] Chapter 3. For the case of Jacobians we also recommend [Fay]. Using the Abel–Jacobi isomorphism (1), one can identify torsion points of order 2 in the Jacobian variety of C with half-periods 0 ¤  2 H1 .C; Z2 /. Given a halfperiod, by taking its orthogonal complement with respect to the intersection product,

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one obtains a subgroup of index 2 in H1 .C; Z/ which determines an unramified double cover of C . Algebraically, given a line bundle  2 Pic0 .C /fOC g together with a sheaf Š isomorphism  W ˝2 ! OC , one associates an unramified double cover f W Cz ! C such that Cz WD Spec.OC ˚ /: The multiplication in the OC -algebra OC ˚ is defined via the isomorphism of sheaves , that is, .a C s/  .b C t / WD ab C .s  t / C a  t C b  s; for a; b 2 OC and s; t 2 . Note that f .OCz / D OC ˚ , in particular H 0 .Cz ; f  L/ D H 0 .C; L/ ˚ H 0 .C; L ˝ /;

(2)

for any line bundle L 2 Pic.C /. The étale double cover f W Cz ! C induces a norm map   Nmf W Pic2g2 .Cz / ! Pic2g2 .C /; Nmf OCz .D/ WD OC .f .D//: It is proved in [M], Section 3, that the inverse image Nmf1 .KC / consists of the disjoint union of two copies Nmf1 .KC /even q Nmf1 .KC /odd of the same abelian variety, depending on the parity of the number of sections of line bundles on Cz . We define the Prym variety of the pair ŒC;  as follows: ˚  Pr.C; / WD Nmf1 .KC /even D L 2 Nmf1 .KC / W h0 .C; L/  0 mod 2 This is a .g  1/-dimensional abelian variety carrying a principal polarization. Precisely, if ‚Cz D W2g2 .Cz /  Pic2g2 .Cz / is the Riemann theta divisor of Cz , then ‚Cz  Pr.C; / D 2 „C ;

(3)

where „C is a principal polarization which can be expressed set-theoretically as ˚  „C D L 2 Nmf1 .KC /even W h0 .Cz ; L/  2 : Example 2.1. We explain how to construct the period matrix of the Prym variety Pr.C; /. Let .a0 ; : : : ; a2g2 ; b0 ; : : : ; b2g2 / be a symplectic basis of H1 .Cz ; Z/ compatible with the involution  W Cz ! Cz exchanging the sheets of f , that is,  .a0 / D a0 ;  .ai / D aiCg1

 .b0 / D b0 ;

 .bi / D biCg1 for i D 1; : : : ; g  1: If .!0 ; : : : ; !2g2 / is the basis of H 0 .Cz ; KCz / dual to the cycles fai g2g2 iD0 , then the forms ui WD !i  !iCg1 are  anti-invariant and the period matrix of Pr.C; / is Z

g1 ui 2 Hg1 : … WD and

bi

i;j D1

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Example 2.2. Having fixed an étale covering f W Cz ! C as above, we denote by ‚C; WD fL 2 Picg1 .C / W h0 .C; L ˝ /  1g the translate of the Riemann theta divisor and by f  W Picg1 .C / ! Pic2g2 .Cz / the pull-back map. The following algebraic form of the Schottky–Jung relation holds: .f  /1 .‚Cz / D ‚C C ‚C; ;

(4)

where ‚C D Wg1 .C /. Indeed, this is an immediate consequence of (2), for if L 2 Picg1 .C / satisfies h0 .Cz ; f  L/  1, then h0 .C; L/  1 or h0 .C; L ˝ /  1. Putting together formulas (3) and (4), one concludes that there exists a proportionality relation between the theta constants of the Jacobian Pic0 .C / having period matrix g 2 Hg , and those of Pr.C; / with corresponding period matrix …g1 2 Hg1 . This is the Schottky–Jung relation [SJ]:  #2

h  i h  .…g1 ; 0/ D # ı ı

h  0 i .g ; 0/  # 0 ı

0 i .g ; 0/ 1

The constant 2 C  is independent of the characteristics ; ı 2 f0; 1gg1 . The moduli space Rg is thus established as a highly interesting correspondence between the moduli space of curves and the moduli space of principally polarized abelian varieties: Rg EE { EEPrg {  {{ EE EE {{ { }{ " Mg Ag1 . Here  is forgetful map whereas Pr g is the Prym map   Rg 3 ŒC;  7! Pr.C; /; „C 2 Ag1 : We denote by Pg1 the closure in Ag1 of the image Pr g .Rg /. It is proved in [FS] that the Prym map is generically injective when g  7. Unlike the case of Jacobians, Pr g is never injective [D2] and the study of the non-injectivity locus of the Prym map is a notorious open problem. Without going into details, we point out [IL] for an important recent result in this direction.

3 Why Rg ? Since Mumford [M] “rediscovered” Prym varieties and developed their algebraic theory using modern techniques, there have been a number of important developments in algebraic geometry where Prym varieties and their generalizations play a decisive role. We mention four highlights:

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3.1 The Schottky problem. The Torelli map tg W Mg ! Ag ; tg .ŒC / WD ŒJac.C /; ‚C ; assigns to a smooth curve its principally polarized Jacobian variety. It is the content of Torelli’s theorem that the map tg is injective, that is, every smooth curve C can be recovered from the pair .Jac.C /; ‚C /, see [An] for one of the numerous proofs. To put it informally, the Schottky problem asks for a characterization of the Jacobian locus Jg WD tg .Mg / inside Ag . Schottky problem (analytic formulation). Characterize the period matrices  2 Hg that correspond to Jacobians. Find equations of the theta constants # Œ ı  .; 0/ of Jacobian varieties of genus g. Van Geemen [vG] has shown the Jacobian locus Jg is a component of the locus J g  Ag consisting of period matrices  2 Hg for which the Schottky–Jung relations are satisfied for all characteristics ; ı 2 f0; 1gg . In the case g D 4 there is a single Schottky–Jung relation, a polynomial of degree 16 in the theta constants, and which cuts out precisely the hypersurface J4  A4 . This is the formula given by Schottky [Sch] in 1888 and one concludes that the following equality holds: J4 D J 4 : Other analytic characterizations of Jg (KP equation, 00 conjecture of van Geemen– van der Geer) have been recently surveyed by Grushevsky [G]. Schottky problem (geometric formulation). Find geometric properties of principally polarized abelian varieties that distinguish or single out Jacobians. The most notorious geometric characterization is in terms of singularities of theta divisors. Andreotti and Mayer [AM] starting from the observation that the theta divisor of the Jacobian of a curve of genus g is singular in dimension at least g  4, considered the stratification of Ag with strata Ng;k WD fŒA; ‚ 2 Ag W dim.‚/  kg and showed that Jg is an irreducible component of Ng;g4 . This is what is called a weak geometric characterization of Jacobians. Unfortunately, Ng;g4 contains other components apart Jg , hence the adjective “weak”. The same program at the level of Prym varieties has been carried out by Debarre [De]. Using the methods of AndreottiMayer, he showed that dim.„C /  g  6 for any ŒC;  2 Rg and for g  7, the Prym locus Pg is a component of Ng;g6 . To give another significant recent example, Krichever [Kr] found a solution to Welters’ Conjecture stating that an abelian variety ŒA; ‚ 2 Ag is a Jacobian if and j2‚j

g

only if the Kummer image Km W A ! P2 1 admits a trisecant line. Using similar methods, Grushevsky and Krichever [GK] found a characterization of the Prym locus Pg in terms of quadrisecant planes in the Kummer embedding.

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By a dimension count, note that dim.Mg / D dim.Rg / D 3g  3 > dim.Jg1 /. One expects to find more Pryms than Jacobians in a given genus, and indeed, it is known that Jg1  Pg1 , that is, Jacobians of dimension g  1 appear as limits of Prym varieties. We refer to [Wi] for the original analytic proof, or to [B1] for a modern algebraic proof. Therefore one has at his disposal a larger subvariety of Ag1 than Jg1 which is amenable to geometric study via the rich and explicit theory of curves and their correspondences. This approach is particularly effective for g 6, when Pr g W Rg ! Ag1 is dominant, hence the study of Rg can be directly used to derive information about Ag1 . It is one of the main themes of these lectures to describe the geometry of Rg when g 8. 3.2 Rationality questions for 3-folds. Prym varieties have been used successfully to detect non-rational Fano 3-folds. If X is a smooth Fano 3-fold (in particular H 3;0 .X/ D 0), its intermediate Jacobian is defined as the complex torus J.X / WD H 2;1 .X /_ =H 3 .X; Z/; with the polarization coming from the intersection product on H 3 .X; Z/. Since H 3;0 .X/ D 0, one obtains in this way a principally polarized abelian variety. Assume now that f W X ! P2 is a conic bundle and consider the discriminant curve ˚  C WD t 2 P2 W f 1 .t / D l1 C l2 ; where li  X are lines : Thus C parametrizes pairs of lines, and assuming that l1 ¤ l2 for every t 2 C , we can consider the étale double cover C 0 ! C from the parameter space of lines themselves to the space classifying pairs of lines. It is then known [B2] that .J.X /; ‚J / Š .Pr.Cz =C /; „/, that is, the intermediate Jacobian of X is a Prym variety. Furthermore, X is rational if and only if .J.X /; ‚J / is a Jacobian. Using the explicit form of the theta divisor of a Prym variety, in some cases one can rule out the possibility that J.X / is isomorphic to a Jacobian and conclude that X cannot be rational. In this spirit, Clemens and Griffiths [CG] proved that any smooth cubic threefold X3  P4 is nonrational. Since J.X3 / is the Prym variety corresponding to a smooth plane quintic, it follows from the study of Sing.„/ carried out in [M], that X3 is not rational. A similar approach, works in a number of other cases, e.g. when X  P6 is a smooth intersection of three quadrics, see [B2]. 3.3 The Hitchin system. This is a topic that has seen an explosion of interest recently since Ngô [N] proved the fundamental lemma in the Langlands program using the topology of the Hitchin system. We place ourselves in a restrictive set-up just to present certain ideas. Let C be a smooth curve of genus g and denote by M WD UC .2; OC / the moduli space of semistable rank 2 vector bundles E on C with det.E/ D OC . For a point ŒE 2 UC .2; OC / with E stable, we have the following identification _ TŒE .M/ D Hom.E; E ˝ KC /0 , where the last symbol refers to the homomorphisms 

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_ can be viewed as the space of Higgs fields of trace zero. The cotangent bundle TM .E; /, where  W E ! E ˝ KC is a homomorphism. The Hitchin map is defined as _ ! H 0 .C; KC˝2 /; H.E; / D det./ 2 H 0 .C; KC˝2 /: H W TM

It is proved in [H] that the map H is a completely integrable system, and for a general quadratic differential q 2 H 0 .C; KC˝2 /, the fibre H 1 .Œq/ equals the Prym variety Pr.Cq =C /, where Cq is the spectral curve whose local equation in the total space of the canonical bundle of C is y 2 D q.x/. x g . Prym level structures have 3.4 Smooth finite Deligne–Mumford covers of M xg . been used by Looijenga [Lo] to construct Deligne–Mumford Galois covers of M These spaces are smooth (as varieties, not only as stacks!), modular and can be used to x g . If S is a comgreatly simplify Mumford’s definition of intersection products on M pact oriented topological surface of genus g, its universal Prym cover is a connected unramified Galois cover Sz ! S corresponding to the normal subgroup of 1 .S; x/ generated by the squares of all elements. The Galois group of the cover is denoted by G WD H1 .S; Z2 /. A Prym level n-structure on a smooth curve of genus g is a class of orientation preserving homeomorphisms f W S ! C , where two such homeomorphisms f; f 0 are identified, if the homeomorphism f 1 B f 0 W S ! S has the property that its lift, viewed as an orientation preserving homeomorphism of Sz, acts  n z Zn /. The moduli space Mg as an element of G on H1 .S; of smooth curves with a 2   xg n Prym level n-structure is a Galois cover of Mg . Remarkably, the normalization M 2 n x g in the function field of Mg of M is a smooth variety for even n  6. Therefore 2 x g is the quotient of a smooth variety by a finite group! M

4 Parametrization of Rg in small genus We summarize the current state of knowledge about the birational classification of Rg for small genus. Firstly, R1 D X0 .2/ is rational. The rationality of R2 is classical and several modern proofs exist in the literature. We sketch the details of one possible approach following [Do1]. Theorem 4.1. R2 is rational. Proof. Suppose that C is a smooth curve of genus 2. The 15 non-trivial points of order 2 on C are in bijective correspondence to sums pCq 2 C2 of distinct Weierstrass points on C . Thus if ŒC;  2 R2 , there exist unique Weierstrass points p ¤ q 2 C such that KC ˝  D OC .p C q/. One considers the line bundle L WD KC˝2 ˝  D KC .p C q/ 2 Pic4 .C /. By applying Riemann–Roch, the image of the map L W C !  P2 is a plane quartic curve with a singular point u WD L .p/ D L .q/. Furthermore, both vanishing sequences of the linear series jLj at the points corresponding to the node u are equal to L L aC .p/ D aC .q/ D .0; 1; 3/;

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that is, the tangent lines at u to the two branches of , intersect the nodal curve with multiplicity 4. Accordingly, setting u WD Œ1 W 0 W 0 2 P2 , the plane equation for in coordinates Œx W y W z can be given as x 2 yz C xyzf1 .y; z/ C f4 .y; z/ D 0; where f1 .y; z/(respectively f4 .y; z/) is a linear (respectively quartic) form. By a judicious change of coordinates, one may assume f1 D 0 and then R2 is birational to a quotient of C 5 by a 2-dimensional torus, which is a rational variety. For the rationality of R3 we mention again [Do1]. There exists an alternative approach due to Katsylo [Ka]. The space R4 is rational [Ca] and we shall soon return to this case. There are two different proofs of the unirationality of R5 in [IGS] and [V2] respectively. The case of R6 is the most beautiful and richest from the geometrical point of view. Observing that dim.R6 / D dim.A5 / D 15, one expects the Prym map Pr 6 W R6 ! A5 to be generically finite, therefore also dominant. By degeneration methods, Wirtinger [Wi] showed this indeed to be the case. Much later, Donagi and Smith [DS] proved that its degree is equal to 27 which suggests a connection to cubic surfaces. We cannot resist quoting from [DS] p. 27: Wake an algebraic geometer in the dead of the night whispering “27”. Chances are, he will respond: “lines on a cubic surface”. Donagi [D2] subsequently showed that the Galois group of R6 over A5 , that is, the monodromy groups of Pr 6 , is equal to the Weyl group W .E6 / S27 . We recall that W .E6 / is the group of symmetries of the set of lines on a cubic surface. Precisely, if X  P3 is a fixed smooth cubic surface and fl1 ; : : : ; l27 g is a numbering of its 27 lines, then W .E6 / WD f 2 S27 W l .i/  l .j / D li  lj for all i; j D 1; : : : ; 27g: The statement that R6 (and hence A5 ) is unirational admits at least three very different proofs due to Donagi [D1] using intermediate Jacobians of Fano 3-folds, Mori and Mukai [MM], and Verra [V1] who used the fact that a general Prym curve ŒC;  2 R6 can be viewed as a section of an Enriques surface. Having reached this point, one might wonder for which values of g is Rg unirational. The following result [FL] (to be explained in some detail in the next chapters), provides an upper bound on the genus g where one may hope to have an explicit unirational description of the general Prym curve ŒC;  2 Rg . x g of Rg is a variety of general Theorem 4.2. The Deligne–Mumford compactification R x 12 /  0, in particular R x 12 cannot be uniruled. type for g  14. Furthermore, .R Allowing us to speculate a little further, by analogy with the case of the spin moduli x g is not of general type for g 11. In what space of curves, it seems plausible that R follows we shall confirm this expectation for g D 7; 8. As for the remaining cases g D 9; 10; 11, hardly anything seems to be known at the moment.

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4.1 Nikulin K3 surfaces. Our next aim is to find a uniform way of parametrizing Rg for g 7. To that end, we consider the following general situation. Let S be a smooth K3 surface and E1 ; : : : ; EN a set of disjoint, smooth rational .2/-curves on S . One may ask when is the class E1 C    C EN divisible by two (even), that is, e ˝2 D OS .E1 C    C EN / for a suitable class e 2 Pic.S /. Equivalently, there exists a double cover  W Sz ! S branched precisely along the curves E1 ; : : : ; EN . Note that  1 .Ei /  Sz are .1/curves which can be blown-down and the resulting smooth surface has an automorphism permuting the sheets of the double covering. The answer to this question is due to Nikulin [Ni] and only two cases are possible: (i) N D 16 and Sz is birational to an abelian surface and S itself is a Kummer surface. (ii) N D 8 and Sz is also a K3 surface. In this case .S; e/ is called a Nikulin K3 surface. For a reference to K3 surfaces with an even set of rational curves, we recommend [Ni], [vGS], whereas for generalities on moduli space of polarized K3 surfaces, see [Do2]. Suppose that .S; e/ is a Nikulin K3 surface and C  S is a smooth curve with C 2 D 2g 2, such that C Ei D 0 for i D 1; : : : ; 8. Then the restriction eC WD e ˝OC is a point of order 2 in the Jacobian of C . This link between Nikulin K3 surfaces and Prym curves prompts us to make the following definition [FV2]: Definition 4.3. The moduli space of polarized Nikulin surfaces of genus g is defined as the following parameter space: ˚ FgN D ŒS; h; e W h 2 Pic.S / is nef ; h2 D 2g  2; Pic.S /  hE1 ; : : : ; E8 ; hi; eD

8  P 1 Ei / 2 Pic.S /; h  Ei D 0; Ei2 D 2; Ei  Ej D 0 for i ¤ j : OS . 2 iD1

Note that FgN is an irreducible variety of dimension 11, see [Do2]. Nikulin surfaces depend on 11 moduli because polarized K3 surfaces of genus g depend on 19 moduli, from which one subtracts 8, corresponding to the number of independent condition being imposed on the lattice Pic.S /. We then consider the Pg -bundle over FgN ˚   PgN WD ŒS; h; e; C W ŒS; h; e 2 FgN ; C  S; C 2 jhj : The restriction line bundle eC WD e ˝ C 2 Pic0 .C /2 induces an étale double cover Cz ! C: As explained above, we have two morphism between moduli spaces

FgN

} }} }} } ~} }

PgN

AA AAg AA AA Rg

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  where g ŒS; e; h; C WD ŒC; eC : Note that dim.PgN / D 11 C g and dim.Rg / D 3g  3; hence dim.PgN /  dim.Rg / exactly for g 7. It is natural to ask whether in this range PgN dominates Rg . Since by construction PgN is a uniruled variety, this would imply (at the very least) the uniruledness of Rg . At this point we would like to recall the following well-known theorem due to Mukai [M1]: Theorem 4.4. A general curve ŒC  2 Mg appears as a section of a K3 surface precisely when g 11 and g ¤ 10. For g D 10, the locus K10 WD fŒC  2 M10 W C lies on a K3 surfaceg is a divisor. The fact that the general curve ŒC  2 M10 does not lie on a K3 surface comes as a surprise, and is due to the existence of the rational homogeneous 5-fold X WD G2 =P  P13 such that KX D OX .3/. Thus codimension 4 linear sections of X are canonical curves of genus 10; if a curve ŒC  2 M10 lies on a K3 surface, then it lies on a 3-dimensional family of K3 surfaces. This affects the parameter count for genus 10 sections of K3 surfaces and one computes that dim.K10 / D 19 C g  3 D 26 D dim.M10 /  1: x g . It is an exThe divisor K10 plays an important role in the birational geometry of M x tremal point of the effective cone of divisors of M10 and it was the first counterexample to the Harris-Morrison Slope Conjecture, see [FP]. In joint work with A. Verra [FV2], we have shown that one has similar results (and much more) for Prym curves, the role of ordinary K3 surfaces being played by Nikulin surfaces. The following result is quoted from [FV2]: Theorem 4.5. We fix an integer g 7, g ¤ 6. A general Prym curve ŒC;  2 Rg lies on a Nikulin surface, that is, the rational map g W PgN ! Rg is dominant. Proof. We discuss the proof only in the case g D 7 and start with a general element ŒC;  2 R7 . We consider the Prym-canonical embedding KC ˝ W C ! P5 : Note that KC ˝ is very ample, for otherwise  2 C2 C2 , in particular C is tetragonal, which contradicts the generality assumption on the pair ŒC; . It is shown in [FV2]

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that h0 .P5 ; C =P5 .2// D 3, that is, jC =P5 .2/j is a net of quadrics and the base locus of this net is a smooth K3 surface S  P5 . We claim that S is a Nikulin K3 surface. Let L WD OS .2H  2C / 2 Pic.S / and consider the standard exact sequence 0 ! L ˝ OS .C / ! L ! L ˝ OC ! 0: Note that L ˝ OC D OC .2H  2KC / D OC . Furthermore h1 .S; L ˝ OS .C // D h1 .S; 2H  C / D 0, because C is quadratically normal. Passing to the long exact sequence, it follows h0 .S; L/ D 1. The numerical characters of L can be computed as follows: L  H D 8 and L2 D 16. After analyzing all possibilities, it follows that L is equivalent to the sum of 8 disjoint lines. Furthermore  D OC .C  H /, which proves that ŒC;  D 7 .ŒS; OS .C  H /, that is, C lies on a Nikulin surface. Theorem 4.5 shows that Rg is uniruled for g 7. As in Mukai’s Theorem 4.4, the genus next to maximal, proves to be exceptional. Let us denote by N6 WD Im. 6 / the locus of Prym curves which are sections of Nikulin surfaces. Theorem 4.6. One has the following identification of effective divisors on R6 : ˚  N6 D ŒC;  2 R6 W Sym2 H 0 .C; KC ˝ / ! H 0 .C; KC˝2 / is not an isomorphism : This locus equals the ramification divisor of the Prym map Pr 6 W R6 ! A5 . The previous method can no longer work when g  8 because Nikulin sections form a locus of codimension at least 2 in Rg . Instead we shall sketch an approach to handle the case of R8 . Full details will appear in the paper [FV3].

5 Prym Brill–Noether loci and the uniruledness of R8 Some preliminaries on Brill–Noether theory for Prym curves and lagrangian degeneracy loci are needed, see [M] and [We] for a detailed discussion. We fix a Prym curve ŒC;  2 Rg and let f W Cz ! C be the induced étale covering map. For a fixed integer r  1, the Prym–Brill–Noether locus is defined as the following determinantal subvariety of Nmf1 .KC /: ˚  V r .C; / WD L 2 Nmf1 .KC / W h0 .Cz ; L/  r C 1; h0 .Cz ; L/  r C 1 mod 2 The expected dimension of V r .C; / as a determinantal variety equals g1 In any event, the inequality !   r r C1 codim V .C; /; Pr.C; / 2

rC1 2

.

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holds. Note that V 1 .C; / D V 0 .C; / D Pr.C; / and V 1 .C; / D „C is the theta-divisor of the Prym variety, in its Mumford incarnation. We fix a line bundle L 2 Nmf1 .KC /. This last condition is equivalent to  L D KCz ˝ L_ , where  W Cz ! Cz is the involution exchanging the sheets of the covering f W Cz ! C . Let us consider the Petri map 0 .L/ W H 0 .Cz ; L/ ˝ H 0 .Cz ; KCz ˝ L_ / ! H 0 .C; KCz /: Using the decomposition H 0 .Cz ; KCz / D H 0 .C; KC / ˚ H 0 .C; KC ˝ /, we can split the Petri map into a  anti-invariant part 2 0 z 0  0 .L/ W ƒ H .C ; L/ ! H .C; KC ˝ /;

s ^ t 7! s   .t /  t   .s/;

and a  invariant part respectively 2 0 z 0   C 0 .L/ W Sym H .C ; L/ ! H .C; KC /; s ˝ t C t ˝ s 7! s   .t / C t   .s/:

Welters calls  0 .L/ the Prym–Petri map. The name is appropriate because analogously to the classical Petri map, via the standard identification   TL Pr.C; / D H 0 .C; KC ˝ /_ coming from Kodaira–Spencer theory, the map  0 .L/ governs the deformation theory of the loci V r .C; /. We mention the following result, see [We] Proposition 1.9: Proposition 5.1. Let L 2 Nmf1 .KC / with h0 .Cz ; L/ D r C 1. The Zariski tangent  ? space TL .V r .C; // can be identified to Im 0 .L/ . In particular, V r .C; / is   smooth and of the expected dimension g  1  rC1 at the point L if and only if  0 .L/ 2 is injective. The main result of [We] states that for a general point ŒC;  2 Rg , the Prym–Petri r map  0 .L/ is injective for every L 2 V .C; /. In particular, ! r C1 r : dim V .C; / D g  1  2 The class of V r .C; / has been computed by De Concini and Pragacz [DP]. If  D  0 =2 2 H 2 .Pr.C; /; Z/ is the class of the principal theta-divisor of Pr.C; /, then  V r .C; / D



rC1 1  . 2 /: 1r  3r1  5r2 : : : .2r  1/   This formula proves that V r .C; / ¤ ; when g  1  rC1 . 2 .L/, enjoying this deformation-theoretic interpretation, has received a The map  0 .L/ seems to have been completely lot of attention. By contrast, its even counterpart C 0

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neglected so far, but this is what we propose to use in order to parametrize Rg when g D 8. We define the universal Prym–Brill–Noether locus ˚   Rgr WD ŒC; ; L W ŒC;  2 Rg ; L 2 V r .C; / :   When g1 rC1  0, the variety Rgr is irreducible, generically smooth of dimension rC1 2 4g  4  2 and mapping dominantly onto Rg . We now fix g  4 and turn our attention to the space Rg2 which has relative   dimension g  4 over Rg . A general point ŒC; ; L 2 Rg2 , corresponds to a general 0 z Prym curve ŒC;   20 Rg _and  a base point free line bundle L such that h .C ; L/ D 3. 2 Setting P WD P H .L/ , we have the following commutative diagram:

Cz

.L;  L/

f

 C

j C 0 .L/j

/ P2  P2 W WWWWW WWWWW WWWWW W+   s P8 D P H 0 .L/_ ˝ H 0 .L/_ . f f f f f f f f s  /P5 D P.Sym2 H 0 .L/_ /

In the above diagram s is a 2 W 1 quasi-étale morphism and Im.s/ D D  P5 is the determinantal cubic hypersurface. The branch locus of s is the Veronese surface Sing.D/ D V4 P5 . It is well-known that D can be identified with the secant variety of V4 . For a general ŒC;  2 Rg as above, one can show that C 0 .L/ is injective, that is, W WD Sym2 H 0 .Cz ; L/  H 0 .C; KC / is a 6-dimensional space of canonical forms on C . The map s is given by   P2  P2 3 Œa; Œb 7! Œa ˝ b C b ˝ a 2 P5 : Equivalently, if P2 is viewed as the space of lines in P.H 0 .L//, then s maps a pair of lines .Œa; Œb/ to the degenerate conic Œa C Œb 2 P5 . Moreover, D is viewed as the space of degenerate conics in P.H 0 .L//. jW j The commutativity of the diagram implies that the map Cz ! P5 induced by the sections in W has degree 2 and factors through C . The image curve, which is a projection of the canonical model of C , lies on the symmetric cubic hypersurface D. Before turning to the case g D 8, we mention the following result: Theorem 5.2 (Verra 2008). Rg2 is a unirational variety for g 7. This of course gives a new proof of the unirationality of Rg when g 7. We turn our attention to the case of R8 and ask when is the image jW j

C ! P5

Prym varieties and their moduli

231

contained in a .2; 2; 3/ complete intersection, that is, we require that C be contained in two additional quadrics. The idea of showing uniruledness of a moduli space of curves by realizing its general point as a section of a canonical surface is not new and has already been used in [BV] to prove that M15 is rationally connected. If S  Pr is a canonical surface and C  S is a curve such that h1 .C; OC .1//  1, then dim jC j  1, in particular C deforms in moduli, and through a general point of the moduli space there passes a rational curve. To estimate the number of moduli of Prym curves lying on a .2; 2; 3/ complete intersection in P5 , we consider the following morphism between two vector bundles over an open subset of Rg2 : E.C; ; L/ D Sym2 .W / PPP PPP PPP PPP PP'



Rg2

/ H 0 .C; K ˝2 / D F .C; ; L/ C mm m m m mmm mmm m m m v m

 Rg .

Both E and F are vector bundles over Rg2 , with fibres over a point ŒC; ; L as in the diagram above. The vector bundle morphism  W E ! F is given by multiplication of sections. Note that when g D 8, both Sym2 .W / and H 0 .C; KC˝2 / have dimension 21, and we expect the corank 2 degeneracy locus of  to be of codimension 4 in Rg2 , and hence map dominantly onto Rg . After some rather substantial work, we can show that a general ŒC;  2 R8 lies on a finite number of surfaces C  S  P5 , where S D Q1 \ Q2 \ D P5 . Singular points of S are the 16 nodes corresponding to the intersections of Q1 \ Q2 with the Veronese surface V4 WD Sing.D/. Furthermore, KS D OS .1/ and from the adjunction formula we find that OC .C / D OC , hence there is an exact sequence 0 ! OS ! OS .C / ! OC ! 0: One finds that dim jOS .C /j D 1, that is, C moves in a pencil of curves on S. Since the torsion line bundle  can be recovered from the projection s, we obtain in fact a pencil in R8 , passing through a general point. One has the following result, full details of which will appear in the forthcoming [FV3]: Theorem 5.3. The moduli space R8 is uniruled.

6 The Kodaira dimension of moduli of Prym varieties The aim of this lecture is to show that Rg is a variety of general type for g  14 and to convey, in an informal setting, some of the ideas contained [FL], [HM], [EH]. First

232

G. Farkas

we discuss a general program of showing that a moduli space is of general type. This x g , by Gritsenko, Hulek strategy has been used by Harris and Mumford in the case of M and Sankaran [GHS] in the case of the moduli space of polarized K3 surfaces and by the author [F3] in the case of the space xgC classifying even theta-characteristics. Any attempt to compute the Kodaira dimension of the moduli spaces of Prym varieties must begin with the construction of a suitable compactification of Rg . This compactification should satisfy a number of minimal requirements: xg ! M xg . • The covering Rg ! Mg should extend to a finite branched covering R x x • Points in Rg ought to have modular meaning. Ideally, Rg should be the coarse moduli space associated to a Deligne–Mumford stack of stable Prym curves of genus g, that is, points in the boundary should correspond to mildly singular curves with some level structure. If this requirement is fulfilled, one can carry out intersection theory on x g and the results have enumerative meaning in terms of curves and their associated R Prym varieties. x g should be manageable, in particular we would like pluri• The singularities of R x g;reg to extend to a resolution canonical forms defined on the locus of smooth points R x x g , defined as of singularities of Rg . This implies that the Kodaira dimension of R the Kodaira dimension of a non-singular model, coincides with the Kodaira–Iitaka x g . In dimension of the canonical divisor KRx g , which is computed at the level of R x practice, this last requirement forces Rg to have finite quotient singularities. In what follows we describe a satisfactory solution to this list of requirements. We fix a genus g  2 and a level l  2. We consider the following generalization of the level l modular curve ˚  Rg;l D ŒC;  W ŒC  2 Mg ;  2 Pic0 .C /  fOC g is a point of order l : 

Obviously Rg;2 D Rg . There is a forgetful map Rg;l  ! Mg of degree l 2g  1. The x g .BZl / viewed as a compactification x g;l WD M moduli space of twisted stable maps R of Rg;l can be fitted into the following commutative diagram, see [ACV]: Rg;l   

 Mg  

x g;l /R 

 xg . /M

x g;l as By analogy with the much studied case of elliptic curves, one may regard R a higher genus generalization of the modular curve X1 .N /. For simplicity we shall explain the construction of Rg;l only in the case l D 2, and refer to [ACV], [CCC] for details for the case l  3. Definition 6.1. If X is a semi-stable curve, a component E X is called exceptional if E ' P1 and E \ .X n E/ D 2, that is, it meets the other components in exactly two

Prym varieties and their moduli

233

points. The curve X is called quasi-stable if any two exceptional components of X are disjoint. If X is a quasi-stable curve, the stable model st.X / of X is obtained by contracting all exceptional components of X . x g of genus g parametrizes Definition 6.2. The moduli space of stable Prym curves R triples ŒX; ; ˇ such that: • X is a quasi-stable curve with pa .X / D g. •  2 Pic0 .X /, that is,  is a locally free sheaf on X of total degree 0. • E ' OE .1/ for all exceptional components E X . • ˇ W ˝2 ! OX is a sheaf homomorphism which is an isomorphism along nonexceptional components; Next we define a stack/functor of stable Prym curves, whose associated coarse xg : moduli space is precisely R Definition 6.3. A family of Prym curves over a base scheme S consists of a triple f

.X ! S; ; ˇ/, where f W X ! S is a flat family of quasi-stable curves,  2 Pic.X/ is a line bundle and ˇ W ˝2 ! OX is a sheaf homomorphism, such that for every point ˝2 s 2 S the restriction .Xs ; Xs ; ˇXs W X ! OXs / is a stable Prym curve of genus g. s Remark 6.4. Note that by replacing in this definition the structure sheaf OX by the dualizing sheaf !X , we obtain the moduli of stable spin curves xg . Different compactifications of the space Rg;l were studied for l  3 by Caporaso–Casagrande–Cornalba [CCC], Jarvis [J] and Abramovich–Corti–Vistoli [ACV]. xg ! M x g which at the level of There exists a forgetful morphism of stacks  W R sets is given by .ŒX; ; ˇ/ D Œst.X /. Even though the morphism Rg ! Mg is xg ! M x g is ramified along the étale (at the level of stacks), the compactification R boundary. This accounts for better positivity properties of the canonical bundle KRxg . xg . x g is expected to become sooner of general type than M As g increases, R x g the closure of the locus of irreducible one-nodal As usual we denote by 0 M x g the boundary divisor whose curves and for 1 i Œg=2 we denote by i  M general point corresponds to the union of two curves of genus i and g  i respectively, meeting transversally at a single point. Example 6.5. Let us take a general point ŒCxy  2 0 , corresponding to a normalization map  W C !  Cxy , where C is a curve of genus g  1 and x; y 2 C are distinct points. We aim to describe all points ŒX; ; ˇ 2  1 .ŒCxy /. Depending on whether X contains an exceptional component or not, one distinguishes two cases:

234

G. Farkas

If X D Cxy , there is an exact sequence 

1 ! Z2 ! Pic0 .Cxy /2 ! Pic0 .C /2 ! 1: Setting C D   ./ 2 Pic0 .C /, there are two subcases to be distinguished: (I) If C ¤ OC , there is a Z2 -ambiguity (coming from the previous sequence) in identifying the fibres C .x/ and C .y/, that is, there exist two possibilities of lifting C to a line bundle on X . Such an identification, together with the choice of line bundle C 2 Pic0 .C /2 uniquely determine a line bundle  on Cxy which is a square root of 0 x g;2 the divisor consisting of such stable Prym the trivial bundle. We denote by 0  R curves together with all their degenerations. (II) If C D OC , then there is exactly one way of identifying C .x/ and C .y/ 00 x g;2 . Points in such that  ¤ OX : The closure of this locus is a divisor denoted 0  R 00 0 are sometimes called Wirtinger double covers [Wi], since they were used in [Wi] to prove that Jacobians of genus g  1 are limits of Prym varieties of genus g. p 1 (III) If X D C [fx;yg E, where E D P , then C 2 OC .x  y/ and E D OE .1/. In this case there is no ambiguity in identifying the fibres and the corresponding x x locus is the ramification divisor ram 0 of the map  W Rg ! Mg . x g / and ı 0 WD Œ0 ; ı 00 WD Keeping the notation above, if ı0 D Œ0  2 Pic.M 0 0 0 x g /, we obtain the following relation: Œ0 ; ı0ram WD Œram  2 Pic. R 0 00

0

00

  .ı0 / D ı0 C ı0 C 2ı0ram : All three cases described in Example 6.5 correspond to certain types of admissible double covers in the sense of [B1]. These coverings are represented schematically as follows: Example 6.6. (Curves of compact type) Let us consider a union of two smooth curves C and D of genus i and g i respectively meeting transversally at a point. We describe x g is a stable Prym curve having as underlying the fibre  1 .ŒC [ D/. If ŒX; ; ˇ 2 R model a curve X with st.X / D C [ D, first we observe that X D C [ D (that is, X has no exceptional components). The line bundle  on X is determined by the choice of two ˝2 line bundles C 2 Pic0 .C / and D 2 Pic0 .D/ satisfying ˝2 C D OC and D D OD respectively. This shows that for 1 i Œg=2 the pull-back under  of the boundary x g splits into three irreducible components divisor i  M   .i / D i C gi C iWgi ; x g is of the form ŒC [ D; C ¤ OC ; D D OD , where the generic point of i  R the generic point of gi is of the form ŒC [ D; C D OC ; D ¤ OD /, and finally iWgi is the closure of the locus of points ŒC [ D; C ¤ OC ; D ¤ OD . The canonical class KRx g can be computed using the Grothendieck–Riemann–Roch x g in the spirit of [HM], or using the Hurwitz formula for the universal curve over R formula for the branched covering .

235

Prym varieties and their moduli

(II)

(III)

y1

y2

x1

x2

Cxy

C

x

y

xN  yN

(I) yN

xN

C

x

Cxy

y

xy

Figure 1. Admissible double covers.

x g /: Theorem 6.7. One has the following formula in Pic.R KRx g D

0

00

13  2.ı0 C ı0 /  3ı0ram  2

Œg=2 X

.ıi C ıgi C ıiWgi /  .ı1 C ıg1 C ı1Wg1 /:

iD1

Proof. We use the Harris–Mumford formula [HM] KMx g  13  2ı0  3ı1  2ı2      2ıŒg=2 ; xg ! M x g , which we together with the Hurwitz formula for the ramified covering  W R ram  recall, is simply branched along 0 . We find that KRx g D  .KMx g / C ı0ram . Without worrying for a moment about the singularities that the moduli space might x g has non-negative Kodaira dimension is equivalent to knowhave, to decide whether R x g of weight 13 and vanishing ing whether there exist any Siegel modular forms on R with order 2 or 3 along infinity.

236

G. Farkas

xg 7 The singularities of R The Kodaira dimension .X / of a complex normal projective variety X is defined as .X/ WD .X 0 /, where  W X 0 ! X denotes an arbitrary resolution of singularities. In general, the fact that the canonical sheaf OX .KX / is big, does not imply that X is of general type. One only has an inequality .X / .X; KX /, relating the Kodaira dimension of X to the Kodaira–Iitaka dimension of its canonical linear series. To give a very simple minded example where equality fails to hold, let C  P2 be a plane quartic curve with three nodes. Then by the adjunction formula, KC D OC .1/, and this divisor is of course big. But .C / D 1 because the normalization of C is a rational curve. The reason is that the singularities of C impose too many adjunction conditions. In x g by working directly with its canonical order to determine the Kodaira dimension of R x g would bundle KRx g (and this is certainly what one wants, for a desingularization of R a priori have no modular interpretation), one must have control over the singularities of the coarse moduli space. x g is governed by Kodaira– x g , the local structure of R Just like in the case of M Spencer deformation theory. Let X be a quasi-stable curve of genus g, and denote by !X (respectively X ) the dualizing sheaf of X (respectively the sheaf of Kähler differentials on X ). Note that !X is locally free, whereas X fails to be locally free at the nodes of X. There is a residue map res W !X !

M

Cp ;

  ! 7! Resp .!/ p2Sing.X/ ;

p2Sing.X/

which is well defined because the residues of a 1-form ! 2 H 0 .X; !X / along the two branches of X corresponding to a node p 2 Sing.X / coincide. There exists an exact sequence M res X ! !X ! Cp ! 0: p2Sing.X/

x g is given by a neighbourhood We also recall that an étale neighbourhood of ŒC  2 M of the origin in the quotient   x g / D Ext1 .C ; OC /=Aut.C / D H 0 .C; !C ˝ C / _ =Aut.C /: TŒC  .M C x g . First of all, note that the versal deformation One has a similar local description of R space of a Prym curve ŒX; ; ˇ coincides with that of its stable model. The concept of an automorphism of a Prym curve has to be defined with some care: x g is an automorphism Definition 7.1. An automorphism of a Prym curve ŒX; ; ˇ 2 R  2 Aut.X/ such that there exists an isomorphism of sheaves  W    !  making

Prym varieties and their moduli

237

the following diagram commutative. .  /˝2

˝2

 ˇ

   OX

/ ˝2 ˇ

'

 / OX .

If C WD st.X / denotes the stable model of X obtained by contracting all exceptional components of X , then there is a group homomorphism Aut.X; ; ˇ/ ! Aut.C / given by  7! C . We call a node p 2 Sing.C / exceptional if it corresponds to an exceptional component that gets contracted under the map X ! C . x g . An étale neighbourhood of ŒX; ; ˇ is We fix a Prym curve ŒX; ; ˇ 2 R isomorphic to the quotient of the versal deformation space C3g3 of ŒX; ; ˇ modulo the action of the automorphism group Aut.X; ; ˇ/. If C t3g3 D Ext1 .1C ; OC / denotes the versal deformation space of C , then the map C3g3 ! C t3g3 is given by ti D i2 , if .ti D 0/  C t3g3 is the locus where an exceptional node pi persists and xg ! M x g is given locally by the map ti D i otherwise. The morphism  W R C3g3 =Aut.X; ; ˇ/ ! C t3g3 =Aut.C /: x g is a space with finite quotient singularThis discussion illustrates the fact that R ities. It is a basic question to describe canonical finite quotient singularities and the answer is provided by the Reid–Shepherd–Barron–Tai criterion [Re]. Definition 7.2. A Q-factorial normal projective variety X is said to have canonical singularities if for any sufficiently divisible integer r  1 and for a resolution of singularities  W X 0 ! X , one has that  .!X˝r0 / D OX .rKX /. If this property is satisfied in a neighbourhood of a point p 2 X , one says that X has a canonical singularity at p. From the definition it follows that a section s of OX .rKX / regular around p 2 X extends regularly to a neighbourhood of  1 .p/. Canonical singularities appear in the Minimal Model Program as the singularities of canonical models of varieties of general type. Assume now V WD C m and let G  GL.V / be a finite group. We fix an element g 2 G with ord.g/ D n. The matrix corresponding to the action of g is conjugate to a diagonal matrix diag. a1 ; : : : ;  am /, where  is an n-th root of unity and 0 ai < n for i D 1; : : : ; m. One defines the age of g as the following sum: age.g/ WD

a1 n

C  C

am n

Definition 7.3. The element g 2 G is said to be junior if age.g/ < 1 and senior otherwise.

238

G. Farkas

We have the following characterization [Re] of finite quotient canonical singularities: Theorem 7.4. Let G  GL.V / be a finite subgroup acting without quasi-reflections. Then the quotient V =G has canonical singularities if and only if each non-trivial element g 2 G is senior. Remark 7.5. If g 2 G acts as a quasi-reflection, then fv 2 V W g  v D vg is a hyperplane and g  diag. a1 ; 1; : : : ; 1/, hence age.g/ D an1 < 1, that is, each quasi-reflection is junior. On the other hand, obviously quasi-reflections do not lead to singularities of V =G, which is the reason for their exclusion from the statement of the Reid–Shepherd–Barron criterion. x g ? Let us fix a point How does one apply Theorem 7.4 to study the singularities of R 3g3 x g as well as the étale neighbourhood C ŒX; ; ˇ 2 R defined above. We denote by H  Aut.X; ; ˇ/ the subgroup generated by automorphism acting as quasi-reflections on C3g3 . The quotient map C3g3 ! C3g3 =H WD Cv3g3 is given by vi WD i2 if the coordinate i corresponds to smoothing out an elliptic tail of X and vi WD i otherwise. By definition, Aut.X; ; ˇ/ acts on Cv3g3 without x g has a canonical quasi-reflections, hence by applying Theorem 7.4, the quotient R singularity at ŒX; ; ˇ if an only if each automorphism is senior. In that is the case, forms defined in a neighbourhood of ŒX; ; ˇ extend locally to any resolution of sinx g does have non-canonical singularities as the following gularities. Unfortunately, R simple example demonstrates: Example 7.6. Let us choose an elliptic curve ŒC1 ; p 2 M1;1 with Aut.C1 ; p/ D Z6 , as well as an arbitrary pointed curve ŒC2 ; p 2 Mg1;1 together with a non-trivial point of order two 2 2 Pic0 .C2 /  fOC2 g. We consider a stable Prym curve xg ; ŒX WD C1 [p C2 ;  2 R where C1 D OC1 and C2 D 2 . We consider an automorphism  2 Aut.X; ; ˇ/, where C2 is trivial and C1 2 Aut.C1 / generates Aut.C1 /. In the versal deformation space C3g3 there exist two coordinates 1 and 2 corresponding to directions which preserve the node p 2 X and deform the j -invariant respectively. One can find a 6-th root of unity 6 such that the action of  on C3g3 is given by   1 D 6 1 ;

  2 D 62 2

and

  i D i for i D 3; : : : ; 3g  3:

The quotient map C3g3 ! Cv3g3 is given by the formulas v1 D 12 ;

v2 D 2

and

vi D i for i D 3; : : : ; 3g  3:

239

Prym varieties and their moduli

Therefore the action of  on Cv3g3 can be summarized as follows:   v1 D 62 v1 ;   v2 D 62 v2 Therefore age. / D

2 6

C

2 6

D

2 3

and

  vi D vi for i D 3; : : : ; 3g  3:

< 1, and this leads to a non-canonical singularity.

The good news is that, in some sense, this is the only source of examples of noncanonical singularities. By a detailed case by case analysis, one proves the following characterization [FL], Theorem 6.7, of the locus of non-canonical singularities: x g is a non-canonical singularity if and Theorem 7.7. Set g  4. A point ŒX; ; ˇ 2 R only if X possesses an elliptic subcurve C1  X with jC1 \ .X  C1 /j D 1, such that the j -invariant of C1 is equal to zero, and the restriction C1 is trivial. x g has a codimension two locus of non-canonical singularities, but the Therefore R key fact is, that this locus of relatively simple and can be easily resolved. Even though x g , Theorem 6.1 there are local obstructions to lifting pluri-canonical forms from R from [FL] shows that these are not global obstructions and in particular the Kodaira x g equals the Kodaira–Iitaka dimension of the canonical linear series dimension of R jKRx g j. This theorem is also a generalization of the result of Harris and Mumford who xg : treated the case of M x g a resolution of singularities. Then zg ! R Theorem 7.8. Let us fix g  4 and " W R for every integer n  1 there is an isomorphism z g ; K ˝n / ' H 0 .R x g ; K ˝n /: " W H 0 .R x z Rg

Rg

x g / equals the Iitaka dimension of the canonical To sum up these considerations, .R x g is as good as a linear series. For all questions concerning birational classification, R smooth variety.

xg 8 Geometric cycles on R For every normal Q-factorial variety X for which an extension result along the lines of Theorem 7.8 holds, in order to show that .X /  0 is suffice to prove that KX is an effective class. Following a well-known approach pioneered by Harris and Mumford x g is of general type [HM] in the course of their proof that the moduli space of curves M for g  24, one could attempt to construct explicitly sections of the pluri-canonical x g by means of algebraic geometry, by considering geometric conditions bundle on R on Prym curves that fail along a hypersurface in the moduli space Rg . Such geometric conditions must be amenable to degeneration to stable Prym curves, for one must be able x g / of the closure of the locus where the condition fails. to compute the class in Pic.R In particular, points in the boundary must have a strong geometric characterization. Finally, one must recognize an empirical geometric principle that enables to distinguish

240

G. Farkas

between divisorial geometric conditions that are likely to lead to divisors of small slope x g /) and divisors x g (ideally to extremal points in the effective cone of divisors Eff.R on R x of high slope which are less interesting. For instance in the case of Mg , one is lead to consider only divisors containing the locus of curves that lie on K3 surfaces, see [FP]: x g be any effective divisor. If the following slope inequality Proposition 8.1. Let D  M s.D/ < 6 C 12=.g C 1/ holds, then D must contain the locus Kg WD fŒC  2 Mg W C lies on a K3 surfaceg: xg This of course sets serious geometric constraints of the type of divisors on M whose class is worth computing, since it is well-known that curves of K3 surfaces behave generically from many points of view (e.g. Brill–Noether theory). Here in contrast we are looking for geometric conditions with respect to which the K3 locus behaves non-generically. We refer to [F1] for a way to produce systematically divisors x g having slope less than 6 C 12=.g C 1/. We close this introductory discussion by on M x g ought to satisfy, summarizing the numerical conditions that an effective divisor on R xg in order to show that the moduli space has maximal Kodaira dimension. Precisely, R x g such that is of general type, if there exists a divisor D  R 0

0

00

00

D  a b0 ı0 b0 ı0 b0ram ı0ram 

Œg=2 X

x g /; .bi ıi Cbgi ıgi CbiWgi ıiWgi / 2 Eff.R

iD1

satisfying the following inequalities: n a n a a o 13 a a a o 13 < ; max ram ; ; ; max 0 ; 00 < 2 b0 b1 bg1 b1Wg1 3 b0 b0 and

na

o

(5)

13 : bi bgi biWgi 2 It is explained in [F2] how one can rederive the results of [HM] using Koszul divisors x g , and how more generally, loci in moduli given in terms of syzygies of the objects on M they parametrize, lead to interesting geometry on moduli spaces. It is thus natural to x g , with the role of the canonical curve try to use the same approach in the case of R maxi1

;

a

;

a

jKC j

<

jKC ˝j

C ! Pg1 being played by the Prym-canonical curve C ! Pg2 . Let us fix a Prym curve ŒC;  2 Rg and the Prym-canonical line bundle L WD g2 KC ˝  2 W2g2 .C / inducing a morphism L W C ! Pg2 : We denote by I.L/  S WD CŒx0 ; : : : ; xg2  the ideal of the Prym-canonical curve and consider the minimal resolution of the homogeneous coordinate ring S.L/ WD S=I.L/ by free graded S -modules:    ! Fi !    F2 ! F1 ! F0 ! S.L/ ! 0;

241

Prym varieties and their moduli

  L where Fi D j S.i j /bi;j .C;L/ . The numbers bi;j .C; L/ D dimC Tor iCj S.L/; C i are the graded Betti numbers of the pair .C; L/ and encode the number of i -th order syzygies of degree j in the equations of the Prym-canonical curve. The graded Betti numbers can be computed via Koszul cohomology, using the resolution of the ground field k WD C by free graded S-modules. Precisely, we write the complex  ! 

iC1 ^

di C1;j 1

H 0 .C; L/ ˝ H 0 .C; L˝.j 1/ / ! di;j

   !

i1 ^

i ^

H 0 .C; L/ ˝ H 0 .C; L˝j /

H 0 .C; L/ ˝ H 0 .C; L˝.j C1/ / !   ;

where di;j .f1 ^    ^ fi ˝ u/ WD

i X

.1/l f1 ^    ^ fyl ^    ^ fi ˝ .fl u/;

lD1

is the Koszul differential, with f1 ; : : : ; fi 2 H 0 .C; L/ and u 2 H 0 .C; L˝i /. One easily checks that di;j B diC1;j 1 D 0, and defines the Koszul cohomology groups: Ki;j .C; L/ WD Ker di;j =Im diC1;j 1 Then dim Ki;j .C; L/ D bi;j .C; L/. The Koszul cohomology theory has been introduced by M. Green [Gr] and can be seen as a highly effective way of packaging geometrically the algebraic information contained in the homogeneous coordinate ring of an embedded variety. We consider the locus in Rg consisting of Prym curves having a non-linear i -th syzygy, that is, ˚  Ug;i WD ŒC;  2 Rg W Ki;2 .C; KC ˝ / ¤ 0 : In order to determine the expected dimension of Ug;i as a degeneracy locus inside Rg , we find a global determinantal presentation of Ug;i . Using a standard argument involving the Lazarsfeld bundle ML defined via the following exact sequence on C 0 ! ML ! H 0 .C; L/ ˝ OC ! L ! 0; one has the following identification, see e.g. [GL2], Lemma 1.10: Ki;2 .C; L/ D

H 0 .C; ^i ML ˝ L˝2 / : Imf^iC1 H 0 .C; L/ ˝ H 0 .C; L/g

(6)

After some diagram chasing explain for instance in detail in [F2], one obtains that Ki;2 .C; L/ ¤ 0 if and only if H 1 .C; ^iC1 ML ˝ L/ ¤ 0. After even more manipulations, this condition is equivalent to requiring that the restriction map     'ŒC;  W H 0 Pg2 ; ^i MPg2 ˝ OPg2 .2/ ! H 0 C; ^i ML ˝ L˝2 (7)

242

G. Farkas

have a kernel of dimension at least

! g  3 .g  1/.g  2i  6/ : dim Ker.'ŒC; /  i C2 i

We refer to [FL], Section 3, for full details. We point out that the dimension of both vector spaces that enter the map 'ŒC;  remain constant as ŒC;  varies in moduli, precisely !   g h0 Pg2 ; ^i MPg2 .2/ D .i C 1/ i C2 and 0

i

2

i

h .C; ^ ML ˝ L / D .C; ^ ML ˝ L

˝2

g2 /D i

!





i.2g  2/ C 3.g  1/ : g2

We get a divisorial condition in moduli, exactly when the vector spaces in (7) have the same dimension, and the required geometric condition is that the map 'ŒC;  be an isomorphism. This happens precisely when g D 2i C 6. Proposition 8.2. Set g WD 2i C 6. There exist vector bundles A and B on R2iC6 with rk.A/ D rk.B/ as well as a vector bundle morphism ' W A ! B such that Ug;i is exactly the degeneracy locus of . In other words, U2iC6;i is a virtual divisor on R2iC6 . By analogy with the case of the classical Green’s Conjecture, it is reasonable to conjecture that the morphism ' W A ! B is generically non-degenerate, and then Ug;i is a genuine divisor on Rg . We recall the statement of the Prym–Green Conjecture [FL] Conjecture 0.7: Conjecture 8.3. For a general curve ŒC;  2 R2iC6 one has the vanishing Ki;2 .C; KC ˝ / D 0 Note that if true, the Prym–Green Conjecture is sharp. For g < 2i C 6 it follows from previous considerations that Ki;2 .C; KC ˝ / ¤ 0 for any ŒC;  2 Rg . Example 8.4. We explain the simplest case of the Prym–Green Conjecture, namely when i D 0 and g D 6. Then one has an identification U6;0 D fŒC;  2 R6 W K0;2 .C; KC ˝ / ¤ 0g  ˚ ¤ D ŒC;  2 R6 W 0 .KC ˝ / W Sym2 H 0 .C; KC ˝ / ! H 0 .C; KC˝2 / : Observe that via Kodaira–Spencer theory, the following identifications hold:  _ TŒC; .R6 / D TŒC  .M6 / D H 0 .C; KC˝2 /

Prym varieties and their moduli

and

243

 _ TPr6 ŒC; .A5 / D Sym2 H 0 .C; KC ˝ / ;

that is, the multiplication map 0 .KC ˝ / is the codifferential of the Prym map and U6;0 is the ramification divisor of the generically finite covering Pr 6 W R6 ! A5 . The Prym–Green Conjecture in genus 6 is equivalent to the infinitesimal Prym–Torelli Theorem! An example of a Prym curve ŒC;  2 R6 for which 0 .KC ˝ / is an isomorphism, that is, ŒC;  2 R6  U6;0 , is provided by Beauville [B2]. Let C  P2 be a smooth plane quintic and choose a quartic X  P2 everywhere tangent to C , that is, X  C D 2.p1 C    C p10 /, where p1 ; : : : ; p10 2 C . Then take  WD OC .2/.p1      p10 /, thus ŒC;  2 R6 . It is not difficult to verify directly that the resulting Prym-canonical curve KC ˝ W C ,! P4 does not lie on a quadric. Example 8.5. As a consequence of the Green–Lazarsfeld non-vanishing theorem [GL1], one can exhibit two codimension two loci in Rg contained in Ug;i , namely 1 Z1 WD   .Mg;i C3 / D fŒC;  2 Rg W gon.C / i C 3g

and Z2 WD fŒC;  2 Rg W  2 CiC2  CiC2  Pic0 .C /g: It is a very interesting open problem to find a codimension one subvariety of Rg which contains both Z1 and Z2 and might be a suitable candidate to be equal to Ug;i . We envisage here a geometric condition in terms of Prym varieties which holds in codimension one in the moduli space, and which a posteriori, should be equivalent to the syzygy condition Ki;2 .C; KC ˝ / ¤ 0. For i D 0 we have seen that this condition is simply that the differential of the Prym map be not bijective. The Prym–Green Conjecture is known to hold in bounded degree. Note that for any integer l  3 one can formulate an analogous level l Prym–Green Conjecture predicting the vanishing Ki;2 .C; KC ˝ / D 0; where  2 Pic0 .C /  fOC g satisfies  ˝l D OC , with C being a general curve of genus 2i C 6. x g . Independent of the validity of the Prym– 8.1 Koszul divisor calculations on R Green Conjecture, one could try to compute the virtual class of a compactification of zg  R x g such Ug;i . It is shown in [FL] that over a partial compactification Rg  R zg ; R x g /  2, there exist extensions A z and B z of the vector bundles xg  R that codim.R z z A and B as well as a homomorphism denoted by 'Q W A ! B such that the degeneracy z g . Furthermore, the vector bundles locus of 'Q is precisely the closure of Ug;i inside R z and B z have modular meaning and one can compute their Chern classes in terms of A tautological classes:

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Theorem 8.6. Set g D 2i C 6. We have the following formula for the virtual class of the Prym–Green degeneracy locus: !

 virt  3.2i C 7/ 3 ram 2i C 2 0 00 S x g /: Ug;i D  .ı0 C ı0 /  ı0     2 Pic.R i C3 2 i  virt  S It is instructive to compare U g;i against the formula of the canonical class: 0 00 x g /: KRx g  13  2.ı0 C ı0 /  3ı0ram     2 Pic.R

Sg;i is a genuine divisor on Assuming the Prym–Green Conjecture in genus g, so that U x g as opposed to a virtual one, one obtains that the class K x is big precisely when R Rg the following equality is satisfied 3.2i C 7/ 13 < () i  3: i C3 2 x g is of general type, see [HM], [EH], [F2]. This implies When g  22 it is known that M x g , as a branched covering of M x g , is of general type as well. Even though the that R validity of the Prym–Green Conjecture for arbitrary g D 2i C 6 remains a challenging x g it is enough to know open problem, for applications to the birational geometry of R that the conjecture holds in bounded even genus g 20. This is something that can be checked (with quite some effort!) by degeneration with the help of the computer algebra program Macaulay2. To summarize we have the following result [FL]: x 2iC6 is a variety of general Theorem 8.7. The moduli space of stable Prym curves R x 12 is non-negative. type for i  4. The Kodaira dimension of R 8.2 Prym curves and the universal difference variety. The problem of determining x g for odd genus has a relatively simpler solution that the the Kodaira dimension of R even genus case. We follow [FL], Section 2. We fix a smooth non-hyperelliptic curve C of genus g. The i -th difference variety of C is defined as the image of the difference map  W Ci  Ci ! Pic0 .C /; .D1 ; D2 / WD OC .D1  D2 /: It is easy to prove, see e.g. [ACGH], that for i < g=2 the map  is birational onto its image. The following definition is due to Raynaud [R]: Definition 8.8. Let E 2 UC .r; d / be a semistable vector bundle on a curve C , such that the slope  WD d=r 2 Z. The theta-divisor of E is defined as the non-vanishing locus: ‚E WD f 2 Picg 1 .C / W H 0 .C; E ˝ / ¤ 0g

Prym varieties and their moduli

245

The locus ‚E is a virtual divisor inside Picg 1 .C /, that is, it is either the full Picard variety when H 0 .C; E ˝ / ¤ 0 for every , or a genuine divisor when there exists a line bundle  2 Picg 1 .C / such that H 0 .C; E ˝ / D 0. In that case, Œ‚E  D r , where  2 H 2 .Picg 1 .C /; Z/ is the class of the “classical” theta divisor. In the latter case, one says that E possesses a theta divisor. Let us assume that g WD 2i C 1, therefore Ci  Ci  Pic0 .C / is a divisor. We denote by QC WD MK_C the dual of the Lazarsfeld bundle, therefore .QC / D 2 2 Z and one may ask whether QC and all its exterior powers have theta divisors, and if so, whether they have an intrinsic interpretation in terms of the geometry of the canonical curve. Using a filtration argument due to Lazarsfeld, one finds that for a generic choice of distinct points x1 ; : : : ; xg2 2 C , there is an exact sequence 0 !

g2 M

OC .xl / ! QC ! KC ˝ OC .x1      xg2 / ! 0:

lD1

This leads to an inclusion of cycles Ci  Ci  ‚^i QC . The main result from [FMP] states that for any smooth curve ŒC  2 Mg the Raynaud locus ‚^i QC is a divisor in Pic0 .C / (that is, ^i QC has a theta divisor), and one has the following equality of cycles: ‚^i QC D Ci  Ci  Pic0 .C / This identification shows that via the difference map, Ci  Ci is a resolution of singularities of ‚^i QC . Having produced a distinguished divisor in the degree zero Jacobian of each curve, we can use it to obtain codimension 1 conditions in Rg by requiring that the point of order 2 belong to this divisor. We define the following locus in Rg : ˚  D2iC1 WD ŒC;  2 R2iC1 W  2 Ci  Ci  ˚ D ŒC;  2 R2iC1 W H 0 .C; ^i QC ˝ / ¤ 0 : Note that D2iC1 has two incarnations, the first one of a more geometric nature showing that points ŒC;  2 D2iC1 are characterized by the existence of a certain secant to the jKC ˝j

Prym-canonical curve C ! P2i1 , the second of a determinantal nature which is x2iC1 of D2iC1 inside very useful if one wishes to compute the class of the closure D x 2iC1 . One has the following formula, see [FL], Theorem 0.2: R x2iC1 inside R x 2iC1 is equal to: Theorem 8.9. The class of the closure D !  2i  i 2i C 1 ram 00 x 2iC1 /: x2iC1  1 .3i C 1/  .ı00 C ı0 /  ı0     2 Pic.R D 2i  1 i 2 4 Comparing this formula against the canonical class KRx g , we prove the following: x 2iC1 is of general type for g  7. Theorem 8.10. The moduli space R

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9 The birational geometry of the moduli of spin curves In this last lecture we propose to treat briefly the moduli space g classifying even theta-characteristics over curves of genus g, that is, the parameter space ˚  gC WD ŒC;  W ŒC  2 Mg ;  2 Picg1 .C /; ˝2 D KC and h0 .C; /  0 mod 2 : At first sight, one might think that the geometry of gC should mirror rather closely that of Rg , since both spaces parametrize curves with level two structures. Indeed there are certain similarities between gC and Rg . Both spaces are covers of Mg and they admit very similar compactifications via stable Prym and spin curves respectively. On the other hand, there are also important differences reflected in birational geometry (to put it loosely, gC seems to be easier to describe than Rg ), as well as in the study of singularities (and here in contrast, the singularities of xgC appear to be more complicated x g ). Both spaces admit obvious higher level generalizations and one can than those of R C talk of moduli spaces Rg;l and g;l for any level l  3. We shall not discuss in these lectures the properties (or even the definition) of these spaces, but the trends observed for level 2 (including Kodaira dimension and singularities) persists and become even more pronounced as the level l increases. A geometrically meaningful compactification of gC by means of stable spin curves, has been found by Cornalba [Co]: Definition 9.1. An even stable spin curve of genus g is a triple ŒX; ; ˇ where: • X is a quasi-stable curve with pa .X / D g. •  2 Picg1 .X /. • E D OE .1/ for all exceptional components E X . • ˇ W ˝2 ! !X is a sheaf morphism which is an isomorphism along each nonexceptional component of X . xg , Hoping this shall not cause confusion with the previously discussed case of R x g the map given by .ŒX; ; ˇ/ WD Œst.X / forgetting we also denote by  W xgC ! M the spin structure and contracting, if necessary, the exceptional components. Note that deg./ D 2g1 .2g C 1/ is the number of even theta-characteristics on any smooth curve of genus g. One has the following commutative diagram: gC   

 Mg  

/ xgC 

 xg . /M

The following is a complete birational classification of xgC in terms of Kodaira dimension.

Prym varieties and their moduli

247

Theorem 9.2. (a) [FV2] The moduli space xgC is uniruled for g 7. (b) [FV2] The Kodaira dimension of xC is equal to zero. 8

(c) [F3] The moduli space xgC is of general type for g  9. Remark 9.3. We observe that as g increases, xgC becomes faster of general type than x g . The two spaces have different Kodaira dimension for genus 8. R It is instructive to repeat in the context of xgC an exercise already carried out for x g and determine in the process the ramification divisor of the covering : R x g denote the closure of the divisor of irreducible nodal Example 9.4. Let 0 M curves. We choose a general point ŒCxy  2 0 and an even stable spin curve ŒX; ; ˇ 2  1 .ŒCxy / with stable model Cxy . Then there are two possibilities, depending on whether X contains an exceptional component or not: • X D Cxy and then  2 Picg1 .X /. Denoting by C 2 Picg1 .C / the pull-back of  to the normalization of X , we observe that ˝2 D KC .x C y/ and the fibers C .x/ and C .y/ can be identified in a unique way such that the resulting line bundle on X satisfies h0 .X; /  0 mod 2. We denote by A0 the closure of the locus of such points in xgC . • X D C [fx;yg E, where E Š P1 is an exceptional component meeting the other component of X in two points. Then by definition E D OE .1/ and an easy application 0 of the Mayer–Vietoris sequence on X gives that ˝2 C D KC with h .C; C /  0 mod 2, that is, C is an even theta-characteristic on C . The closure of such points ŒX; ; ˇ 2 xgC will be denoted by B0 : Both A0 , B0 are irreducible boundary divisors of xgC and  is simply branched over B0 . Setting ˛0 WD ŒA0  and ˇ0 WD ŒB0  2 Pic.xgC /, the following formula holds:   .0 / D ˛0 C 2ˇ0 :

(8)

We leave it as an exercise to verify using Example 9.4 that indeed, deg.A0 =0 / C 2deg.B0 =0 / D 2g1 .2g C 1/: For 1 i Œg=2, we denote by Ai  xgC the closure of the locus corresponding to pairs of even pointed spin curves   C C ŒC; y; C ; ŒD; y; D  2 i;1  gi;1 and by Bi  xgC the closure of the locus corresponding to pairs of odd spin curves      gi;1 : ŒC; y; C ; ŒD; y; D  2 i;1 Setting ˛i WD ŒAi  2 Pic.xgC /; ˇi WD ŒBi  2 Pic.xgC /, one has the relation   .ıi / D ˛i C ˇi :

(9)

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Again, we invite the reader to check that deg.Ai =i / C deg.Bi =i / D 2g1 .2g C 1/. Applying the Riemann-Hurwitz formula to the covering  coupled with formulas (8) and (9), one obtains: g

KxC  13  2˛0  3ˇ0  3.˛1 C ˇ1 /  2 g

Œ2 X

.˛i C ˇi / 2 Pic.xgC /:

iD2

jKC j

9.1 The theta-null divisor. Let C ! Pg1 be a non-hyperelliptic canonically embedded curve. The space of quadrics containing C ˚  I2 .C / WD Ker Sym2 H 0 .C; KC / ! H 0 .C; KC˝2 /   has dimension g2 . The space of rank three quadrics inside Sym 2 H 0 .C; KC / also 2g2 has codimension 2 , therefore the condition that there exist a rank three quadric in PI2 .C / is expected to be divisorial in moduli. Let Q 2 I2 .C / be a rank three quadric, hence Sing.Q/ is a .g  4/-dimensional linear space. Assume that C \ Sing.Q/ D fx1 ; : : : ; xn g. Then the unique ruling of Q cuts out a pencil A of degree g  1  n2 on C , such that KC D A˝2 ˝ OC .D/. If n D 0, that is, C \ Sing.Q/ D ;, then 1 A 2 Wg1 .C / is a theta-characteristic, which prompts us to define the following subvariety of gC : Definition 9.5. The theta-null divisor on gC is defined as the locus ‚null WD fŒC;  2 gC W h0 .C; /  2g: The locus in Mg consisting of curves whose canonical model lies on a rank three quadric breaks-up into components depending on the cardinality #.C \ Q/. For each integer gC2 n g  1, we define the Gieseker–Petri divisor 2 G P 1g;k WD fŒC  2 Mg W 9A 2 Wk1 .C / such that H 0 .C; KC ˝ A˝.2/ / ¤ 0g: Note that .‚null / D G P g;g1 and one has a set-theoretic equality 

˚

ŒC  2 Mg W there exists Q 2 PI2 .C / with rank.Q/ 3 D

g1 [

G P 1g;k :

kDΠgC3 2 

The following result is proved in [F3], Theorem 0.2: Theorem 9.6. The class of the closure of the locus of vanishing theta-characteristics in xgC is equal to Œg

x null ‚

2 1 1 1X   ˛0  ˇi 2 Pic.xgC /: 4 16 2

iD1

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Quite remarkably, the formula in Theorem 9.6 contains no terms involving ˇ0 or ˛i with i > 0! We can compare this formula against KxC and observe that KxC is g g not expressible as a combination of ‚null and boundary divisors and one need another effective divisor to offset the negative coefficient of ˇ0 in the expression of KxC . g

x g . The most classical divisors on M x g are loci of 9.2 Brill–Noether divisors on M r curves carrying a certain linear series of type gd . Let us fix integers r; d  1 such that the Brill–Noether number .r; g; d / D g  .r C 1/.g  d C r/ D 1: Recalling that .g; r; d / is the expected dimension of the determinantal subvariety Wdr .C / of the Jacobian Picd .C / consisting of linear series of dimension at least r, see [ACGH] Chapter 4, one expects when .g; r; d / D 1 the locus of curves with a grd to be a divisor. r D fŒC  2 Mg W Wdr .C / ¤ ;g is an irreducible divisor Indeed, the subvariety Mg;d and the class of its compactification has been computed [EH]: g

Œ2   X gC1 r x g /: x i.g  i /ıi 2 Pic.M ı0  ŒMg;d   cg;r;d .g C 3/  6 iD1

After a bit of linear algebra in the vector space Pic.xgC /, one finds that there exist constants a; b 2 Q>0 such that Œg

2 X  x r /  11g C 29  2˛0  3ˇ0  x null C b  .M a‚ ai ˛i C bi ˇi /; g;d gC1

iD1

where ai ; bi  2 for i  2 and a1 ; b1  3. Therefore gC is of general type if 11g C 29 < 13 () g > 8: gC1 Theorem 9.6 also shows that x8C cannot be uniruled, since we have found an explicit canonical divisor x2 / C KxC D a‚null C b  .M 8;7 8

4 X

.ai ˛i C bi ˇi /;

iD1

2 where M8;7 D fŒC  2 M8 W W72 .C / ¤ ;g. To complete the proof of Theorem 9.2 and show that KxC is rigid, we use the g following strategy and prove that the following statements hold, see [FV2]:

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x null is a uniruled extremal effective divisor. • ‚ 2 / are extremal • The boundary divisors Ai and Bi where i  1, as well as   .M8;7 and rigid.

x null < 0, x null such that R  ‚ • there exists a covering family of rational curves R ‚  x2 R   .M8;7 / D 0 and R  ˛i D R  ˇi D 0 for 1 i 4. Assuming this, for all integers n  1, we can write x null /j C na‚ x null : jnKxC j D jn.KxC  a‚ 8

8

We repeat this argument for the remaining divisors to get smaller and smaller linear systems, then we conclude that .x8C / D 0. x 8 . Of the three conditions listed above, the last one is by far 9.3 Mukai geometry of M the most difficult to realize. The fact that it can be achieved is rather counter-intuitive. x null therefore it should consist The curve R  x8C on one hand, should be contained in ‚ of Brill–Noether special spin curves. On the other hand, we require  .R/ be disjoint x 2 , that is, R should consist of spin curves which are general from the point from M 8;7 of view of another Brill–Noether theoretic condition. The fact that such an R  x8C exists and one can separate in such a fine way two distinct Brill–Noether conditions is x 8 as a GIT quotient of a certain due to the existence of a second birational model of M Grassmannian. Let V WD C 6 and consider the Grassmannian in the Plücker embedding G WD G.2; V / ,! P.ƒ2 V / D P14 : Then KG D OG .6/ and a general 7-plane P7  P14 intersects G along a smooth canonical curve of genus 8 with general moduli, see [M2]. Let us fix a point ŒC;  2 ‚null . The canonical model C P7 lies on a rank three quadric QC 2 H 0 .P7 ; C =P7 .2//. The quadric QC lifts to a quadric QG 2 H 0 .P14 ; G=P14 .2// containing the Grassmannian in its Plücker embedding. There is a 6-dimensional space of extensions of C by a K3 surface C \ P7



S \  P8



G \  P14

and for each such extension, the quadric QC lifts to a quadric QS 2 H 0 .P8 ; S=P8 .2//. Note that rank.QS / rank.QC / C 2 D 5, and for a general K3 extension S  C , the equality rank.QS / D 5 holds. Proposition 9.7. There is a pencil of K3 extensions of C S G such that rank.QS / D 4.

251

Prym varieties and their moduli

This result is proved in [FV2] and it plays a crucial role in the proof that .x8C / D 0. One has the following commutative diagram, showing that such a K3 surface S is a 7 W 1 cover of a smooth quadric Q0  P3 : S

P1

f

/ Q0 Š P1  P1 P3 OOO o OOO2 1 oooo OOO o o o OOO o o O' wooo

P1 .

The K3 surface S carries two elliptic pencils jE1 j and jE2 j corresponding to the projections 1 and 2 and such that Ei2 D 0 for i D 1; 2. Moreover, C  E1 C E2 and E1  E2 D 7. Let R be the pencil in x8C obtained by pulling-back via f planes passing through a general line l0  P3 . Then following [FV2] we write that R  D  .R/  D g C 1 D 9 and

R  .˛0 C 2ˇ0 / D R    .ı0 / D  .R/  ı0 D 6.g C 3/ D 66:

There are two reducible fibers in the pencil R corresponding to the planes through l0 spanned by the pairs of rulings of Q0 passing through the points of intersection of l0 \ Q0 . Each of them is counted with multiplicity 72 , hence R  ˇ0 D

7 7 C D 7: 2 2

Therefore we find that R  ˛0 D 52, hence x null D R‚ and

1 9 52 1 R   R˛0 D  D 1 < 0 4 16 4 16 x 2 / D 0: R    .M 8;7

This completes the proof of the fact that .x8C / D 0.

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Gavril Farkas, Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany E-mail: [email protected]

On generalized Wronskians ´ Letterio Gatto and Inna Scherbak

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Wro´nskians, in general . . . . . . . . . . . . . . . . . . . 2 Wro´nskians and linear ODEs . . . . . . . . . . . . . . . . 3 Wro´nski sections of line bundles . . . . . . . . . . . . . . 4 Wro´nskians of sections of Grassmann bundles (in general) 5 Wro´nskians of sections of Grassmann bundles of jets . . . 6 Linear systems on P 1 and the intermediate Wro´nskians . . 7 Linear ODEs and Wro´nski–Schubert calculus . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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257 260 262 265 274 280 282 288 292

Introduction Let f WD .f0 ; f1 ; : : : ; fr / be an .r C1/-tuple of holomorphic functions in one complex variable. The Wro´nskian of f is the holomorphic function W .f / obtained by taking the determinant of the Wro´nski matrix whose entries of the j -th row, 0  j  r, are the j -th derivatives of .f0 ; f1 ; : : : ; fr /. The first appearance of Wro´nskians dates back to 1812, introduced by J. M. Hoene-Wro´nski (1776–1853) in the treatise [28] – see also [45]. The ubiquity of the Wro´nskian in nearly all the branches of mathematics, from analysis to algebraic geometry, from number theory to combinatorics, up to the theory of infinite dimensional dynamical systems, is definitely surprising if compared with its elementary definition. The present survey aims to outline links between some different Wro´nskian materializations to make evident their common root. The emphasis will be put on the mutual relationships among linear ordinary differential equations (ODEs), the theory of ramification loci of linear systems (e.g. Weierstrass points on curves) and the intersection theory of complex Grassmann varieties, ruled by the famous calculus [51] elaborated in 1886 by H. C. H. Schubert (1848–1911), to which the Italians M. Pieri (1860–1913) and G. Z. Giambelli (1879–1953) contributed too – see [24], [40]. The notion of Wro´nskian belongs to mathematicians’ common background because of its most popular application, which provides a method (sketched in Section 2) to find a particular solution of a non-homogeneous linear ODE. It relies on the following  Work partially sponsored by PRIN “Geometria sulle Varietà Algebriche" (Coordinatore A. Verra), Politecnico di Torino. The second author was sponsored by an INDAM-GNSAGA grant (2009) for Visiting Professors at the Politecnico di Torino.

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key property of the Wro´nskian of a fundamental system of solutions of a linear homogeneous ODE: the derivative of the Wro´nskian is proportional to the Wro´nskian itself, whose proof is due to J. Liouville (1809–1882) and N. H. Abel (1802–1829). This apparently innocuous property should be considered as the first historical appearance of Schubert calculus. To see it, one must embed the Wro´nski determinant into a full family of generalized Wro´nskians, already used in 1939 by F. H. Schmidt [50] to study Weierstrass points and, in recent times and with the same motivation, by C. Towse in [52]. For a sample of applications to number theory see also [3] and [35]. If  D .0  1      r / is a partition, the generalized Wro´nskian W .f / is the determinant of the matrix whose j -th row, for 0  j  r, is the row of the derivatives of order j C rj of .f0 ; f1 ; : : : ; fr /. Clearly W .f / D W0 .f /, where the subscript 0 stands for the null partition .0; : : : ; 0/. The derivative of W .f /, appeared in the proof of Liouville’s–Abel’s theorem, is the first example of a generalized Wro´nskian, W.1/ .f /, corresponding to the partition .1; 0; : : : ; 0/. The bridge to Schubert calculus is our generalization of Liouville’s and Abel’s theorem (see [22]): Giambelli’s formula for generalized Wro´nskians holds. More precisely, if f is a fundamental system of solutions of a linear ODE with constant coefficients, then W .f / N 0 .f /, where is proportional to the usual Wro´nski determinant, W .f / D  .h/W N N  .h/ is the Schur polynomial associated to a sequence h D .h0 ; h1 ; : : : / of explicit polynomial expressions in the coefficients of the given ODE and to the partition  – see Section 7. If the characteristic polynomial of the linear differential equation splits into the product of distinct linear factors, then hj is nothing else than the j -th complete symmetric polynomial in its roots. Let us now change the landscape for a while. Take a smooth complex projective curve C of genus g  0 and an isomorphism class L 2 Picd .C / of line bundles of degree d on C . A gdr on C is a pair .V; L/, where V is a point of the Grassmann variety G.r C 1; H 0 .L// parameterizing .r C 1/-dimensional vector subspaces of the global holomorphic sections of L. If v D .v0 ; v1 ; : : : ; vr / is a basis of V; the Wro´nskian r.rC1/ W .v/ is a holomorphic section of the bundle Lg;r;d WD L˝rC1 ˝ K ˝ 2 – see Section 3. It can be constructed by gluing together local Wro´nskians W .f /, where f D .f0 ; f1 ; : : : ; fr / is an .r C 1/-tuple of holomorphic functions representing the basis v in some open set of C that trivializes L. As changing the basis of V amounts to multiply W .v/ by a non-zero complex number, one obtains a well-defined point W .V / WD W .v/ .mod C  / in P H 0 .Lg;r;d / called the Wro´nskian of V . The Wro´nski map G.r C 1; H 0 .L// ! P H 0 .Lg;r;d / mapping V to W .V / is a holomorphic map; two extremal cases show that, in general, it is neither injective nor surjective. Indeed, if C is hyperelliptic and L 2 Pic2 .C / is the line bundle defining its unique g21 , then G.2; H 0 .L// is just a point, and the Wro´nski map to P H 0 .Lg;1;2 / is trivially injective and not surjective. On the other hand, if C D P 1 and L D OP 1 .d /, then the Wro´nski map G.r C 1; H 0 .OP 1 .d /// ! P H 0 .L0;r;d / is a finite surjective morphism whose degree is equal to the Plücker degree of the Grassmannian G.r C 1; d C 1/, thence in this case the Wro´nski map is not injective, cf. [9].

On generalized Wro´nskians

259

The problem of determining the pre-image of an element of P H 0 .L0;r;d / through the Wro´nski map defined on G.r C 1; H 0 .OP 1 .d /// leads to an intriguing mixing of Geometry, Analysis and Representation Theory. It turns out that certain non-degenerate elements of G.r C1; H 0 .OP 1 .d ///, defined through suitable intermediate Wro´nskians, correspond to the so-called Bethe vectors appeared in representation theory of the Lie algebra sl rC1 .C/. The correspondence goes through critical points of a remarkable rational function related to Knizhnikov–Zamolodchikov equation on correlation functions of the conformal field theory, [37], [47], [48], [49]. Interestingly, the critical points of the mentioned rational function in the case r D 1 were examined in the XIX century, in works of Heine and Stieltjes on second order Fuchsian differential equations having a polynomial solution of a prescribed degree. Schubert calculus on Grassmannians has been introduced even before. However, the relationship between these items – in the case r D 1 – was conceived a decade ago in [46], [49]. In the real framework, the relationship between Wro´nskians, Schubert calculus and rational curves was discovered and studied by L. Goldberg, A. Eremenko & A. Gabrielov, V. Karlhamov & F. Sottile, and others – see [25], [10], [11], [30] and references therein. More links between linear differential equations, projective curves and Schubert varieties appeared in a local context in the investigations of M. Kazarian on singularities of the boundary of fundamental systems of solutions of linear differential equations, [29]. Here, we take another point of view. A. Nigro proposes to extend the notion of ramification locus of a linear system on a curve to that of ramification locus of a holomorphic section of a Grassmann bundle [39]. The construction was motivated by the following observation (see also [8]): Let triv .r;d / be the set of all the sections  W C ! G.r C 1; J d L/ such that the pull back of the tautological bundle r over G.r C 1; J d L/ is trivial. Then each gdr WD .V; L/ induces a holomorphic section V 2 triv .r;d /, via the bundle monomorphism C  V ! J d L (cf. Section 5.3). The point is that the space triv .r;d / is larger than the space of linear systems, and so the theory becomes richer. A distinguished subvariety indwells in G.r C 1; J d L/, called Wro´nski subvariety in [39]. It is a Cartier divisor which occurs as the zero locus of a certain Wro´nski section W . The Wro´nskian of any section  2 triv .r;d / is defined to be W0 ./ WD   W .mod C  /; if  D V for some V 2 G.r C 1; H 0 .L//, it coincides with the usual Wro´nskian of V – see Section 5. In particular, if M is a line bundle defining the unique g21 over a hyperelliptic curve of genus g  2, the extended Wro´nski map triv .1;2 / ! P H 0 .M˝2 ˝ K/ is dominant (see [8]), its behavior is closer to the surjectivity of the Wro´nski map defined on the space of gdr s on P 1 . The latter, in this case, coincides with triv .r;d / modulo identification of V with V . In general, the construction works as follows. Let % W F ! X be a vector bundle of rank d C 1 and %r;d W G.r C 1; F / ! X be the Grassmann bundle of .r C 1/dimensional subspaces of fibers of %. Consider 0 ! r ! %r;d F ! Qr ! 0, the universal exact sequence over G, and denote by  .c t .Qr  %r;d F // the Schur polynomial, associated to the partition , in the coefficients of the Chern polyno-

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mial of Qr  %r;d F . As is well known (see e.g. [15, Ch. 14]), the Chow group A .G/ of cycles modulo rational equivalence is a free A .X /-module generated by B WD f .c t .Qr  %r;d F // \ ŒG j  2 P .rC1/.d r/ g, where P .rC1/.d r/ denotes the set of the partitions  of length at most r such that 0  d  r, and  \ ŒG denotes the cap product with the fundamental class of G. Let F WD .Fi /d i0 be a filtration of F by quotient bundles, such that Fi has rank i . Schubert varieties f .F / j  2 P .rC1/.d r/ g associated to F (the definition is in Section 4.4) play the role of generalized Wro´nski subvarieties. In particular .1/ .F / is what in [39] was called the F -Wro´nski subvariety of G. It is a Cartier divisor, that is the zero locus V V of a section W of the bundle rC1 %r;d FrC1 ˝ rC1 r_ over G. We say that W is the F -Wro´nskian. If  W X ! GVis a holomorphic section, its Wro´nskian is, by V definition, W0 . / WD   W 2 H 0 . rC1 F ˝ rC1   r_ /. Its class in A .X / is nothing else than   Œ.1/ .F / \ ŒG. The generalized Wro´nski class of  in A .X / is   Œ .F / \ ŒX , which is the class of  1 . .F //, provided that the codimension of the locus coincides with the expected codimension jj WD 0 C    C r . Recall that Œ .F / can be easily computed as an explicit linear combination of the elements of the basis B above, for instance by the recipe indicated in Section 4, especially Theorem 4.13. Let now "i WD ci .r / 2 A .G/ be the Chern classes of the tautological bundle r ! G. Consider a basis v WD .v0 ; v1 ; : : : ; vr / of solutions of the differential equation y .rC1/  "1 y .r/ C    C .1/rC1 "rC1 y D 0; (1) taken in the algebra .A .G/ ˝ Q/ŒŒt  of formal power series in an indeterminate t with coefficients in the Chow ring of G with rational coefficients. In Section 7.12 we show that, for each partition  2 P .d r/.rC1/ ,  F // D  .c t .Qr  r;d

W .v/ ; W0 .v/

i.e. each element of the A .X /-basis of the Chow ring of G is the quotient of generalized Wro´nskians associated to a fundamental system of solutions of an ordinary linear ODE with constant coefficients taken in A .G/. This will be a consequence of Giambelli’s formula for generalized Wro´nskians, proven in [22], which so provides another clue to the ubiquity of Wro´nskians in mathematics. The survey was written with an eye on a wide range of readers, not necessarily experts in algebraic geometry. We thank the referees for substantial efforts to improve the presentation.

1 Wronskians, ´ in general 1.1. In the next two sections let K be either the real field R or the complex field C together with their usual euclidean topologies. If U  K is an open connected subset

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of K, we shall write O.U / for the K-algebra of regular K-valued functions defined over U : here regular means either C 1 differentiable if K D R or complex holomorphic if K D C. Let v WD .v0 ; v1 ; : : : ; vr / 2 O.U /rC1 : (2) If t is a local parameter on U , we denote by D W O.U / ! O.U / the usual derivation d=dt. The Wro´nski matrix associated to the .r C 1/-tuple (2) is the matrix valued regular function: 1 0 1 0 v0 v v1 ::: vr B Dv C B Dv0 Dv1 : : : Dvr C C B C B W M.v/ WD B : C D B : :: :: C : :: @ :: A @ :: : : : A Dr v

D r v0

D r v1

:::

D r vr

The determinant W0 .v/ WD det.W M.v// is the Wro´nskian of v WD .v0 ; v1 ; : : : ; vr /. It will be often written in the form: W0 .v/ WD v ^ Dv ^    ^ D r v:

(3)

In this paper, however, we want to see Wro´nskians as a part of a full family of natural functions generalizing them. They will be called, following the few pieces of literature where they have already appeared ([3], [52]) generalized Wro´nskians. 1.2. Generalized Wronskians. ´ Let r  0 be an integer. A partition  of length at most r C 1 is an .r C 1/-tuple of non-negative integers in the non-increasing order: (4)  W 0  1      r  0: Pr The weight of  D .0 ; 1 ; : : : ; r / is jj WD iD0 j , that is  is a partition of the integer jj. In this paper we consider only partitions of length r C 1. To each partition one may associate a Young–Ferrers diagram, an array of left justified rows, with 0 boxes in the first row, 1 boxes in the second row, …, and r boxes in the .r C1/-th row. We denote by P .rC1/.d r/ the set of all partitions whose Young diagram is contained in the .r C 1/  .d  r/ rectangle, i.e. the set of all partitions  such that d  r  0  1      r  0: If the last r  h entries of  2 P .rC1/.d r/ are zeros, then we write simply  D .0 ; 1 ; : : : ; h /, omitting the last zero parts. For more on partitions see [34]. 1.3. Definition. Let v as in (2) and  as in (4). The generalized Wro´nski matrix associated to v and to the partition  is, by definition, 1 0 1 0 D r v1 ::: D r vr D  r v0 D r v BD 1Cr1 v0 D 1Cr1 v1 : : : D 1Cr1 vr C BD 1Cr1 vC C B C B W M .v/ WD B C WD B C: :: :: :: :: : : A @ A @ : : : : : D rC0 v

D rC0 v0

D rC0 v1

:::

D rC0 vr

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The -generalized Wro´nskian is the determinant of the generalized Wro´nski matrix: W .v/ WD det W M .v/: Coherently with (3) we shall write the -generalized Wro´nskian in the form: W .v/ WD D r v ^ D 1Cr1 v ^    ^ D rC0 v:

(5)

The usual Wro´nskian corresponds to the partition of 0, that is W .v/  W0 .v/. 1.4. Remark. Notation (3) and (5) is convenient because the derivative of any generalized Wro´nskian can be computed via Leibniz’s rule with respect to the product “^”: D.W .v// D D.D r v ^ D 1Cr1 v ^    ^ D rC0 v/ X D i0 Cr v ^ D 1Ci1 Cr1 v ^    ^ D rCir C0 v: D i0 C i 1 C    C i r D 1 ij  0

A simple induction shows that any derivative of W .v/ is a Z-linear combination of generalized Wro´nskians. Recall, as in Section 1.2, that partitions can be described via Young–Ferrers diagrams, and that a standard Young tableau is a numbering of the boxes of the Young–Ferrers diagram of  with integers 1; : : : ; jj arranged in an increasing order in each column and each row [17]. The following observation has convinced us that the Schubert calculus can be recast in terms of Wro´nskians, see Section 7. 1.5. Theorem. We have D h W .v/ D

X

c W .v/;

jjDh

where c is the number of the standardYoung tableaux of theYoung–Ferrers diagram . The coefficients c ’s and their interpretation in terms of Schubert calculus are very well known; in particular, they can be calculated by the hook formula: c D

jjŠ ; k1 : : : kjj

where the kj ’s, 1  j  jj, are the hook lengths of the boxes of , see [17, p. 53].

2 Wronskians ´ and linear ODEs Wro´nskians are usually introduced when dealing with linear ordinary differential equations (ODEs).

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2.1. We use notation of Section 1.1. For a.t / D .a1 .t/; : : : ; arC1 .t // 2 O.U /rC1 and f 2 O.U /, consider the linear ODE D rC1 x  a1 .t /D r x C    C .1/rC1 arC1 .t /x D f

(6)

and the corresponding linear differential operator Pa .D/ 2 EndK .O.U //, Pa .D/ WD D rC1  a1 .t /D r C    C .1/rC1 arC1 .t /:

(7)

The set of solutions, f;a , of (6) is an affine space modelled over KrC1 : if xp is a particular solution , then f;a D xp C ker Pa .D/: The celebrated Cauchy theorem ensures that given a column c D .cj /0j r 2 KrC1 , there exists a unique element xc 2 ker Pa .D/ such that D j f .0/ D cj for all 0  j  r. Assume now that v as in (2) is a basis of ker Pa .D/. A particular solution of (6) can be found through the method of variation of arbitrary constants. Assume that 0 1 c0 .t / Bc1 .t /C B C c D c.t / D B : C 2 O.U /rC1 ; @ :: A cr .t / and look for a solution of (6) of the form xp WD .v  c/.t / D v.t /  c.t / D

r X

ci .t /vi .t /;

iD0

where “” stands for the usual row-by-column product. The condition that D j v  Dc D 0 for all 0  j  r means that D j xp D D j v  c for all 0  j  r and D rC1 xp D D rC1 v  c C D r v  Dc. The equality Pa .D/xp D f implies, by substitution, the equation D r v  Dc D f: The unknown functions c D c.t / must then satisfy the differential equations 0 1 0 1 Dc0 0 BDc1 C B 0 C B C B C W M.v/ B : C D B : C : @ :: A @ :: A Dcr

f

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The key remark is that the Wro´nski matrix is invertible in O.U /. Thus we get a system of first order ODEs, 0 1 0 B0C B C Dc D .W M.v//1  B : C ; @ :: A f

which can be solved by usual methods. To show the invertibility, one usually shows that if the Wro´nski matrix does not vanish at some point of U , then it does vanish nowhere on U (recall that U is a connected open set). Assume W0 .v/.P / ¤ 0 for some P 2 U . Let us choose a local parameter t on U which is 0 at P , identifying the open set U with a connected neighborhood of the origin. Computing the derivative of the Wro´nskian one discovers the celebrated 2.2. Liouville’s theorem ([4], p. 195, §27.6). The Wro´nskian W D W0 .v/ satisfies the differential equation (8) DW D a1 W: The proof of Theorem 2.2 is as follows. By defining Dv as the row whose entries are the derivatives of the entries of v, one notices that Pa .D/v D .Pa .D/v0 ; Pa .D/v1 ; : : : ; Pa .D/vr / D 0: Thence D rC1 v D a1 .t /D r v  a2 .t /D r1 v C    C .1/r arC1 .t /v and one gets DW0 .v/ D D.v ^ Dv ^    ^ D r v/ D v ^ Dv ^    ^ D r1 v ^ D rC1 v D v ^ Dv ^    ^ .a1 .t /D r v  a2 .t /D r1 v C    C .1/r v/ D a1 .t /v ^ Dv ^    ^ D r v D a1 .t /W0 .v/: The Wro´nskian then takes the form (Abel’s formula) Z t  W0 .v/ D W0 .v/.0/  exp a.u/du ;

(9)

0

where W0 .v/.0/ denotes the value of the Wro´nskian at t D 0. Equation (9) shows that if W .v/.0/ ¤ 0 then W .v/.t / ¤ 0 for all t 2 U . We shall see in Section 7 why the proof of Liouville’s theorem is a first example of the Schubert calculus formalism governing the intersection theory on Grassmann schemes. 2.3. Generalized Wronskians ´ of solutions of ODEs. Using generalized Wro´nskians as in 1.2, Liouville’s theorem (8) can be rephrased as W.1/ .v/ D a1 .t /W0 .v/; and generalized as follows.

On generalized Wro´nskians

265

2.4. Proposition. Let 1k WD .1; 1; : : : ; 1/ be the primitive partition of the integer 1  k  r C 1. If v WD .v0 ; v1 ; : : : ; vr / is a basis of ker Pa .D/ then W.1k / .v/ D ak .t /W .v/:

(10)

Indeed, consider (7). If v 2 ker Pa .D/, then it is a K-linear combination of v0 ; v1 ; : : : ; vr , and hence the Wro´nskian of these r C 2 functions vanishes: W .v; v0 ; v1 ; : : : ; vr / D 0: By expanding the Wro´nskian along the first column one obtains W .v/D rC1 v  W.1/ .v/D r v C    C .1/rC1 W.1rC1 / .v/v D 0;

(11)

and combining with Pa .D/ D 0 this implies rC1 X

.1/k .W.1k / .v/  ak .t /W .v//D k v D 0:

(12)

kD1

For general v 2 ker Pa .D/, the .r C1/-tuple .v; Dv; : : : ; D r v/ is linearly independent, and then (12) implies (10) for all 1  k  r C 1. 2.5. A natural question arises: Can we conclude that any generalized Wro´nskian W .v/ associated to a basis of ker Pa .D/ is a multiple of the Wro´nskian W0 .v/? The answer is obviously yes. In fact whenever one encounters one exterior factor in the generalized Wro´nskian of the form D j Crj v with j C rj  r C 1, one uses the differential equation to express D j Crj v as a linear combination of lower derivatives of the vector v, with coefficients polynomial expressions in a and its derivatives, W .v/ D G .a; Da; D 2 a; : : : /W .v/: The coefficient G .a; Da; D 2 a; : : : / assumes a particular interesting form in the case the coefficients a of the equation are constant (so D i a D 0, for i > 0). We will address this case in Section 7.

3 Wronski ´ sections of line bundles 3.1. A holomorphic vector bundle of rank d C 1 on a smooth complex projective variety X is a holomorphic map % W F ! X, where the complex manifold F is locally a product of X and a complex .d C 1/-dimensional vector space, cf. [26, page 69]. For P 2 X, we denote by FP WD %1 .F / F the fiber. Consider the vector space H 0 .F / WD H 0 .X; F / of global holomorphic sections of F (omitting the base variety when clear from the context). For s 2 H 0 .X; F / we will denote the value of s at P 2 X by s.P / 2 FP . The image of s in the stalk of the sheaf of sections of F at P will be denoted by sP .

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A line bundle over X is a vector bundle of rank 1. The set of isomorphism classes of line bundles on X is a group under the tensor product; this group is denoted by Pic.X/. If W X ! S is a proper flat morphism, then we define a relative line bundle as an equivalence class of line bundles on X , where L1 and L2 are declared equivalent  if L1 ˝ L1 2 Š N , for some N 2 Pic.S /. The group of isomorphism classes of relative line bundles on X is denoted by Pic.X=S / WD Pic.X /=  Pic.S /. 3.2. In the attempt to keep the paper self-contained, we recall a few basic notions about line bundles on a smooth projective complex curve. From now on, we denote the curve by C . It will often be identified with a compact Riemann surface, i.e. with a complex manifold of complex dimension 1 equipped with a holomorphic atlas A WD f.U˛ ; z˛ / j ˛ 2 Ag, where z˛ is a local coordinate on an open U˛ . In this context, denote by OC the sheaf of holomorphic functions on C : for .U˛ ; z˛ / 2 A the sheaf OC .U˛ / is the C-algebra of complex holomorphic functions in z˛ . The canonical line bundle of C is the line bundle K ! C whose transition functions are the derivatives of the coordinate changes,

˛ˇ W U˛ \ Uˇ ! C  ;

˛ˇ D dz˛ =dzˇ :

The holomorphic functions f ˛ˇ g obviously form a cocycle: ˛ˇ ˇ D ˛ . A global holomorphic section ! 2 H 0 .C; K/ is a global holomorphic differential, i.e. a collection ff˛ dz˛ g, where f˛ 2 O.U˛ / and f˛ jU˛ \U D ˛ˇ fˇ jU \U . We shall write ˇ

˛

ˇ

f˛ dz˛ D !jU˛ . The integer g D h0 .K/ WD dimC H 0 .K/ is the genus of the curve.

3.3. Jets of line bundles. Let W X ! S be a proper flat family of smooth projective curves of genus g  1 parameterized by some smooth scheme S . Let X S X ! S be the 2-fold fiber product of X over S and let p; q W X S X ! X be the projections onto the first and the second factor respectively. Denote by ı W X ! X S X be the diagonal morphism and by  the ideal sheaf of the diagonal in X S X. The relative canonical bundle of the family is by definition K WD ı  .= 2 /. For each L 2 Pic.X=S /, see 3.1, and each h  0 let   OXS X h  ˝q L (13) J L WD p  hC1 be the bundle of jets (or principal parts) of L of order h. As X is smooth, J h L is a vector bundle on X of rank h C 1. By definition, J 0 L D L. Set, by convention, J 1 L D 0 – the vector bundle of rank 0. The fiber of J h L over P 2 X – a complex vector space of dimension h C 1 – will be denoted by JPh L. The obvious exact sequence OXS X OXS X h ! hC1 ! ! 0 hC1   h gives rise to an exact sequence (see [32, p. 224] for details) 0 !

th;h1

0 ! L ˝ KhC1 !J h L ! J h1 L ! 0:

(14)

On generalized Wro´nskians

267

If 0 W C ! fptg is a trivial family over a point, i.e. reduced to a single curve, and if L is any line bundle, then the exact sequence (14) for J h L remains the same: in this case the relative canonical bundle coincides with the canonical bundle of the curve. 3.4. In the notation of Section 3.2, let v D .v˛ / be a non-zero holomorphic section of a line bundle L, i.e. v˛ 2 O.U˛ / and v˛ D `˛ˇ  vˇ on U˛ \ Uˇ , where f`˛ˇ g are transition functions. Let .U˛ ; z˛ / be a coordinate chart of C trivializing L. Denote by D˛ W O.U˛ / ! O.U˛ / the derivation d=dz˛ and by D˛j the j -th iterated of D˛ . Then 9 80 1 > ˆ > ˆ v˛ > ˆ = ˆ @ : A > ˆ > ˆ ; : D˛h v˛ is a section of J h L – see [8]. It may thought of as a global derivative of order h of the section v. In fact it is a local representation of v together with its first h derivatives. The truncation morphism occurring in (14), th;h1 W J h L ! J h1 L, is defined in such a way that th;h1 .Dh v.P // D .Dh1 v/.P /. See [8] for further details. 3.5. One says that v 2 H 0 .L/ vanishes at P 2 C with multiplicity at least h C 1 if .Dh v/.P / D 0. Concretely, if v˛ 2 OC .U˛ / is the local representation of v in the open set U˛ , then v vanishes at P 2 U˛ with multiplicity at least h C 1 if v˛ vanishes at P together with all of its first h derivatives. The fact that Dh v is a section of J h L says that the definition of vanishing at a point P does not depend on the open set U˛ containing it. We also say that the order of v at P is h  0 if Dh1 v.P / D 0 and Dh v.P / ¤ 0. To each 0 ¤ v 2 H 0 .L/ one may attach a divisor on C : X .ordP v/P: (16) .v/ D P 2C

The sum (16) is finite because v is locally a holomorphic function and hence its zeros are isolated P and the compactness of C implies that they are finitely many. The degree of v is P 2C ordP v  0. This number does not depend on a holomorphic section of L and, by definition, is the degree of L. The degree of the canonical bundle is 2g  2 [1, p. 8]. The set of isomorphism classes of line bundles of degree d is denoted by Picd .C /. If W X ! S is a smooth proper family of smooth curves of genus g, then Picd .X=S / denotes the relative line bundles of relative degree d . A bundle L 2 Pic.X=S/ has relative degree d if deg.LjXs / D d for each s 2 S. 3.6. If U is a (finite dimensional complex) vector space, G.k; U / will denote the Grassmannian parameterizing the k-dimensional vector subspaces of U . Let gdr .L/ be a point of G.r C 1; H 0 .L//, where L 2 Picd .C /. We write gdr for gdr .L/ and some P L 2 Picd .C /. If E D eP P is an effective divisor on C , and V is a gdr .L/, let V .E/ WD fv 2 V j ordP v  eP g:

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Clearly V .E/ is a vector subspace in V ; it is not empty because it contains at least the zero section. If dim V .P / D r for all P 2 C , then the gdr .L/ is said to be base point free. It is very ample if dimC V .P  Q/ D r  1 for all .P; Q/ 2 C  C . If V is base point free and v WD .v0 ; v1 ; : : : ; vr / is a basis of V , the map v W C ! P r ;

P 7! .v0 .P / W v1 .P / W : : : W vr .P //;

(17)

is a morphism whose image is a projective algebraic curve of degree d . Although the complex value of a section at a point is not well defined, the ratio of two sections is. Thus the map (17) is well defined. If V is very ample, (17) is an embedding, i.e. a biholomorphism onto its image. 3.7. Let ! WD .!0 ; !1 ; : : : ; !g1 / be a basis of H 0 .K/. The map ! WD .!0 W !1 W : : : W !g1 / W C ! P g1 sending P 7! .!0 .P / W !1 .P / W : : : W !g1 .P // is the canonical morphism, that is, its image in P g1 is a curve of degree 2g  2. If the canonical morphism is not an embedding, the curve is called hyperelliptic. 3.8. Definition. Let V be a gdr .L/. A point P 2 C is a V -ramification point if there exists 0 ¤ v 2 V such that Dr v.P / D 0, i.e. iff there exists a non-zero v 2 V vanishing at P with multiplicity r C 1 at least. Ramification points of a gdr can be detected as zero loci of suitable Wro´nskians. Let v WD .v0 ; v1 ; : : : ; vr / be a basis of V and let vi;˛ W U˛ ! C be holomorphic functions representing the restriction Pof the section vi to U˛ , for 0  i  r. If P 2 U˛ is a V -ramification point, let v D riD0 ai vi be such that Dr v.P / D 0. The last condition translates into the following linear system: 0 10 1 0 1 0 1 v0;˛ a0 0 a0 v1;˛ ::: vr;˛ BD˛ v0;˛ D˛ v1;˛ : : : D˛ vr;˛ C Ba1 C B0C Ba1 C B CB C B C B C (18) W M˛ .v/ B : C WD B : :: :: C B :: C D B :: C : : : : : @ : @:A : : : A @ : A @:A ar

D˛r v0;˛

D˛r v1;˛

:::

D˛r vr;˛

ar

0

It admits a non-trivial solution if and only if the determinant W0 .v˛ / D v˛ ^ D˛ v˛ ^    ^ D˛r v˛ 2 OC .U˛ / vanishes at P . It is easy to check that on U˛ \ Uˇ one has (see e.g. [14, Ch. 2–18] or [8]) r.r1/ W0 .v˛ / D `˛ˇ rC1 . ˛ˇ / 2 W0 .vˇ /;

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On generalized Wro´nskians

and thus the data fW0 .v˛ / j ˛ 2 Ag glue together to give a global holomorphic section W0 .v/ 2 H 0 .C; L˝rC1 ˝ K ˝

r.rC1/ 2

/;

(19)

said to be the Wro´nskian of the basis v of V . The Wro´nskian of any such a basis cannot vanish identically. Indeed, write the section W0 .v/ as W0 .v/ WD Dr v0 ^    ^ Dr vr ; where Dr v is as in (15), i.e. Dr vj is locally represented by the j -th row of the matrix (18). Assume that W0 .v˛ / vanished everywhere along the smooth connected curve C . Then the sections Dr vj , for 0  j  r, corresponding to the columns of the matrix (18), are linearly dependent, that is, up to a basis renumbering, Dr v0 D a1 Dr v1 C    C ar Dr vr 2 H 0 .J r L/: However Dr W H 0 .L/ ! H 0 .J r L/ is a section associated to the surjection H 0 .J r L/ ! H 0 .L/, induced by the truncation map J r L ! L ! 0 (see e.g. [8, Section 2.7]) and one would get the non-trivial linear relation v0 D a1 v1 C    C ar vr 2 H 0 .L/; against the assumption that .v0 ; v1 ; : : : ; vr / is a basis of V . As a consequence the ramification locus of the given gdr occurs in codimension 1. The construction does not depend on the choice of a basis v of V . Indeed, if u were another one, then u D Av for some invertible A 2 GlrC1 .C/, and thence W0 .u/ D r.rC1/ det.A/W0 .v/. Thus any basis of V defines the same point of P H 0 .L˝rC1 ˝K ˝ 2 /, which we denote by W0 .V /. 3.9. The Wronski ´ map. We have so constructed a map G.r C 1; H 0 .L// ! P H 0 .L˝rC1 ˝ K ˝

r.rC1/ 2

/;

V 7! W0 .V /;

(20)

which associates to each gdr .L/ its Wro´nskian W0 .V /. Adopting the same terminology used in the literature when C D P 1 and L WD OP 1 .d / (see e.g. [10], [11]), the map (20) will be called Wro´nski map. Its behavior depends on the curve and on the choice of the linear system. It is, in general, neither injective nor surjective as the following two extremal cases show. If C D P 1 , the unique bundle of degree d is OP 1 .d /, K D OP 1 .2/ and the Wro´nski map G.r C 1; H 0 .OP 1 .d /// ! P H 0 .OP 1 ..r C 1/.d  r///; in this case defined between two varieties of the same dimension, is a finite surjective morphism of degree equal to the Plücker degree of the Grassmannian G.r C 1; d C 1/. In particular it is not injective – see [9], [48] and [10], [11] over the real numbers.

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At a general point of P H 0 .OP 1 ..r C 1/.d  r/// (represented by a form of degree .r C 1/.d  r/) there correspond as many distinct linear systems V as the degree of the Grassmannian. For a closer analysis of the fibers of such a morphism see [47]. On the other hand if C is hyperelliptic and M 2 Pic2 .C / is the line bundle defining its unique g21 , cf. Section 3.7, then G.2; H 0 .M// is just a point and the Wro´nski map: G.2; H 0 .M// ! P H 0 .M˝2 ˝ K/ is trivially injective and not surjective, since by the Riemann–Roch formula we have h0 .M˝2 ˝ K/ > 1. Later on we shall see how to make the situation more uniform, by enlarging in a natural way the notion of linear system on a curve. It will be one of the bridges connecting this part of the survey with the first one, regarding Wro´nskians of differential equations. 3.10. The V -weight of a point. Let V be a gdr and P 2 C . The V -weight at P is wt V .P / WD ordP W0 .V / D ordP W0 .v/; for some basis v of V . The total weight of the V -ramification points is X wt V D wt V .P /; P 2C

where the above sum is clearly finite. The total weight coincides with the degree of r.rC1/ the bundle L˝rC1 ˝ K ˝ 2 , i.e. the degree of its first Chern class: Z r.rC1/ wtV D c1 .L˝rC1 ˝ K ˝ 2 / \ ŒC  C Z Z r.r C 1/ (21) c1 .K/ \ ŒC  D .r C 1/ .c1 .L/ \ ŒC / C 2 C D .r C 1/d C .g  1/r.r C 1/; which is the so-called Brill–Segre formula. For example, a smooth plane curve of degree d can be thought of as an abstract curve (compact Riemann surface) embedded in P 2 via some V 2 G.3; H 0 .L// for some L 2 Picd .C /: .v0 W v1 W v2 / W C ! P 2 where v WD .v0 ; v1 ; v2 / is a basis of V . The V -ramification points correspond, in this case, to flexes of the image of C in P 2 . According to the genus-degree formulae, the total number of flexes, keeping multiplicities into account, is given by (21) for r D 2 f D 3d.d  2/; which is one of the famous Plücker formulas for plane curves.

On generalized Wro´nskians

271

3.11. Wronskians ´ on Gorenstein curves. Let C be an irreducible plane curve of degree d with ı nodes and cusps. Using the extension of the Wro´nskian of a linear system defined on a Gorenstein curve, due to Widland and Lax [53], the celebrated Plücker formula f D 3d.d  2/  6ı  8

can be obtained from the tautological identity (see [18] for details): ].smoothV -ramification points/ D ].ramification points/  ].singular ramification points/: For more on jets and Wro´nskians on Gorenstein curves see [12] and [13]. 3.12. The V -weight of a point P coincides with the weight of its order partition. We say that n 2 N is a V -order at P 2 C if there exists v 2 V such that ordP v D n. Each point possesses only r C 1 distinct V -orders. In fact n is a V -order if dim V .nP / > dim V ..n C 1/P /. We have the following sequence of inequalities: r C 1 D dim V  dim V .P /  dim V .2P /      dim V .dP /  dim V ..d C 1/P / D 0: The last dimension is zero because the unique section of V vanishing at P with multiplicity d C 1 is zero. At each step the dimension does not drop more than one unit and then there must be precisely r C 1 jumps. If 0  i0 < i1 <    < ir  d is the order sequence at some P 2 C , the V -order partition at P is .P / D .ir  r; ir1  .r  1/; : : : ; i1  1; i0 /: One may choose a basis .v0 ; v1 ; : : : ; vr / of V such that ordP vj D ij . The use of such a basis shows that the Wro´nskian W0 .v/ vanishes at P with multiplicity wt V .P / D

r X

.ij  j / D j.P /j.

j D0

The following result is due to [42] (unpublished) and to [52]. 3.13. Proposition. A partition  is the V -order partition at P 2 C if and only if W .v˛ /.P / D 0, for all  ¤  such that jj  jj, and W .v˛ /.P / ¤ 0 (here v˛ is any local representation of a basis of V around P ). In this case W0 .V / vanishes at P with multiplicity exactly jj.

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3.14. A more intrinsic way to look at Wro´nskians and ramification points, which can be generalized to the case of families of curves, is as follows. For V 2 G.r C 1; H 0 .L// one considers the vector bundle map Dr W C  V ! J r L

(22)

defined by Dr .P; v/ D Dr v.P / 2 JPr L. Both bundles have rank r C 1 and since V has only finitely many ramification points, there is a non-empty open subset of C where the map Dr has the maximal rank r C 1. Then P 2 C is a V -ramification point if rkP Dr  r. The rank of Dr is smaller than the maximum if and only if the determinant map of (22) rC1 ^

Dr W OC !

rC1 ^

J rL

vanishes at P . The Wro´nski section rC1 ^

Dr 2 H 0

 rC1 ^

 r.rC1/ J r L D H 0 .L˝rC1 ˝ K ˝ 2 /

vanishes precisely where the map Dr drops rank. If v D a0 v0 C  Car vr , with respect to the basis v D .v0 ; v1 ; : : : ; vr / of V , then Dr v D a0 Dr v0 Ca1 Dr v1 C  Car Dr vr . On a trivializing open set U˛ of C one has the expression 1 0 0 1 a0 v0;˛ C a1 v1;˛ C    C ar vr;˛ a0 Ba0 D˛ v0;˛ C a1 D˛ v1;˛ C    C ar D˛ vr;˛ C Ba1 C C B B C .Dr v/jU˛ D B C D W0 .v˛ /  B :: C : :: A @ @:A : a0 D˛r v0;˛ C a1 D˛r v1;˛ C    C ar D˛r vr;˛

ar

In other words, the local representation of the map Dr is W0 .v˛ / W U˛  C rC1 ! U˛  C rC1 ; from which

det.Dr jU˛ / D v˛ ^ D˛ v˛ ^    ^ D˛r v˛ ;

V i.e. rC1 Dr is represented by the Wro´nskian W0 .v/. Changing the basis v of V , the Wro´nski section gets multiplied by a non-zero complex number and hence rC1 ^

Dr mod C  D W0 .V / 2 P H 0 .L˝rC1 ˝ K ˝

r.rC1/ 2

/

i.e. precisely the Wro´nskian associated to the linear system V . 3.15. How do generalized Wro´nskians come into play in this picture? Here the question is more delicate. We have already mentioned that if the V -order partition of a point P is .P / then the generalized Wro´nskians W .V / must vanish for all  such that

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jj < j.P /j and W .P / ¤ 0. It is however well known that the general gdr on a general curve C has only simple ramification points, i.e. all the points have weight 1. This says that if a gdr has a ramification point with weight bigger than 1, the generalized Wro´nskians do not impose independent conditions, as the locus occurs in codimension 1 while the expected codimension is bigger than 1. To look for more geometrical content one can move along two directions. The first, that we just sketch here, consists in considering families of curves. Let W X ! S be a proper flat family of smooth curves of genus g and let .V ; L/ be a relative gdr , i.e. V is a locally free subsheaf of  L and L 2 Picd .X=S /. One can then study the ramification locus of the relative gdr which fiberwise cuts the ramification locus of Vs 2 G.r C 1; H 0 .LjXs // through the degeneracy locus of the map Dr W  V ! Jr L; where Jr L denotes the jets of L along the fibers (see e.g. [23]). The map above V V induces a section OX ! rC1 Jr L ˝ rC1  V , which is the relative Wro´nskian W0 .V/ of the family. Because of the exact sequence (14): rC1 ^

Jr L ˝

rC1 ^

˝ r.rC1/ 2

 V D L˝rC1 ˝ K

˝

rC1 ^

V :

In this case the class in A .X/ of the ramification locus of V is ˝ r.rC1/ 2

ŒZ.W0 .V// D c1 .L˝rC1 ˝ K

/   c1 .V /:

A second approach to enrich the phenomenology of ramification points consists in keeping the curve fixed and varying the linear system. This is the only possible approach with curves of genus 0: all the smooth rational curves are isomorphic, and all the gdr s, with base points or not, are parameterized by the Grassmannian G.r C 1; H 0 .OP 1 .d ///. Here the situation is as nice as one would desire: all what may potentially occur it occurs indeed. For instance, if 1 ; : : : ; h are partitions such that P ji j D .r C 1/.d  r/ (= the total weight of the ramification points of a gdr ) and P1 ; : : : ; Ph are arbitrary points on P 1 one can count the number of all of the linear system such that the V order partition at Pi is precisely i . However if C has higher genus, such a kind of analysis is not possible anymore. For instance the general curve C of genus g  2 has only simple Weierstrass points, i.e. all have weight 1, but each curve carries one and only one canonical system. The picture holding for linear systems on the projective line can be generalized in the case of higher genus curves provided one updates the notion of gdr .L/ to that of a section of a Grassmann bundle, a path which was first indicated in [20] and then further developed in [39] and [8]. Go to the next two sections for a sketch of the construction.

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4 Wronskians ´ of sections of Grassmann bundles (in general) This section is a survey of the construction appeared in [39], partly published in [8], with some applications in [20]. 4.1. Let %d W F ! X be a vector bundle of rank d C1 over a smooth complex projective variety X of dimension m  0. For each 0  r  d , let %r;d W G.r C 1; F / ! X be the Grassmann bundle of .r C 1/-dimensional subspaces of the fibers of F . For r D 0 we shall write %0;d W P .F / ! X , where P .F / WD G.1; F / is the projective bundle associated to F . The bundle G.r C 1; F / carries a universal exact sequence (cf. [15, Appendix B.5.7]): r 0 ! r ! %r;d F ! Qr ! 0; (23) where r is the universal subbundle of %r;d F and Qr is the universal quotient bundle. Let .%r;d / WD fholomorphic  W X ! G.r C 1; F / j %r;d ı  D idX g be the set of holomorphic sections of %r;d . The choice of  2 .%r;d / amounts to specify a vector sub-bundle of F of rank r C 1. In fact the pull-back   r via  2 .%r;d / is a rank r C 1 subbundle of F . Conversely, given a rank r C 1 subbundle V of F , one may define the section V 2 .%r;d / by V .P / D VP 2 G.r C 1; FP /. The set .%r;d / is huge and may have a very nasty behavior: even the case when X D P 1 and F D J d OP 1 .d /, is far from being trivial. In fact it is related with the small quantum cohomology of Grassmannians, see [2]. A first simplification is to fix 2 Pic.X/ to study the space V  .%r;d / D f 2 .%r;d / j rC1   r D g. Again, if D OP 1 .n/ and F D J d OP 1 .d /, then n .%r;d / WD OP 1 .n/ .%r;d / can be identified with the space of the holomorphic maps P 1 ! G.r C 1; d C 1/ of degree n, compactified in [2] via a Quot-scheme construction. We shall see the easiest case (n D 0) in Section 6. In the following, for our limited purposes, we shall restrict the attention to the definitely simpler set triv .%r;d / WD f 2 .%r;d / j   r is a trivial rank .r C 1/ subbundle of F ! Xg: 4.2. Proposition. The set triv .%r;d /, if non empty, can be identified with an open set of the Grassmannian G.r C 1; H 0 .F //. Proof. If  2 triv .%r;d /, there is an isomorphism  W X  C rC1 !   r . Then WD   . r / ı  W X  C rC1 ! F is a bundle monomorphism. Let i W X ! F defined by i .P / D .P; ei /. It is clearly a holomorphic section of F . Furthermore 0 ; 1 ; : : : ; r span an .r C 1/-dimensional subspace U of H 0 .F / which does not depend on the choice of the isomorphism . Thus   r is isomorphic to X  U and

On generalized Wro´nskians

275

.P / D fu.P / j u 2 U g 2 G.r C 1; FP /. Conversely, if U 2 G.r C 1; H 0 .F //, one constructs a vector bundle morphism  W X  U ! F via .P; u/ 7! u.P /. This V morphism drops rank if rC1  D 0, this is a closed condition and so there is an open set U  G.r C 1; F / such that for U 2 U, the map U makes X  U into a vector subbundle of F . One so obtains a section U by setting U .P / D UP 2 G.r C 1; FP /. The easy check that U D  and that UU D U is left to the reader. 4.3. Assume now that F comes equipped with a system F of bundle epimorphisms qij W Fi ! Fj , for each 1  j  i  d , such that Fd D F , where Fi has rank i C 1, qi i D idFi and qij qj k D qik for each triple d  i  j  k  1. We set F1 D 0 by convention. The map qdj W F ! Fj will be simply denoted by qj and fker.qi /g gives a filtration of F by subbundles of rank d  i . Let @i W r ! %r;d Fi be the composition of the universal monomorphism r ! %r;d F with the map qi . The universal morphism r can be so identified with @d . 4.4. For each  2 P .rC1/.d r/ the subscheme  .%r;d F / D fƒ 2 G.r C 1; F / j rk ƒ @j Crj 1  j; 0  j  rg

(24)

of G.r C1; F / is the -Schubert variety associated to the system F and to the partition . The Chow classes modulo rational equivalence fΠ.%r;d F / j  2 P .rC1/.d r/ g freely generate A .G.r C 1; F // as a module over A .X / through the structural map %r;d . qd h

4.5. For each 0  h  d C 1, let Nh .F / WD ker.F ! Fd h /. It is a vector bundle of rank h. One can define Schubert varieties according to such a kernel flag N .F / by setting, for each partition  of length at most r C 1,  .%r;d N .F // D fƒ 2 G.r C 1; F / j ƒ \ Nd C1.j Crj / .F /  r C 1  j g: It is a simple exercise of linear algebra to show that  .%r;d F / D  .%r;d N .F //: Both descriptions are useful according to the purposes. The first description is more suited to describe Weierstrass points as in Section 3 (it gives an algebraic generalization of the rank sequence in a Brill–Noether matrix, see [1, p. 154]), while the second is useful when dealing with linear systems on the projective line (see Section 6 below). 4.6. Definition. The F -Wro´nskian subvariety of G.r C 1; F / is W0 .%r;d F / WD .1/ .%r;d F /:

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By (24), the F -Wro´nski variety W0 .%r;d F / of G.r C 1; F / is the degeneracy scheme of the natural map @r W r ! %r;d Fr , i.e. the zero scheme of the map rC1 ^

rC1 ^

@r W

r !

rC1 ^

%r;d Fr :

The map W0 .%r;d F / WD

rC1 ^

@r 2 Hom



D H 0 X;

rC1 ^

 rC1 ^

%r;d r ;

%r;d Fr ˝

rC1 ^

rC1 ^

%r;d Fr

%r;d r_





(25)

V V is the Wro´nski section (of the line bundle rC1 %r;d Fr ˝ rC1 %r;d r_ ). The F Wro´nski variety is then a Cartier divisor, because it is the zero scheme of the Wro´nski section (25). In this setting, the Schubert subvariety  .%r;d F / of G.r C 1; F /, associated to the partition  2 P .rC1/.d r/ , plays the role of a generalized Wro´nski subvariety associated to the system F . 4.7. Among all such Schubert varieties associated to F one can recognize some distinguished ones. It is natural to define the F -base locus subvariety of G.r C 1; F / as B.%r;d F / D .1rC1 / .%r;d F /I and the F -cuspidal locus subvariety as C .%r;d F / D .1r / .%r;d F /: Each Schubert subvariety  .%r;d F / has codimension jj in G.r C 1; F /. In particular, the base locus variety B.%r;d F / has codimension r C 1. 4.8. Let  2 .%r;d /. The F -ramification locus of  is the subscheme  1 .W0 .%r;d F // of X, its F -base locus is and its F -cuspidal locus is

 1 .B.%r;d F //  1 .C .%r;d F //:

The definition of Wro´nski map defined on sections of Grassmann bundles equipped with filtrations, as in Section 4.3, is very natural too. 4.9. Definition. For  2 .%r;d /, the section rC1  rC1  ^ ^ Fr ˝   r_ W0 . / WD   .W0 .%r;d F // 2 H 0 X;

will be called the F -Wro´nskian of  .

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The class in A .X / of the ramification locus of  is ŒZ.W0 .// D Œ 1 .W0 .%r;d F // D   ŒW0 .%r;d F / D c1

 rC1 ^

Fr ˝

rC1 ^

(26)    r_ \ ŒX  D .c1 .Fr /    c1 .r // \ ŒX :

If X is a curve, the expected dimension of the ramification locus is 0 and so, when  is not entirely contained in the Wro´nski variety, the total weight w of the ramification points of  is by definition the degree of the cycle Œ 1 .W0 .%r;d F //: Z w D

X

.c1 .Fr /    c1 .// \ ŒX :

According to the definitions above, a point P 2 X is a ramification point of  2 .%r;d / if W0 ./.P / D 0, which amounts to say that the map   @r W   r ! Fr drops rank at P . 4.10. Definition. Fix 2 Pic.X /. The holomorphic map  .%r;d / ! P H 0

 rC1 ^

 Fr ˝ _ ;

 7! W0 . /

.mod C  /;

is the Wro´nski map defined on  .%r;d /. V V V Indeed W0 . / is a section of   . rC1 %r;d Fr ˝ rC1 r_ / D rC1 Fr ˝ _ . The class of the ramification locus of  , as in (26), can be now expressed as ŒZ.W0 . // D .c1 .Fr /  // \ ŒX  2 A .X /: 4.11. The extended Wronski ´ map. It is particularly easy to express the Wro´nskian of a section  2 triv .%r;d /. Let U 2 G.r C 1; F / such that  D U . The pull-back of the map @r W r ! %r;d Fr is   @r W X  U ! Fr :

(27)

The Wro´nskian is the determinant of the map (27): rC1 ^

  @r W

rC1 ^

.X  U / !

rC1 ^

Fr :

Once a basis .u0 ; u1 ; : : : ; ur / of U is chosen, the Wro´nskian rC1 ^

 rC1 ^    @r 2 H 0 X; Fr

is represented by the holomorphic section X !

VrC1

Fr given by

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P 7! qr .u0 /.P / ^ qr .u1 /.P / ^    ^ qr .ur /.P / 2

VrC1

FP ;

where qr is the epimorphism introduced in 4.3. Changing basis the section gets multiplied by a non-zero constant, and so the Wro´nski map V triv .%r;d / ! P H 0 .X; rC1 Fr / V defined by  7! W0 . / mod C  2 P H 0 .X; rC1 Fr / coincides with the map 0

G.r C 1; H .F // ! P H

0



X;

rC1 ^

 Fr ;

U 7! qr .u0 / ^ qr .u1 / ^    ^ qr .ur / mod C  : where u D .u0 ; u1 ; : : : ; ur / is any basis of U . 4.12. Here is a quick review of intersection theory on G.r C 1; F / which is necessary for enumerative geometry purposes. First recall some basic terminology and notation. P Let a D a.t / D n0 an t n be a formal power series with coefficients in some ring A and  be a partition as in (4). Set an D 0 for n < 0. The -Schur polynomial associated to a is, by definition, ˇ ˇ ˇ a ar1 C1 ::: a0 Cr ˇˇ ˇ r ˇa 1 ar1 : : : a0 Cr1 ˇˇ ˇ r  .a/ D det.aiCri j /0i;j r D ˇ : ˇ : (28) :: :: :: ˇ ˇ :: : : : ˇ ˇ ˇa r a .r1/ : : : a0 ˇ r r1 The Chern polynomial of a bundle E is denoted by c t .E/. Write c t .Qr  %r;d F // for the ratio c t .Qr /=c t .%r;d F / of Chern polynomials. According to the Basis Theorem [15, p. 268], the Chow group A .G.rC1; F // is a free A .X /-module (via the structural morphism %r;d W A .X / ! A .G.r C 1; F //) generated by f  .c t .Qr  %r;d F // \ ŒG.r C 1; F / j  2 P .rC1/.d r/ g: If r D 0, let

i WD .1/i c1 .0 /i \ ŒP .F /

for each i  0. Then, by [15, Ch. 14], .0 ; 1 ; : : : ; d / is an A .X /-basis of A .P .F // and for each j  0 the following relation, defining the Chern classes of F , holds: d C1Cj C %0;d c1 .F /d Cj C    C %0;d cd C1 .F /j D 0:

(29)

V A main result of [21] says that rC1 A .P .F // can be equipped with a structure of A .G.r C 1; F //-module of rank 1. It is generated by 0 ^ 1 ^    ^ r in such a way that, for each  2 P .rC1/.d r/ ,  .c t .Qr  %r;d F //  0 ^ 1 ^    ^ r D r ^ 1Cr1 ^    ^ rC0 : (30)

On generalized Wro´nskians

279

We shall see in the last section that  .c t .Qr  %r;d F // are related to Wro´nskians associated to a fundamental system of solutions of a suitable differential equation. Define now:  i WD Œ.i/ .%0;d F / 2 A .P .F //; 0  i  d; where .i/ .%0;d F / is nothing but the zero locus in codimension i of the map @i1 W 0 ! Fi1 . Because of the relation i D

i X

%0;d cj .Fi1 /ij ;

(31)

j D0

it follows that . 0 ;  1 ; : : : ;  d / is an A .X /-basis of A .PV .F // as well. For  2 P .d C1/.d r/ let   WD  r ^  1Cr1 ^    ^  rC0 2 rC1 A .P .F //. Again by [21], the set f  j  2 P .rC1/.d r/ g is an A .X /-basis of A .G.r C 1; F //. Denote by Œ .%r;d F / the class in A .G.r C 1; F // of the F -Schubert variety  .%r;d F /. 4.13. Theorem. The following equality holds: Œ .%r;d F / D Œ.r / .%0 F / ^ Œ.1Cr1 / .%1 F / ^    ^ Œ.rC0 / .%r F / D   (32) V modulo the identification of A .G.r C 1; F // with rC1 A .P .F //. Equality (32) is an elegant and compact re-interpretation of the determinantal formula of Schubert calculus proven by Kempf and Laksov in [31] to compute classes of degeneracy loci of maps of vector bundles. This formula was first generalized in [43, (8.3)], see also [44, Example 3.5 and Appendix 4]. Then a far reaching generalization was obtained in [16] with help of the correspondences in flag bundles. In fact in [43], the P -ideals of polynomials supported on degeneracy loci were studied, giving a deeper insight in enumerative geometry of these loci. Formula (32) was basically discovered in [19] for trivial bundles. The present formulation is as in [39]. Let us sketch the proof of Theorem 4.13. Set j WD d  r  rj , then  WD .0 ; 1 ; : : : ; r / 2 P .rC1/.d r/ : Denote Aj WD Nrj Cj C1 (see Section 4.5), i.e Aj fits into the exact sequence 0 ! Aj ! F ! Fd .j Crj /1 ! 0: Then 0 ¨ A0 ¨ A1 ¨    ¨ Ar is a flag of subbundles of Fd . The Schubert variety .A0 ; A1 ; : : : ; Ar / D fƒ 2 G.r C 1; F / j ƒ \ Ai  i g coincides with  .%r;d F / defined by (24), as a simple check shows. Formula 7.9 in [33], which translates the determinantal formula proven in [31], implies Œ.A0 ; A1 ; : : : ; Ar / D Œ.A0 / ^ Œ.A1 / ^    ^ Œ.Ar /; which is thence equivalent to (32).

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5 Wronskians ´ of sections of Grassmann bundles of jets 5.1. The general framework of Section 4 shows that the notion of linear system can be generalized into that of pairs .; F /, where F is a vector bundle on X equipped with a filtration and  a section of the Grassmann bundle G.r C 1; F /. This picture can be fruitfully applied in the case of (families of) smooth complex projective curves of genus g  0. For the time being let C be any one such, and let L 2 Picd .C /. In this section we shall denote by %d W J d L ! C the bundle of jets of L ! C up to the order d . Accordingly, for each 0  r  d , we shall denote %r;d W G.r C 1; J d L/ ! C the Grassmann bundle of .r C 1/-dimensional subspaces of fibers of %. The natural filtration of J d L given by the quotients J d L ! J i L ! 0 , for 1  i  d , will be denoted J  L (setting J 1 L D 0). 5.2. The kernel filtration of J d L, N .L/: 0 N1 .L/    Nd .L/ Nd C1 .L/ D J d L;

(33)

is defined through the exact sequence of vector bundles 0 ! Nh .L/ ! J d L ! J d h L ! 0, where Nh .L/ is a vector bundle of rank h. It will be also called the osculating flag – see below and Section 6. The fiber of Nh .L/ at P 2 C will be denoted by Nh;P .L/. As in the previous section, the -generalized Wro´nskian subvariety of G.r C 1; J d L/ is  .%r;d J  L/, which has codimension jj in G.r C 1; J d L/. By virtue of Proposition 4.2, the space triv .%r;d / of sections  of %r;d such that   r is a trivial subbundle of J d L, can be identified with an open subset of G.r C 1; H 0 .J d L//. Hence   r is of the form C  U for some U 2 G.r C 1; H 0 .J d L//. As in Section 4 we gain a Wro´nski map, triv .%r;d / ! P H 0 .L˝rC1 ˝ K ˝

r.rC1/ 2

/;

(34)

defined by  7! W0 . / .mod C  /. As we said, this map is the restriction to the open subset triv .%r;d /  G.r C 1; H 0 .J d L// of the determinant map G.r C 1; H 0 .J d L// ! P H 0 .L˝rC1 ˝ K ˝

r.rC1/ 2

/;

sending U to tr .u0 / ^ tr .u1 / ^    ^ tr .ur / .mod C  /, where .u0 ; u1 ; : : : ; ur / is a basis of U and tr denotes the epimorphism J d L ! J r L. 5.3. We notice now that each gdr .L/, i.e. V 2 G.r C 1; H 0 .L//, can be seen in fact as an element of triv .%r;d /, because Dd W C  V ! J d L realizes C  V as a (trivial) vector subbundle of J d L. Indeed Dd V WD fDd v j v 2 V g is an .r C 1/dimensional subspace of H 0 .J d L/ because the map J d L ! L ! 0 induces the surjection H 0 .J d L/ ! H 0 .L/ ! 0, see e.g. [8], and then Dd v D 0 implies v D 0. We have thus an injective map G.r C 1; H 0 .L// ,! triv .%r;d /  G.r C 1; H 0 .J d L//;

On generalized Wro´nskians

281

sending V to Dd V , and W0 .Dd V / WD Dr u0 ^ Dr u1 ^    ^ Dr ur mod C  D W0 .V / which proves that our Wro´nski map defined on triv .%r;d /, which is in general strictly larger than G.r C1; H 0 .L//, coincides with the Wro´nskian W0 .V / defined in Section 3. We are now in a position to define generalized Wro´nskian subloci. Recall the natural evaluation map ev W C  triv .%r;d / ! G.r C 1; J d L/ sending .P; / 7! .P /. If  .%r;d J  L/ is a generalized Wro´nski variety of G.r C1; J d L/, then ev1 . .%r;d J  L// cuts the locus of pairs .P;  / such that .P / 2  .J  L/. We also set evP . / D .P /, for each P 2 C . It follows that the general section of any irreducible component of evP1 . .J  L// is a section having  as a ramification partition. 5.4. The map Dd;P W H 0 .L/ ! JPd L sending v 7! Dd v.P / is a vector space 1 monomorphism. If V 2 G.r C 1; H 0 .L//, then v 2 V \ Dd;p Nh;P .L/ if and h only if D v.P / D 0, i.e. if and only if v vanishes at P with multiplicity at least h. This explains the terminology osculating flag used in Section 5.2. 5.5. Example. More details about the present example are in [20]. Let W X ! S be a proper flat family of smooth projective curves of genus g  2. The Hodge bundle of the family is E WD  K . The vector bundle map over X  E ! J 2g2 K is injective and then it induces a section K W X ! G.g; J 2g2 K /. In this case the cuspidal locus of K , which is by definition K1 .1g1 .J  K//, coincides with the locus in X of the Weierstrass points of the hyperelliptic fibers of . With the notation as in 4.12 and 4.13, its class in Ag1 .X/ is given by ŒK1 .1g1 .J  K/ D K Œ1g1 .J  K/ D K . 0 ^  2 ^    ^  g / and can be easily computed through a straightforward computation (see [20, Section 3], where the computation was performed for g D 4). Since on each hyperelliptic fiber there are precisely 2g C 2 Weierstrass points, the class of the hyperelliptic locus in Ag2 .S/ is given by ŒH  D

1   K . 0 ^  2 ^    ^  g /; 2g C 2

which yields precisely the formula displayed in [38, p. 314]. 5.6. If C D P 1 and L D OP 1 .d /, then triv .%r;d / coincides in this case with G.r C 1; H 0 .L// and our picture allows to rephrase in an elegant way the situation exposed in the

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first part of [9]. The Wro´nski map triv .%r;d / ! P H 0 .OP 1 ..r C 1/.d  r/// coincides with (20), modulo the identification of triv .%r;d / with G.r C 1; H 0 .OP 1 .d ///. In other words, when C is not rational, the theory exposed up to now is a generalization of the theory of linear systems on the projective line, for which we want to spend some additional words in a separate section.

6 Linear systems on P 1 and the intermediate Wronskians ´ In the case of linear systems gdr defined on the projective line, the picture outlined in Section 5 gets simpler. However, even this case is particularly rich of nice geometry interacting with other parts of mathematics. For the sake of brevity, denote by Ld the invertible sheaf OP 1 .d /, i.e. the unique line bundle on P 1 of a fixed degree d . The elements of a basis x WD .x0 ; x1 / of H 0 .L1 / can be regarded as homogeneous coordinates .x0 W x1 / on P 1 . Furthermore H 0 .Ld / D Symd H 0 .L1 /, i.e. H 0 .Ld / can be identified with the C-vector space generated by the monomials fx0i x1d i g0id , and a gdr on P 1 is a point of G.r C 1; H 0 .Ld //. Any basis v WD .v0 ; v1 ; : : : ; vr / of V 2 G.r C 1; H 0 .Ld // defines a rational map 'V W P 1 ! P r ;

P 7! .v0 .P / W v1 .P / W : : : W vr .P //:

(35)

If V has no base points (that is, if dim V .P / D dim V  1 for each P 2 P 1 ), then the image of (35) is a non-degenerated (that is, not contained in any hyperplane) rational curve of degree d in P r . In particular, if r C 1 D dim H 0 .Ld /, then V D H 0 .Ld / and 'V .P 1 / is nothing else than the rational normal curve of degree d . Each curve of degree d in P r can be seen as the rational normal curve in P H 0 .Ld / composed with a projection P H 0 .Ld / ! P r whose center is a complementary linear subvariety of V 2 G.r C 1; H 0 .Ld // (see e.g. [9], [30]). Keeping the notation of Section 5, let %d W J d Ld ! P 1 be the bundle of d -jets of Ld . Then Dd W P 1  H 0 .Ld / ! J d L (cf. (22) is an injective morphism between vector bundles of the same rank, that is, an isomorphism. In particular, the map Dd;P W H 0 .Ld / ! JPd Ld ;

P 7! Dd v.P /;

(36)

is an isomorphism of vector spaces, for each P 2 P 1 . We define the osculating flag at P of H 0 .Ld /, F;P W 0 F1;P    Fd;P Fd C1;P D JPd L; by setting (cf. 5.2)

1 Fh;P D Dd;p .Nh;P .L//  H 0 .Ld /:

In other words, v 2 V \ Fh;P if and only if v vanishes at P with multiplicity at least h, that is, Dh v.P / D 0. In fact, Fh;P may be identified with the vector subspace of the homogeneous polynomials of H 0 .Ld / that vanish at P with multiplicity at least h. Yet another interpretation of Fh;P is the set of all hyperplanes of P H 0 .Ld / intersecting the rational normal curve in P H 0 .Ld // at P with multiplicity at least d  h.

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On generalized Wro´nskians

6.1. The Riemann–Roch formula shows that h0 .Ld / D h0 .J d Ld /; thus the injective “derivative map” Dd W H 0 .Ld / ! H 0 .J d Ld / is an isomorphism which itself induces a biholomorphism: G.r C 1; H 0 .Ld // ! G.r C 1; H 0 .J d Ld //: So one concludes that triv .%r;d / D G.r C 1; H 0 .J d Ld // Š G.r C 1; H 0 .Ld // parameterizes all the gdr ’s on P 1 (with base points or not). In particular it is compact. For V 2 G.r C1; H 0 .Ld //, denote by V the corresponding element of triv .%r;d /. The evaluation morphism P 1  G.r C 1; H 0 .Ld // ! G.r C 1; J d Ld / maps .P; V / to V .P / 2 G.r C 1; JPd Ld /. 6.2. By 6.1, the Wro´nski map  7! W0 . / (see (34)) coincides with the Wro´nski map (20) of Section 3.9: G.r C 1; H 0 .Ld // ! P H 0 .L.rC1/.d r/ /;

V 7! W0 .V /:

(37)

It is a finite surjective morphism (see e.g. [9], [30], [47]). Its degree Nr;d is precisely the Plücker degree of the Grassmannian G.r C 1; d C 1/: Z 1Š2Š : : : rŠ  .r C 1/.d  r/Š .rC1/.d r/ Nr;d D .1/ \ ŒG.r C 1; d C 1/ D : .d  r/Š.d  r C 1/Š : : : d Š Thus, given a homogeneous polynomial W of degree .d  r/.r C 1/ in two indeterminates .x0 ; x1 /, there are at most Nr;d distinct gdr ’s having W as a Wro´nskian. The number Nr;d was calculated by Schubert himself in 1886, cf. [51] and [15, p. 274]. In the case of real rational curves, the degree of the Wro´nski map was obtained by L. Goldberg for r D 1 ([25]), and for any r  1 by A. Eremenko and A. Gabrielov ([10]). For more considerations on real Wro´nski map see also [30]. 6.3. For any partition  2 P .rC1/.d r/ define  .P / WD  .F;P /  G.r C 1; H 0 .Ld //: It is a Schubert variety of codimension jj in G.r C 1; H 0 .Ld //. If .V; P / is the order partition of V at P (see Section 3.12) then V 2 ı.V;P / .P /  .V;P / .P /; and P is a V -ramification point if and only if j.V; P /j > 0. The Wro´nskian W0 .V / of V vanishes exactly at the V -ramification points. The total weight of the V -ramification points equals the dimension of G.r C 1; H 0 .Ld // (one can see that by putting g D 0 in (21)). Let f.P ; w/g WD f.P0 ; w0 /; .P1 ; w1 /; : : : ; .Pk ; wk // be a kC1-tuple of pairs where Pi 2 P 1 and wi ’s are positive integers such that k X iD1

wi D .r C 1/.d  r/:

(38)

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Thus, in notation of Section 3.14, if Dwi 1 W0 .V / 2 H 0 .J wi 1 L.rC1/.d r/ / vanishes at Pi , for every 0  i  k, then P0 ; P1 ; : : : ; Pk are exactly the ramification points of V , each one of weight wt V .Pi / D wi D j.Pi ; V /j. We have V 2 ı.V;P0 / .P0 / \ ı.V;P1 / .P1 / \    \ ı.V;Pk / .Pk / D .V;P0 / .P0 / \ .V;P1 / .P1 / \    \ .V;Pk / .Pk /:

(39)

Condition (38) means that the “expected dimension” of the intersection (39) is zero. Intersections of Schubert varieties associated with the osculating flags of the normal rational curve were first studied by D. Eisenbud and J. Harris in the eighties, [9]. In particular, they showed that the intersection (39) is zero-dimensional indeed, and hence the number of distinct elements in the intersection is at most Z .P0 ;V /  .P1 ;V / : : : .Pk ;V / \ ŒG.r C 1; H 0 .Ld //; G.rC1;H 0 .Ld //

where  is the Schubert cycle defined by the equality  \ŒG.rC1; H 0 .Ld // D Œ . This fact was used in [7] to deduce explicit formulas (and a list up to n D 40) for the number of space rational curves of degree n  3 having 2n hyperstalls at 2n prescribed points. 6.4. Preimages of the Wronski ´ map. Notice that if P 2 P 1 is a base point of V , it occurs in the V -ramification locus as well, and the Wro´nskian vanishes at it with weight r C 1. The set BP of linear systems having P as base point is a closed subset of G.r C 1; H 0 .Ld // of codimension r C 1. In fact BP WD evP1 .B.%r;d J  Ld //, which is a closed subset of codimension r C 1 (cf. Section 4.7). Let f.P ; w/g be as in 6.3. Denote by Gr;d .P / the set of all V 2 G.r C 1; H 0 .Ld // whose base locus contains no Pi , 0  i  r. It is an open dense subset of codimension r C 1, Gr;d .P / D G.r C 1; H 0 .Ld // n .BP0 [ BP1 [    [ BPk /: Consider now a .k C 1/-tuple of partitions E D .0 ; 1 ; : : : ; k /; 

jj j D wj ; 0  j  k:

We shall write: j WD j;0  j;1      j;r : The elements of E P / D  .P0 / \  .P1 / \    \  .Pk / \ Gr;d .P / G.r C 1; H 0 .Ld // I.; 0 1 k (40) E correspond to the base point free linear systems ramifying at P according to .

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E P / leads to interesting analytic considerations The problem of determining I.; related with Wro´nskians. Up to a projective change of coordinates, it is not restrictive to assume that P0 D 1 WD .0 W 1/. Using the coordinate x D x1 =x0 , the osculating flag at 1 shall be denoted by F;1 . Accordingly, the partition 0 will be renamed 1 . Notice that Fj;1 coincides with the vector space Polyj of the polynomials of degree at most j in the variable x: in fact a polynomial P .x/ (thought of as the affine representation of a homogeneous polynomial of degree d in two variables) vanishes at E P /, let 1 with multiplicity j if and only if it has degree d  j . For V 2 I.; WV .x/ WD

W0 .V / .rC1/.d r/ x0

be the representation of the W0 .V / in the affine open subset of P 1 defined by x0 ¤ 0. The degree of the polynomial WV .x/ is less or equal than .r C 1/.d  r/, because of possible ramifications of V at 1. We have WV .x/ D .x  z1 /w1 : : : .x  zk /wk ;

(41) Pk

where zi WD x.Pi / are the values of the coordinate x at Pi 2 P 1 ; iD1 wi D deg WV .x/  .r C 1/.d  r/. For a basis v D .v0 ; v1 ; : : : ; vr / of V , consider fi WD vi =x0d and write f D .f0 ; f1 ; : : : ; fr /. According to (3), one writes WV .x/ D f ^ Df ^    ^ D r f , where  j  d fi j D f D : dx j 0ir The space V can be realized as the solution space of the following differential equation ˇ ˇ ˇ g f0 f1 ::: fr ˇˇ ˇ ˇ Dg Df1 ::: Dfr ˇˇ Df0 ˇ ˇ ˇ :: : : :: :: :: :: (42) EV .g/ D ˇ : ˇ D 0: : : ˇ ˇ r r r r ˇ ˇ D g D f D f : : : D f 0 1 r ˇ ˇ rC1 ˇD g D rC1 f0 D rC1 f1 : : : D rC1 fr ˇ E P /, denote by V the flag obtained by 6.5. Intermediate Wronskians. ´ For V 2 I.; the intersection of V and F;1 : V D fV0 V1 V2    Vr D V g ;

dim Vj D j C 1:

(43)

All the polynomials in Vj have degree  dj , where 0  d0 < d1 <    < dr  d is the order sequence of V at P0 (cf. Section 3.12). Recall that V has no base point and WV .x/ as in (41). Define the j -th intermediate Wro´nskian of V as Wj .x/ WD WVj .x/, the Wro´nskian of Vj , 0  j  r. In particular, the r-th intermediate Wro´nskian coincides with WV .x/.

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Non-vanishing properties of intermediate Wro´nskians have been recently investigated in an analytic context in [5] and [6] to study factorizations of linear differential operators with non-constant C-valued coefficients. Intermediate Wro´nskians are important because every V 2 G.r C 1; Polyd / is completely determined by the set of its intermediate Wro´nskians W0 .x/; : : : ; Wr .x/. Indeed, the ODE (42) can be rewritten as follows: W22 .x/ W12 .x/ Wr2 .x/ d d d g.x/ d :::   D 0: dx Wr1 .x/WrC1 .x/ dx W3 .x/W1 .x/ dx W2 .x/W0 .x/ dx W1 .x/ By [41, Part VII, Section 5, Problem 62], one can take as a basis of V the following set of r C 1 linearly independent solutions of (42): g0 .x/ D W0 .x/; Z g1 .x/ D W0 .x/

x

Z

x

g2 .x/ D W0 .x/ :: : gr .x/ D W0 .x/

Z

x

W0 W2 ; W12

Z

W0 . /W2 . / W12 . / W0 . /W2 . / W12 . /



W1 W3 W22

Z 

! ;

W1 . /W3 . / W22 . /

Z



Z



:::

Wr1 WrC1 Wr2

!

 :::

:

Define now polynomials Z0 .x/; Z1 .x/; : : : ; Zr .x/ through the following formula: Zi .x/ D

k Y

.x  zj /mj .i/ ;

0  i  r;

(44)

j D1

where mj .i / D j;r C j;r1 C    C j;ri ;

1  j  k:

In particular Zr .x/ D WV .x/. 6.6. Lemma ([48]). The ratio Tri .x/ WD Wi .x/=Zi .x/ is a polynomial of degree .i C 1/.d  i / 

i X lD0

rl;1 

k X

mj .i /:

(45)

j D1

In particular, T0 .x/ D 1. Thus we have Wrj .x/ D Tj .x/Zrj .x/, 0  j  r. The roots of Tj .x/ are said to be the additional roots of the .r  j /-th intermediate Wro´nskian. If (40) contains more than one element, then the intermediate Wro´nskians of these elements all differ by the additional roots.

On generalized Wro´nskians

287

6.7. Non-degenerate planes ([48]). The intersection (40) contains some distinguished elements, called non-degenerate planes. Denote by .f / the discriminant of a polynomial f .x/ and by Res.f; g/ the resultant of polynomials f .x/; g.x/. E P / a non-degenerate plane if the polynomials T0 .x/; : : : ; Definition. We call V 2 I.; Tr1 .x/ i) do not vanish at the ramification points P1 ; : : : ; Pk , i.e. Ti .zj / ¤ 0 for all 0  i  r  1 and all 1  j  k; ii) do not have multiple roots: .Ti / ¤ 0, for all 0  i  r; iii) for each 1  i  r , Ti and Ti1 have no common roots: Res.Ti ; Ti1 / ¤ 0. 6.8. Relative discriminants and resultants. Non-degenerate planes correspond to critical points of a certain generating function which can be described in terms of relative discriminants and resultants. For fixed z D .z1 ; : : : ; zk /, any monic polynomial f .x/ can be written in a unique way as the product of two monic polynomials T .x/ and Z.x/ satisfying T .zj / ¤ 0;

f .x/ D T .x/Z.x/;

Z.x/ ¤ 0;

for any x ¤ zj ; 1  j  k: (46)

One defines the relative discriminant of f .x/ with respect to z as z .f / D

.f / D .T /.Res.Z; T //2 ; .Z/

and the relative resultant of fi .x/ D Ti .x/Zi .x/, i D 1; 2, with respect to z as Resz .f1 ; f2 / D

Res.f1 ; f2 / D Res.T1 ; T2 /Res.T1 ; Z2 /Res.T2 ; Z1 /: Res.Z1 ; Z2 /

E P / given by (40), then the decomposition If V is a non-degenerate plane in I.; Wi .x/ D Tri .x/Zi .x/ is exactly the same as displayed in (46). The generating E P / is a rational function such that its critical points determine the function of I.; non-degenerate elements in such an intersection. Its expression is (see [48] ) .T1 ; : : : ; Tp1 / D ˆ.;z/ E

z .W0 / : : : z .Wr1 / : Resz .W1 ; W2 / : : : Resz .Wr1 ; Wr /

(47)

Part of the following theorem was originally obtained by A. Gabrielov (unpublished), along his investigations of the Wro´nski map. 6.9. Theorem ([48]). There is a one-to-one correspondence between the critical .T0 ; : : : ; Tr1 / and the nonpoints with non-zero critical values of the function ˆ.;z/ E E P / given by (40). degenerate planes in the intersection I.;

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In other words, every such critical point defines the intermediate Wro´nskians, and hence a non-degenerate plane, see 6.5. Conversely, for every non-degenerate plane one can calculate the intermediate Wro´nskians, and the corresponding polynomials Ti .x/ supply a critical point with a non-zero critical value of the generating function (47). 6.10. Relation to Bethe vectors in the Gaudin model (see [37], [46], [48]). Once one re-writes (47) in terms of unknown roots of the polynomials Tj ’s, the generating function turns into the master function associated with the Gaudin model of statistical mechanics. In the Gaudin model, the partitions j , 1  j  k, of Section 6.4 are the highest weights of sl rC1 -representations, and the j -th representation is marked by the point Pj . Recall that 1 is the partition related to P0 WD .0 W 1/ 2 P 1 , after renaming 0 , see Section 6.4. Denote by 1 the partition dual to 1 . Certain commuting linear operators, called Gaudin Hamiltonians, act in the subspace of singular vectors of the weight 1 in the tensor product of the sl rC1 -representations of the weights 1 ; : : : ; k , and one looks for a common eigenbasis of the Gaudin Hamiltonians. The Bethe Ansatz is a method to look for common eigenvectors. It gives a family of vectors of the required weight 1 meromorphically depending on a number of auxiliary complex parameters. The Bethe system is a system of equations on these parameters, and any member of the family that corresponds to a solution of the Bethe system is a common singular eigenvector of the Gaudin Hamiltonians called the Bethe vector. It turns out that the Bethe system coincides with the system on critical points with non-zero critical value of the function ˆ;z; E . In other words, the auxiliary complex parameters are exactly the additional roots of the intermediate Wro´nskians! Thus every non-degenerate plane of (40) defines a Bethe vector and vice versa. This link has led to an essential progress in studies of the Gaudin model as well as in algebraic geometry (e.g., Shapiro–Shapiro conjecture), see [36] and references therein.

7 Linear ODEs and Wronski–Schubert ´ calculus This last section surveys and announces the results of [22], an attempt to reconcile the first part of this survey, regarding Wro´nskians of fundamental systems of solutions of linear ODEs, with the geometry described in the last four sections. The main observation is that Schubert cycles of a Grassmann bundle can be described through Wro´nskians associated with a fundamental system of solutions of a linear ODE. 7.1. Let us work in the category of (not necessarily finitely generated) associative commutative Q-algebras with unit. Let A be such a Q-algebra. We denote by AŒT  and AŒŒt the corresponding A-algebras of polynomials and of formal power series, P respectively (here t and T are indeterminates over A). For  D n0 an t n 2 AŒŒt , we write .0/ for the “constant term” a0 . If P .T / 2 AŒT  is a polynomial of degree

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r C 1, we denote by .1/i ei .P / the coefficient of T rC1i , for each 0  i  r C 1; for instance, if P is monic, e0 .P / D 1, we have P .T / D T rC1  e1 .P /T r C    C .1/rC1 erC1 .P /: Let B be another Q-algebra. Each 2 HomQ .A; B/ induces two obvious Q-algebra homomorphisms, AŒT  ! BŒT  and AŒŒt  ! BŒŒt , the both are also by P denoted n . The former is defined by e . .P // D .e .P // and the latter by a t ! 7 i i n0 n P n n0 .an /t . 7.2. Let Er WD QŒe1 ; e2 ; : : : ; erC1  be the polynomial Q-algebra in the set of indeterminates .e1 ; : : : ; erC1 /. We call UrC1 .T / D T rC1  e1 T r C    C .1/rC1 erC1 the universal monic polynomial of degree r C 1. Thus ei .UrC1 .T // D ei for all 0  i  r C 1. Let hN WD .h0 ; h1 ; h2 ; : : : ; hr ; hrC1 ; : : : / be the sequence in Er defined by the equality of formal power series: X

1 1  e1 t C    C .1/rC1 t rC1 X .e1 t  e2 t 2 C    C .1/r erC1 t rC1 /n : D1C

hn t n D

n0

n1

One gets h0 D 1, h1 D e1 , h2 D e12  e2 , … . In general hn D det.ej iC1 /1i;j n (see [15, p. 264]). For any .r C1/-tuple or a sequence aN D .a0 ; a1 ; : : : / of elements of any Er -module, we set N D ai  e1 ai1 C    C .1/i ei a0 ; Ui .a/

N D a0 ; U0 .a/

1  i  r:

(48)

Although only a0 ; a1 ; : : : ; ar appear in (48), we prefer to define Uj also for sequences. N D 0 for all 1  i  r. We have Ui .h/ 7.3. Let x WD .x0 ; x1 ; : : : ; xr / and fN WD .fn /n0 be two sets of indeterminates over Q. Let Er Œx; fN  WD Er Œx0 ; x1 ; : : : ; xr I f0 ; f1 ; : : :  be the Q-polynomial algebra and Er Œx; fN ŒŒt the corresponding algebra of formal power series. Denote by D WD d=dt the usual formal derivative of formal power series. Its j -th iterated is D

j

X

tn an nŠ n0



D

X n0

anCj

tn ; nŠ

am 2 Er Œx; fN :

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Evaluating the polynomial UrC1 at D we get the universal differential operator: UrC1 .D/ D D rC1  e1 D r C    C .1/rC1 erC1 : P n Let f WD n0 fn tnŠ 2 QŒfN ŒŒt   Er Œx; fN ŒŒt . Consider the universal Cauchy problem for a linear ODE with constant coefficients: UrC1 .D/y D f;

D i y.0/ D xi ;

0  i  r:

We look for solution of (49) in Er Œx; fN ŒŒt . P 7.4. Theorem ([22]). Let n0 pn  t n 2 Er Œx; fN ŒŒt  be defined by P X U0 .x/ C U1 .x/t C    C Ur .x/t r C nrC1 fnr1 t n n pn t D ; 1  e1 t C    C .1/rC1 erC1 t rC1 n0 where Uj are as in (48). Then g WD

X n0

pn

tn nŠ

(49)

(50)

(51)

is the unique solution of the Cauchy problem (49). The universality of UrC1 .D/ means the following.

P 7.5. Theorem. Let A be a Q-algebra, P 2 AŒT ,  D n0 n t n =nŠ 2 AŒŒt  and .b0 ; b1 ; : : : ; br / 2 ArC1 any .r C1/-tuple. Then the unique Q-algebra homomorphism, defined by xi 7! bi , ei 7! ei .P / and fi 7! i , maps the universal solution g, as in (51), to the unique solution of the Cauchy problem P .D/y D ;

D i y.0/ D bi ;

0  i  r:

(52)

For each 0  i  r, let i W Er Œx; fN  ! Er be the unique Er -algebra homomorphism over the identity sending x 7! .0; : : : ; 0; 1; h1 ; : : : ; hri / and fN 7! .0; 0; : : : /. „ ƒ‚ … i

7.6. Corollary. If ui WD i .g/ 2 Er ŒŒt, where g is the unique solution of the universal Cauchy problem (49), then u D .u0 ; u1 ; : : : ; ur / is an Er -basis of ker UrC1 .D/. Proof. Using the same arguments as in Theorem 7.4, one shows that ui is a solution of UrC1 .D/y D 0. Furthermore, if u WD a0 u0 C a1 u1 C    C ar ur D 0, then u is the unique solution of UrC1 .D/y D 0, with the zero initial conditions. Then by uniqueness u D 0, i.e. .u0 ; : : : ; ur / are linearly independent. 7.7. Corollary. Let A be any Q-algebra and P 2 AŒT . Let W Er ! A be the unique morphism mapping ei 7! ei .P /. Then . .u0 /; .u1 /; : : : ; .ur // is an A-basis of ker P .D/.

On generalized Wro´nskians

291

In other words, ker P .D/ Š ker UrC1 .D/ ˝Er A. 7.8. Let n  0 be an integer and  a partition of length at most r C 1 with weight n. n 0 1 r Denote by  the coefficient of x0 x1 : : : xr in the expansion of .x0 C x1 C    C xr /n . With the usual convention 0Š D 1, one has ! n nŠ D :  0 Š1 Š : : : r Š In Section 4.12 the Schur polynomials P .a/ associated to partition  and to (the coefficients of) a formal power series a D n0 an t n were defined, see (28). In our N notation, the coefficients form sequence aN D .a0 ; a1 ; : : : /; below we will write  .a/ instead of  .a/. 7.9. Theorem. For each partition , the following equality holds: ! n X X n N t : W .u/ D C .h/  nŠ n0 jjDn

N In particular, the “constant term” is W .u/.0/ D  .h/. It is a straightforward combinatorial exercise made easy by the use of the basis u found in 7.6. See [22] for details. 7.10. Proposition. Giambelli’s formula for Wro´nskians holds: N  W0 .u/: W .u/ D  .h/ Proof. First of all, by Remark 2.5, W .u/ is proportional to W0 .u/, i.e. W .u/ D c W0 .u/ for some c 2 Er . Next, two formal power series are proportional if and only if the coefficients of the same powers of t are proportional, with the same factor of proportionality. Finally, c D

W .u/.0/ N D  .h/; W0 .u/.0/

according to Theorem 7.9. 7.11. Corollary. Pieri’s formula for generalized Wro´nskians holds: P hi W .u/ D  W .u/, where the sum is over the partitions  D .0 ; 1 ; : : : ; r / such that jj D i C jj and 0  0  1  1      r  r :

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It is well known that Giambelli’s and Pieri’s implies each other. See e.g. [15, Lemma A.9.4]. 7.12. Let now %r;d W G ! X be a Grassmann bundle, where G WD G.r C 1; F / and F is a vector bundle of rank d C 1. As recalled in Section 4.12, A .G/ is freely generated as A .X/-module (see [15, Proposition 14.6.5]) by  .c t .Qr  %r;d F // \ ŒG: The exact sequence (23) implies that c t .r /c t .Qr / D c t .%r;d F /, which is equivalent to c t .Qr / D c t .r /c t .Qr  %r;d F /: 1 D c t .r / c t .%r;d F / Set "i D .1/i ci .r / and consider the differential equation D rC1 y  "1  D r y C    C .1/rC1 "rC1  y D 0:

(53)

We look for solutions in .A .G/ ˝ Q/ŒŒt . By Corollary (7.7) the unique morphism W Er ! A .G/ ˝ Q, sending ei 7! "i , maps the universal fundamental system .u0 ; u1 ; : : : ; ur / to v D .v0 ; v1 ; : : : ; vr /, where vi D .ui / and, as a consequence, it maps hi to ci .Qr  %r;d F / and W .u/ to W .v/. Then we have proven that  .c t .Qr  %r;d F // D

W .v/ : W0 .v/

In other words, the Chow group A .G/ can be identified with the A .X /-module generated by the generalized Wro´nskians associated to the basis v of solutions of the differential equation (53). In particular we have shown that the class Œ .%r;d F / of the generalized Wro´nski variety  .%r;d F / is an A .X /-linear combination of ratios of generalized Wro´nskians associated to the basis v of (53), by virtue of (30), (31) and (32).

References [1] E. Arbarello, M. Cornalba, Ph. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. 1, Grundlehren Math. Wiss. 267, Springer-Verlag, Berlin 1984. [2] A. Bertram, Quantum Schubert calculus. Adv. Math. 128 (1997), 289–305. [3] G. W. Anderson, Lacunary Wro´nskians on genus one curves. J. Number Theory 115 (2005), no. 2, 197–214. [4] V. I. Arnold, Ordinary differential equations. Translated from the Russian by Richard A. Silverman, MIT Press, Cambridge, Mass., 1978. [5] R. Camporesi, Linear ordinary differential equations. Revisiting the impulsive response method using factorization. Internat. J. Math. Ed. Sci. Tech. 42 (2011), no. 4, 497–514.

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[6] R. Camporesi and A. J. Di Scala, A generalization of a theorem of Mammana. Colloq. Math. 122 (2011), 215–223. [7] J. Cordovez, L. Gatto, and T. Santiago, Newton binomial formulas in Schubert calculus. Rev. Mat. Complutense 22 (2009), no. 1, 129–152. [8] C. Cumino, L. Gatto, and A. Nigro, Jets of line bundles on curves and Wro´nskians. J. Pure Appl. Algebra 215 (2011), 1528–1538. [9] D. Eisenbud and J. Harris, Divisors on general curves and cuspidal rational curves. Invent. Math. 74 (1983), 371–418. [10] A. Eremenko and A. Gabrielov, Degrees of real Wro´nski maps. Discrete Comput. Geom. 28 (2002), 331–347. [11] A. Eremenko and A. Gabrielov, The Wro´nski map and Grassmannians of real codimension 2 subspaces. Comput. Methods Funct. Theory 1 (2001) 1–25. [12] E. Esteves, Wro´nski algebra systems on families of singular curves. Ann. Sci. École Norm. Sup. (4) 29 (1996), 107–134. [13] E. Esteves, Wro´nski algebra systems and residues. Bol. Soc. Brasil. Mat. (N.S.) 26 (1995), 229–243. [14] O. Forster, Lectures on Riemann surfaces. Grad Texts in Math. 81, Springer-Verlag, New York 1999. [15] W. Fulton, Intersection theory. Ergeb. Math. Grenzgeb. (3) 2, Springer-Verlag, Berlin 1984. [16] W. Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas. Duke Math. J. 65 (1992), 381–420. [17] W. Fulton, Young tableaux. London Math. Soc. Stud. Texts 35, Cambridge University Press, Cambridge 1997. [18] L. Gatto, Weierstrass loci and generalizations, I. In Projective geometry with applications, E. Ballico, ed., Lecture Notes in Pure and Appl. Math. 166, Dekker, New York 1994, 137–166. [19] L. Gatto, Schubert calculus via Hasse–Schmidt derivations. Asian J. Math. 9 (2005), no. 3, 315–322. [20] L. Gatto and P. Salehyan, Families of special Weierstrass points. C. R. Acad. Sci. Paris Ser. I 347 (2009), 1295–1298 [21] L. Gatto and T. Santiago, Schubert calculus on a Grassmann algebra. Canad. Math. Bull. 52 (2009), no. 2, 200–212. [22] L. Gatto and I. Scherback, Linear ODEs, Wro´nskians and Schubert calculus. Moscow Math. J. 12 (2012), no. 2, 275–291. [23] L. Gatto and F. Ponza, Derivatives of Wro´nskians with applications to families of special Weierstrass points. Trans. Amer. Math. Soc. 351 (1999), no. 6, 2233–2255. [24] G. Z. Giambelli, Risoluzione del problema degli spazi secanti. Mem. R. Accad. Torino 52 (1902), 171–211. [25] L. Goldberg, Catalan numbers and branched coverings by the Riemann sphere. Adv. in Math. 85 (1991), 129–144.

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[26] Ph. Griffiths and J. Harris, Principles of algebraic geometry. Wiley Classics Lib., John Wiley & Sons, Inc., New York 1994, Addison Wesley. [27] R. Hartshorne, Algebraic geometry. Grad Texts in Math. 52, Springer-Verlag, New York 1977. [28] J. M. Hoene-Wro´nski, Réfutation de la théorie des fonctions analytiques de Lagrange. Blankenstein, Paris 1812. [29] M. Kazarian, Singularities of the boundary of fundamental systems, flat points of projective curves, and Schubert cells. J. Soviet. Math. 52 (1990), 3338–3349. [30] V. Kharlamov and F. Sottile, Maximally inflected real rational curves. Moscow Math. J. 3 (2003), no. 3, 947–987. [31] G. Kempf and D. Laksov, The determinantal formula of Schubert calculus. Acta Math. 32 (1974), 153-162. [32] D. Laksov, Weierstrass points on curves. In Young tableaux and Schur functors in algebra and geometry (Toru´n, 1980), Astérisque 87–88, Soc. Math. France, Paris 1981, 221–247. [33] D. Laksov and A. Thorup, Schubert calculus on Grassmannians and exterior products. Indiana Univ. Math. J. 58 (2009), no. 1, 283–300. [34] I. G. Macdonald, Symmetric functions and Hall poynomials. Clarendon Press, Oxford 1979. [35] A. Milas, E. Mortenson, and K. Ono, Number theoretic properties of Wro´nskians of Andrews–Gordon series. Internat. J. Number Theory 4 (2008), 323–337. [36] E. Mukhin, V. Tarasov, and A. Varchenko, Schubert calculus and representations of the general linear group. J. Amer. Math. Soc. 22 (2009), no 4, 909–940. [37] E. Mukhin andA.Varchenko, Critical points of master functions and flag varieties. Commun. Contemp. Math. 6 (2004), no. 1, 111–163. [38] D. Mumford, Towards an enumerative geometry of moduli space of curves. In Arithmetic and geometry, Vol. II, Progr. Math. 36, Birkhäuser, Boston, Mass., 1983, 271–328. [39] A. Nigro, Sections of Grassmann bundles. Ph.D. Thesis, Politecnico di Torino, 2010. [40] M. Pieri, Formule di coincidenza per le serie algebriche 1n di coppie di punti dello spazio ad n dimensioni. Rend. Circ. Mat. Palermo 5 (1891), 252–268. [41] G. Pólya and G. Szego, Problems and theorems in analysis II. Springer-Verlag, Heidelberg 1976. [42] F. Ponza, Sezioni Wronskiane generalizzate a famiglie di punti di Weierstrass speciali. Ph.D. Thesis, Consorzio Universitario Torino–Genova, 1996. [43] P. Pragacz, Enumerative geometry of degeneracy loci. Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 3, 413–454. [44] P. Pragacz, Symmetric polynomials and divided differences in formulas of intersection theory. In Parameter spaces, Banach Center Publications 36, Polish Academy of Sciences, Warsaw 1996, 125–177. [45] P. Pragacz, La vita e l’opera di Józef Maria Hoene-Wro´nski. Atti Accad. Peloritana dei Pericolanti 89 (2011), no. 1, C1C8901001. [46] I. Scherbak, Rational functions with prescribed critical points. Geom. Funct. Anal. 12 (2002), 1365–1380.

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[47] I. Scherbak, Gaudin’s model and the generating function of the Wro´nski map. In Geometry and topology of caustics (Warsaw, 2002), Banach Center Publ. 62, Academy of Sciences, Warsaw 2004, 249–262. [48] I. Scherbak, Intersections of Schubert varieties and critical points of the generating function. J. London Math. Soc. (2) 70 (2004), 625–642. [49] I. Scherbak and A. Varchenko, Critical points of functions, sl2 representations, and Fuchsian differential equations with only univalued solutions. Moscow Math. J. 3 (2003), no. 2, 621–645. [50] F. H. Schmidt, Die Wronskische Determinante in beliebigen differenzierbaren Funktionenkörpern. Math. Z. 45 (1939), 62–74. [51] H. Schubert, Anzahl-Bestimmungen für lineare Räume beliebiger Dimension. Acta Math. 8 (1886), 97–118. [52] C. Towse, Generalized Wro´nskians and Weierstrass weights. Pacific J. Math. 193 (2000), no. 2, 501–508. [53] C. Widland and R. Lax, Weierstrass points on Gorenstein curves. Pacific J. Math. 142 (1990), no. 1, 197–208. Letterio Gatto, Dipartimento di Matematica, Politecnico di Torino, 10129 Torino, Italy E-mail: [email protected] Inna Scherbak, School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel E-mail: [email protected]

Lines crossing a tetrahedron and the Bloch group Kevin Hutchinson and Masha Vlasenko

1 Lines crossing a tetrahedron Let k be an arbitrary infinite field. Consider the projective spaces P n .k/ with fixed sets of homogenous coordinates .t0 W t1 W : : : W tn / 2 P n .k/. We call a subspace L  P n .k/ of codimension r admissible if   codim L \ fti1 D    D tis D 0g D r C s for every s and distinct i1 ; : : : ; is . (Here codim.X/ > n means X D ;.) Let   Cnr D Z admissible L  P n .k/; codim.L/ D r be the free abelian group generated by all admissible subspaces of P n .k/ of codimension r. Then for every r we have a complex d

d

d

r r ! CrC1 ! Crr ! 0 !       ! CrC2

(we assume that Cnr D 0 when n < r) with the differential X d ŒL D .1/i ŒL \ fti D 0g

(1)

where every fti D 0g  P n .k/ is naturally identified with P n1 .k/ by throwing away the coordinate ti . We are interested in the homology groups of these complexes Hnr D Hn .Cr /. P For example, one can easily see that H11 Š k  . Indeed, a hyperplane f ˛i ti D 0g is admissible whenever all the coefficients ˛i are nonzero, and if we identify h˛ i 1 ; Œf˛0 t0 C ˛1 t1 D 0g 7! C11 Š ZŒk  ; ˛0 h ˛ ˛ i (2) 1 2 ; Œf˛0 t0 C ˛1 t1 C ˛2 t2 D 0g 7! ; C21 Š ZŒk   k  ; ˛0 ˛1 then the differential d W C21 ! C11 turns into Œ.x; y/ 7! Œx  Œxy C Œy:  The second autor is grateful to Anton Mellit, who taught her the idea of passing from linear subspaces to configurations (Lemma 1 below) and pointed out the K-theoretical meaning of Menelaus’ theorem, and to the organizers of IMPANGA summer school on algebraic geometry for their incredible hospitality and friendly atmosphere.

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(One can recognize Menelaus’ theorem from plane geometry behind this simple computation.) Hence we have ı˚  H11 Š ZŒk   Œx  Œxy C Œy W x; y 2 k  Š k  : Continuing the identifications of (2), C1 turns into the bar complex for the group k  (with the term of degree 0 thrown away) and therefore Hn1 D Hn .k  ; Z/;

n  1:

Now we switch to r D 2 and try to compute H32 . The four hyperplanes fti D 0g form a tetrahedron  in the 3-dimensional projective space P 3 .k/ and the line ` is admissible if it 1) intersects every face of  transversely, i.e. at one point Pi D ` \ fti D 0g; 2) does not intersect edges fti1 D ti2 D 0g of , i.e. all four points P0 ; : : : ; P3 2 ` are different . Therefore it is natural to associate with ` a number, the cross-ratio of the four points P0 ; : : : ; P3 on `. Namely, there is a unique way to identify ` with P 1 .k/ so that P0 , P1 and P2 become 0, 1 and 1 respectively, and we denote the image of P3 by .`/ 2 P 1 .k/ X f0; 1; 1g D k  X f1g. We extend  linearly to a map 

C32 ! ZŒk  X f1g; X X ni Œ.`i /: ni Œ`i  7! Theorem 1. Let  W k  ˝ k  ! k  ˝ k  be the involution  .x ˝ y/ D y ˝ x. P P (i) If d. ni Œ`i / D 0 then ni .`i / ˝ .1  .`i // D 0 in .k  ˝ k  / . (ii) Let L  P 4 .k/ be an admissible plane and `i D L \ fti D 0g, i D 0; : : : ; 4. If we denote x D .`0 / and y D .`1 / then .`2 / D

y ; x

.`3 / D

1  x 1 1  y 1

and

.`4 / D

1x : 1y

(iii) The map induced by  on homology  W H32 ! B.k/ is surjective, where

  Ker ZŒk  X f1g ! .k  ˝ k  / ; Œa 7! a ˝ .1  a/ D h i h i h i E B.k/ D 1x 1 1x Œx  Œy C yx  1y C 1y ;x¤y 1

is the Bloch group of k ([5]).

(3)

Lines crossing a tetrahedron and the Bloch group

299

(iv) We have H32 Š H3 .GL2 .k//=H3 .k  / and the kernel of (3)    K D Ker H32 ! B.k/ fits into the exact sequence 0 ! Tor.k  ; k  / ! K=T .k/ ! k  ˝K2 .k/ ! K3M .k/=2 ! 0; (4) where Tor.k  ; k  / is the unique nontrivial extension of Tor.k  ; k  / by Z=2, and T .k/ is a 2-torsion abelian group (conjectured to be trivial). We remark that Tor.k  ; k  / D Tor..k/; .k// is a finite abelian group if k is a finitely-generated field. Furthermore, it is proved in [5] that B.k/ has the following relation to K3 .k/: let K3ind .k/ be the cokernel of the map from Milnor’s K-theory K3M .k/ ! K3 .k/, then there is an exact sequence 0 ! Tor.k  ; k  / ! K3ind .k/ ! B.k/ ! 0:

(5)

In particular, if k is a number field then as a consequence of (5) and Borel’s theorem ([1]) we have dim B.k/ ˝ Q D r2 ; where r2 is the number of pairs of complex conjugate embeddings of k into C. Proof of (i) and (ii). One can check that the diagram C32

d

/ C2 2 Œt0 Wt1 Wt2 7!t0 ˝.t1 /C.t1 /˝t2 Ct2 ˝t0 Ct0 ˝t0



  Œa7!a˝.1a/ / .k  ˝ k  / ZŒk  X f1g

is commutative, and therefore (i) follows. It is another tedious computation to check (ii). In the next section we will prove the remaining claims (iii) and (iv) and also show that Hn2 Š Hn .GL2 .k/; Z/=Hn .k  ; Z/; n  3: (6)

2 Complexes of configurations We say that n C 1 vectors v0 ; : : : ; vn 2 k r are in general position if every  r of them are linearly independent. Let C.r; n/ be the free abelian group generated by .n C 1/-tuples of vectors in k r in general position. For fixed r we have a complex d

d

d

   ! C.r; 2/ ! C.r; 1/ ! C.r; 0/

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with the differential d Œv0 ; : : : ; vn  D

X

.1/i Œv0 ; : : : ; vL i ; : : : ; vn :

(7)

The augmented complex C.r; / ! Z ! 0 is acyclic. Indeed, if  X d ni Œv0i ; : : : ; vni  D 0 and v 2 k r is such that all .n C 2/-tuples Œv; v0i ; : : : ; vni  are in general position (such vectors v exist since k is infinite) then X  X ni Œv; v0i ; : : : ; vni  : ni Œv0i ; : : : ; vni  D d Lemma 1. Cnr Š C.r; n/GLr .k/ for the diagonal action of GLr .k/ on tuples of vectors. Moreover, the complex Cr is isomorphic to the truncated complex C.r; /GLr .k/;r . Proof. For n  r there is a bijective correspondence between subspaces of codimension r in P n .k/ and GLr .k/-orbits on .n C 1/-tuples Œv0 ; : : : ; vn  of vectors in k r satisfying the condition that vi span k r . It is given by z Š kr ; L  P n 7! Œv0 ; : : : ; vn ; vi D image of ei in k nC1 =L z D KerŒv0 ; : : : ; vn T  k nC1 ; Œv0 ; : : : ; vn  7! L z is the unique lift of L to a linear subspace in k nC1 and e0 ; : : : ; en is a standard where L basis in k nC1 . An admissible point in P r .k/ is a point which doesn’t belong to any of the r C 1 hyperplanes fti D 0g, and for the corresponding vectors Œv0 ; : : : ; vr  it means that every r of them are linearly independent. For n > r a subspace L of codimension r in P n .k/ is admissible whenever all the intersections L \ fti D 0g are admissible in P n1 .k/. Hence it follows by induction that admissible subspaces correspond exactly to GLr .k/orbits of tuples “in general position”. Obviously, differential (1) is precisely (7) for tuples. The tuples of vectors in general position in k r modulo the diagonal action of GLr .k/ are called configurations, so C.r; n/GLr .k/ is the free abelian group generated by configurations of n C 1 vectors in k r . Proof of (iii) and (iv) in Theorem 1. For brevity we denote C.2; n/ by Cn and GL2 .k/ by G. Since the complex of G-modules C is quasi-isomorphic to Z we have the 1 hypercohomology spectral sequence with Epq D Hq .G; Cp / ) HpCq .G; Z/. Since 1 1 all modules Cp with p > 0 are free we have Epq D 0 for p; q > 0 and Ep0 D .Cp /G .  1 1 If G1  G is the stabilizer of 0 then E0q D Hq .G; ZŒG=G1 / D Hq .G1 ; Z/ by Shapiro’s lemma. We have k   G1 and Hq .k  ; Z/ D Hq .G1 ; Z/ (see Section 1

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2 2 1 in [6]), so E0q D Hq .k  ; Z/. Further, Ep0 D Hp ..C /G / and E0q D Hq .k  ; Z/. This spectral sequence degenerates on the second term. Indeed, the embedding

1 0  ; k ,! G; ˛ 7! 0 ˛

is split by determinant, and therefore all maps Hq .k  ; Z/ ! Hq .G; Z/ are injective. 1 2 Consequently, Epq D Epq and for every n  2 we have a short exact sequence   0 ! Hn .k  ; Z/ ! Hn .G; Z/ ! Hn .C /G ! 0: It follows from Lemma 1 that   Hn2 D Hn .C /G D Hn .G; Z/=Hn .k  ; Z/;

n  3:

Let Dn be the free abelian group generated by .n C 1/-tuples of distinct points in P 1 .k/. Again we have the differential like (7) on D and the augmented complex D ! Z ! 0 is acyclic. We have a surjective map from C to D since a nonzero vector in k 2 defines a point in P 1 .k/ and the group action agrees. The spectral 1 D Hq .G; Dp / ) HpCq .G; Z/ was considered in [5]. In particular, sequence Ezpq 1 z Ep0 D .Dp /G is the free abelian group generated by .p  2/-tuples of different points since G-orbit of every .p C 1/-tuple contains a unique element of the form 1 1 .0; 1; 1; x1 ; : : : ; xp2 /, and the differential d 1 W Ez04 ! Ez03 is given by Œx; y 7! Œx  Œy C

hy i x











1  x 1 1x C : 1  y 1 1y

(8)

2 with small indices are According to [5], terms Ezpq

H3 .k  ˚ k  / H2 .k  / ˚ .k  ˝ k  /

.k  ˝ k  /

k

0

0

Z

0

0

p.k/

where p.k/ is the quotient of ZŒk  X f1g by all 5-term relations as in right-hand side of (8), and the only non-trivial differential starting from p.k/ is d 3 W p.k/ ! H2 .k  / ˚ .k  ˝ k  / D ƒ2 .k  / ˚ .k  ˝ k  / ; Œx 7! x ^ .1  x/  x ˝ .1  x/:

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4 1 D Ez30 D B.k/ and we have a commutative triangle Therefore Ez30

/ / E1 D H 2 H3 .G/ 3 30 LLL LLL LLL LL& &  Ez 1 D B.k/ 30

where both maps from H3 .G/ are surjective, hence the vertical arrow is also surjective. It remains to check that the vertical arrow coincides with  . A line ` in P 3 .k/ is given by two linear equations and for an admissible line it is always possible to chose them in the form ´ t0 C x1 t2 C x2 t3 D 0; t1 C y1 t2 C y2 t3 D 0: This line corresponds to the tuple of vectors ! ! ! 1 0 x1 ; ; ; y1 0 1

x2 y2

!

which can be mapped to the points 0, 1, 1, xy11 yx22 in P 1 .k/, hence the vertical arrow maps it to Πxy11 yx22  (actually we need to consider a linear combination of lines which vanishes under d but for every line the result is given by this expression). On the other hand, four points of its intersection with the hyperplanes are P0 P1 P2 P3

D .0 W y1 x2  y2 x1 W x2 W x1 /; D .y2 x1  y1 x2 W 0 W y2 W y1 /; D .x2 W y2 W 0 W 1/; D .x1 W y1 W 1 W 0/;

and if we represent every point on ` as ˛P0 C ˇP1 then the corresponding ratios ˇ˛ will be 0, 1,  yx22 ,  yx11 . Hence .`/ D xy11 yx22 again and (iii) follows. To prove (iv) we first observe that the Hochschild–Serre spectral sequence associated to det 1 ! SL2 .k/ ! GL2 .k/ ! k  ! 1 gives a short exact sequence     det 1 ! H0 k  ; H3 .SL2 .k/; Z/ ! Ker H3 .GL2 .k/; Z/ ! H3 .k  ; Z/   ! H1 k  ; H2 .SL2 .k/; Z/ ! 1:

(9)

The first term here maps surjectively to K3ind .k/ (see the last section of [2]), and the map is conjectured by Suslin to be an isomorphism (see Sah [4]). It is known that its kernel is at worst 2-torsion (see Mirzaii [3]).

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Thus we let

  T .k/ WD Ker H0 .k  ; H3 .SL2 .k/; Z// ! K3ind .k/ :

By the preceding remarks, this is a 2-torsion abelian group. Since the embedding k  ! GL2 .k/ is split by the determinant, the middle term in (9) is isomorphic to H32 . Then applying the snake lemma to the diagram   / T .k/ / H0 k  ; H3 .SL2 .k/; Z/ / K ind .k/ /0 0 _ 3 _ 0

 /K

 / H3 2

 / B.k/

/0

gives the short exact sequence

  0 ! Tor.k  ; k  / ! K=T .k/ ! H1 k  ; H2 .SL2 .k/; Z/ ! 0:

Finally, it follows from [2] that there is a natural short exact sequence   0 ! H1 k  ; H2 .SL2 .k/; Z/ ! k  ˝ K2M .k/ ! K3M .k/=2 ! 0: This proves (4).

References [1] A. Borel, Cohomologie de SLn et valeurs de fonctions zêta aux points entiers. Ann. Scuola Norm. Sup. Pisa Cl. Sc. (4) 4 (1977), no. 4, 613–636. [2] K. Hutchinson and L. Tao, The third homology of the special linear group of a field. J. Pure. Appl. Algebra 213 (2009), no. 9, 1665–1680 [3] B. Mirzaii, Third homology of general linear groups. J. Algebra 320 (2008), no. 5, 1851–1877. [4] C.-H. Sah, Homology of classical Lie groups made discrete. III. J. Pure Appl. Algebra 56 (1989), no. 3, 269–312. [5] A. Suslin, K3 of a field and the Bloch group. Proc. Steklov Inst. Math. 183 (1991), no. 4, 217–239. [6] A. Suslin, The homology of GLn , characteristic classes and Milnor K-theory. Proc. Steklov Inst. Math. 165 (1985), 207–226. [7] B. Totaro, Milnor K-theory is the simplest part of algebraic K-theory. K-theory 6 (1992), 177–189. Kevin Hutchinson, School of Mathematical Sciences, Belfield Office Park 9/10, Belfield, Dublin 4, Ireland E-mail: [email protected] Masha Vlasenko, School of Mathematics, Trinity College, Dublin 2, Ireland E-mail: [email protected]

On complex and symplectic toric stacks Andreas Hochenegger and Frederik Witt

Introduction Toric varieties play an important rôle both in symplectic and complex geometry. In symplectic geometry, the construction of a symplectic toric manifold from a smooth polytope is due to Delzant [D]. In algebraic geometry, there is a more general construction using fans rather than polytopes. However, in case the fan is induced by a smooth polytope Audin [Au] showed both constructions to give isomorphic projective varieties. For rational but not necessarily smooth polytopes the Delzant construction was refined by Lerman and Tolman [LT], leading to symplectic toric orbifolds or more generally, symplectic toric DM stacks [LM]. We show that the stacks resulting from the Lerman–Tolman construction are isomorphic to the stacks obtained by Borisov et al. [BCS] in case the stacky fan is induced by a polytope. No originality is claimed (cf. also the article by Sakai [S]). Rather we hope that this text serves as an example driven introduction to symplectic toric geometry for the algebraically minded reader.

1 Delzant’s theorem We briefly describe the symplectic construction of a toric variety starting from a rational polytope. Good references are [Au] and [Gu]. In this section we assume manifolds, tensors, maps between manifolds etc. to be differentiable (i.e. of class C 1 ) unless mentioned otherwise.

³

Symplectic toric manifolds. Let U be a manifold. A symplectic form for U is a closed, non-degenerate 2-form !, that is d! D 0, and the natural map sending a vector field v 2 X.U / to the 1-form v ! D !.v; / 2 1 .U / is a linear isomorphism. In particular, U must be even dimensional. We call the pair .U; !/ a symplectic manifold. The automorphism group of a symplectic manifold, the group of symplectomorphisms Symp.U; !/, consists of diffeomorphisms preserving the symplectic form under pullback. Symplectomorphisms exist in abundance. Indeed, take any smooth function H 2 C 1 .U / and define the associated Hamiltonian vector field vH by !.vH ; / D dH . Then for U compact the flow of vH gives a curve in Symp.U; !/. Definition 1.1. Let G be a Lie group. An action of G on a symplectic manifold .U; !/ is hamiltonian if

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• there is a G-equivariant map  W U ! g from the manifold to the dual of the Lie algebra g of G (G acting via the coadjoint representation). We call  the moment map of the action; ³

• the fundamental vector fields v ] induced by v 2 g satisfy v ] ! D d h; vi (where h ; i denotes evaluation of  2 C 1 .U; g / on v 2 g). P Example. Consider C d with its standard symplectic form !0 D i dzk ^d zN k =2. Let T d D S 1      S 1 D .R=Z/d denote the compact (as opposed to algebraic) torus of dimension d . Then t D .t1 ; : : : ; td / 2 T d acts on z 2 C d via t:z D .t1 z1 ; : : : ; td zd /. For the standard basis e1 ; : : : ; ed of the Lie algebra td Š Rd , the induced fundamental vector fields are ek] .z/ D i.zk @zk  zNk @zN k /, hence ek] ! D .zk d zNk C zNk dzk /=2 D d jzk j2 . Therefore, the action is hamiltonian with moment map ³

0 .z/ D 12 .jz1 j2 ; : : : ; jzd j2 /: More generally, if G  T d acts on C d as a subgroup of T d , then the action is hamiltonian with moment map G D  B , where  is the dual of the natural inclusion of Lie algebras  W g ,! td . Hamiltonian actions by compact tori are particularly interesting because of the following Theorem 1.2 (Atiyah [At], Guillemin–Sternberg [GS]). For the hamiltonian action of a compact torus with moment map  on a compact, connected symplectic manifold, the set of fixed points of the action is a finite union of submanifolds C1 ; : : : ; Cr . On each of these submanifolds, .Cj /  j is constant and the image of  is the convex hull of the points j . Since the convex hull of a finite set of points in a real vector space is a polytope, one refers to the image of the moment map as the moment polytope. Example. In continuation of the previous example, consider the complex projective space P d Š S 2d C1 =S 1 . The T d C1 -action on C d C1 preserves the unit sphere S 2d C1 on which t 2 S 1 acts via .t; : : : ; t/ 2 T d C1 . The standard symplectic form on C d C1 descends to the quotient S 2d C1 =S 1 and induces the well-known Fubini–Study form !FS . We get a hamiltonian T d -action for .P d ; !FS / by sending .t1 ; : : : ; td / 2 T d to .1; t1 ; : : : ; td / 2 T d C1 and using the T d C1 -action on the sphere. Indeed, if s W td ! td C1 denotes the resulting inclusion at Lie algebra level, then T d B D s  B0 jS 2d C1 (with  W S 2d C1 ! P d the natural projection), that is, T d .Œz0 W : : : W zd / D

2

Pd

1

kD0 jzk j

2

.jz1 j2 ; : : : ; jzd j2 /

is a moment map for this action. In particular, the moment polytope is the simplex given by the images under T d of the fixed points Œ1 W 0 W : : : W 0; : : : ; Œ0 W 0 W : : : W 1.

On complex and symplectic toric stacks

307

Definition 1.3. A symplectic toric manifold is a symplectic manifold .U; !/ of dimension 2d together with an effective hamiltonian action by a compact d -torus. This is the symplectic counterpart of a complex toric variety (defined at the beginning of Section 4). Note that for an effective action of T l we need l  dim U=2 (cf. for instance Theorem 1.3 in [Gu]) so that d is the maximal dimension. Furthermore, since in this case, the moment map must be a submersion at some point (i.e. the differential is surjective at that point), the moment polytope is d -dimensional. For a compact torus T d we denote by N Š Zd the natural lattice inside the Lie algebra t Š Rd . The dual lattice Hom.N; Z/ is written M . Further, let   t be a polytope with m facets (i.e. codimension 1 faces) and open interior (the vertices are not necessarily lattice points). Definition 1.4. The polytope  will be called rational if it can be written D

m \

f˛ 2 t j h˛; uj i  j 2 Rg

(1)

j D1

for uj 2 N , j D 1; : : : ; m. In this case, we take the uj 2 N to be primitive and inward pointing. Furthermore, we say that  is smooth if for any vertex w 2 , the subset of vectors uj1 ; : : : ; ujd corresponding to facets meeting at w, forms a basis of N . For example, the moment polytope of P d , or more generally of any other toric symplectic manifold, is smooth. Conversely: Theorem 1.5 (Delzant [D]). Any smooth polytope  arises as the moment polytope of a symplectic toric manifold U . Furthermore, two symplectic toric manifolds are equivariantly symplectomorphic if and only if their associated moment polytopes can be mapped to each other by translation. Remark. For a given polytope  it follows from Delzant’s construction that U admits a natural compatible complex structure and is therefore Kähler. In fact, U is biholomorphic to any complex projective toric variety associated with a polytope with vertices in N and inducing the same normal fan as . Furthermore, the euclidean volume of U is proportional to the euclidean volume of  (cf. for instance Theorem 2.10 in [Gu]). The Lerman–Tolman theorem. From the view point of toric geometry it is natural to extend Delzant’s theorem to the case of rational polytopes. Namely, any rational polytope (with vertices in N ) gives a complex projective toric variety which is an orbifold, i.e. has at worst quotient singularities. Any effective orbifold is of the form U=G, where U is a manifold and G a compact connected Lie group acting effectively and locally freely on U , that is, with finite isotropy groups (cf. Corollary 2.16 and Theorem 2.19 in [MM]; for the noneffective case, see [HM]).

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On an orbifold X , differential forms, vector fields etc. can be defined either by using the isomorphism with U=G or in terms of an orbifold atlas. This is a collection .U˛ ; G˛ ; f˛ W U˛ ! X / such that G˛ is a discrete group acting effectively on the manifold U˛ and f˛ descends to a homeomorphism U˛ =G˛ ! X onto an open subset V˛  X. These data are required to satisfy the following conditions: • The collection fV˛ g is an open covering of X . • If x˛ 2 U˛ and xˇ 2 Uˇ get mapped to the same point in X , i.e. f˛ .x˛ / D fˇ .xˇ /, then there exists a germ of a diffeomorphism f˛ˇ from some connected open neighbourhood of x˛ to an open neighbourhood of xˇ such that fˇ B fˇ ˛ D f˛ . A differential form of degree k on an orbifold is then given by a collection of ˛ 2 k .U˛ / which agree on overlaps and which are invariant under the induced action of G˛ . Similarly, one can define vector fields. Hamiltonian group actions are more delicate to define (cf. [HS]), but nevertheless there is a natural notion of a symplectic toric orbifold (see also Definition 3.7 of a symplectic toric stack). Now with each point y 2 V˛  X we can associate the isotropy group of the G˛ -orbit f˛1 .y/, which is well defined up to conjugation. In the case of a symplectic toric orbifold for instance, there exists an integer nF for any each open facet F (i.e. the relative interior of the facet Fx) in  such that the isotropy group of any y in the preimage of F under the moment map is Z=nF Z. Attaching this integer to the open facet as an additional datum associates a labelled polytope  with any symplectic toric orbifold. Two such labelled polytopes are isomorphic if they differ only by a translation such that the corresponding facets carry the same labels. Theorem 1.6 (Lerman–Tolman [LT]). There is a 1-1 correspondence between isomorphism classes of labelled rational polytopes and symplectic toric orbifolds up to equivariant symplectomorphism. Example. Consider the polytope   R given by the interval Œ0; 1 with non-trivial labels n1 D k and n2 D 1 at the facets 0 and 1, whence 1 D 0 and 2 D 1 in (1). Let e1;2 be the standard basis of R2 and define ˇ W R2 ! R by ˇ.ej / D nj vj . This fits into the exact sequence ˇ



0!g y ! R2 ! R ! 0 which descends to the sequence on torus level: ˇN

y Š R=Z N! .R=Z/2 ! R=Z ! 0; 0 ! G

t 7! .t; k t /:

Now in general one can show that D  .1 ; 2 / is a regular value for the induced y Š S 1 acts via moment map Gy D  0 . In our case, 1 .1/ Š S 3 on which G y G

the inclusion N (written multiplicatively) t 2 S 1 7! .t; t k / 2 T 2 . By the symplectic

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reduction principle (see for instance Section 23 in [CdS] and also the example after Definition 3.4), S 3 =S 1 is a symplectic orbifold which inherits a toric structure from the y In particular, the unlabelled polytope (where k D 1) gives U D P 1 action of T 2 =G. with the Fubini–Study form. The general construction (without proof) will be outlined in Section 3. Note that the point corresponding to .0; 1/ in S 3 =S 1 is the only one with non-trivial stabiliser group (which is isomorphic to the group of k-th roots of unity Zk  S 1 ). As an orbifold, S 3 =S 1 is the so-called k-conehead (an explicit orbifold atlas will be exhibited in the next section). In the symplectic category labelled polytopes thus occur rather naturally. As we have mentioned before and seen in the previous example, the labels give rise to codimension 1 singularities. This cannot happen for algebraic toric varieties coming from a fan – they are necessarily normal and have thus singularities of codimension at least 2. This is where the idea of a stacky fan – due to Borisov et al. [BCS] – comes in. In the next few sections we will explain how theses concepts are related.

2 Lie groupoids and stacks In order to compare the results of [LT] and [BCS] we need to pass from orbifolds to (differentiable) stacks. These can be thought of either as categories fibred into groupoids [BX] or as pseudofunctors from the category of manifolds to the category of groupoids [H] which in both cases satisfy additional gluing conditions. We stick to the former approach, but to keep the exposition elementary we will not give a complete definition of a stack. Instead, we rather emphasise their description by means of (equivalence classes of) Lie groupoids using the dictionary established in [BX]. Stacks. In the following we will consider the category M of (differentiable) manifolds (not necessarily Hausdorff) with (differentiable) maps as morphisms. Good references for this section are the aforementioned texts by Behrend and Xu [BX] and Heinloth [H]. A short introduction to the idea of a stack which is sufficient for our purposes is given in [F]. Definition 2.1. (i) A category is called a groupoid if every morphism is invertible. (ii) A category fibred in groupoids (CFG) over M, written X ! M, is a category X together with a functor  W X ! M satisfying the following property. For every morphism U ! V in M and every object y of X lying over V (i.e. .y/ D V ), there exists a morphism f W x ! y lying over U ! V (i.e. .f / D U ! V ) which is unique up to unique isomorphism. This means that for any other morphism fQ W xQ ! y lying over U ! V there exists a unique isomorphism ˛ W xQ ! x lying over the identity of U such that fQ D f B ˛. A morphism between CFGs X and X 0 is a functor X ! X 0 which commutes with the projections to M. An isomorphism is a morphism X ! X 0 which is an equivalence of categories.

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Remark. (i) One often refers to the x of the definition as the pull-back of y via U ! V . It is unique up to unique isomorphism. It follows that for a manifold U , the subcategory X.U / of X consisting of objects lying over U and morphisms lying over the identity, is a groupoid. We call X.U / the fibre of  W X ! M. (ii) Similarly one could consider CGFs over base categories other than M such as topological spaces, complex (analytic) spaces …. The collection of CGFs over some base category defines itself a 2-category [Gr]. Example. (i) Let U be a manifold. We define U to be the category whose objects are maps X ! U between manifolds. A morphism between f W X ! U and g W Y ! U is a map h W X ! Y such that f D g B h. The projection  W U ! M is given by .X ! U / D X so that U.X / D C 1 .X; U /. The pull-back of f W Y ! U via g W X ! Y is obtained by the usual pull-back of maps g  f D f B g W X ! U . (ii) Let G be a Lie group. We define the stack BG to be the category consisting of objects .U; P / where pP W P ! U is a principal G-fibre over U . Morphisms .U; P / ! .V; Q/ consist of pair of maps .f W U ! V; fO W P ! Q/ such that fO is G-equivariant map and pQ B fO D f B pP . The projection  W BG ! M is defined by .U; P / 7! U . The fibre BG.U / is thus the subcategory of principal G-fibre bundles over U with bundle maps as morphisms. (iii) More generally, let G act on a manifold U . We define a CFG ŒU=G with fibres ŒU=G.X / WD f.P ! X; u W P ! U / j P 2 BG.X /; u is G-equivariantg by taking the same morphisms as in (ii) subject to the additional condition that fO must form a commutative triangle with the G-equivariant maps to U . We then recover the previous examples. Indeed, if the action of G is proper and free, then U=G is again a manifold so that any pair .P ! X; u W P ! U / in ŒU=G is determined by uQ W P =G Š X ! U=G, and therefore ŒU=G Š U=G. Secondly, taking a one point space U D , then .P ! X; u W P ! / 2 Œ =G.X / is determined by P ! X , that is, ŒU=G Š BG. A stack is a CFG X ! M which satisfies certain gluing conditions. The previous examples of CFGs all define stacks (see [BX] or [H]). A(n) (iso)morphism between stacks X and X 0 is a(n) (iso)morphism between CFGs. If U is a manifold, then Mor M .U; X/ Š X.U / where a functor F W U ! X corresponds to u D F .IdU / 2 X.U / (see Lemma 1.3 in [H]). Example. (i) A morphism F W U ! V is induced by a map f W U ! V . Any g W X ! U is mapped to f B g D g  f 2 V .X /. (ii) A morphism U ! BG is given by a principal fibre bundle P ! U . Any g W X ! U is mapped to the pull-back bundle g  P 2 BG.X /. (iii) A morphism U ! ŒV =G is given by a principal G-fibre bundle P ! U together with a G-equivariant map u W P ! V . Any g W X ! U is mapped to .g  P; gO  u/ 2 ŒV =G.X /, where gO is the induced bundle map g  P ! P covering gW X ! U.

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To define a differentiable stack we need one more notion. Definition 2.2. Given three CFGs X, Y, and Z over M and morphisms f W X ! Y and g W Y ! Z, the fibre product X Z Y is defined to be the following category. Objects are triples .x; y; ˛/, where x and y are objects in X and Y lying over the same object U in M and ˛ W f .x/ ! g.y/ is an isomorphism in Z lying over the identity of U . A morphism from .x 0 ; y 0 ; ˛ 0 / to .x; y; ˛/ is given by morphisms a W x 0 ! x in X and b W y 0 ! y in Y lying over the same morphism U 0 ! U in M such that ˛ B f .a/ D g.b/ B ˛ 0 W f .˛ 0 / ! g.y/. With the obvious projection X Z Y ! M, the fibre product becomes itself a CFG over M. Definition 2.3. (i) A stack X is representable if it is isomorphic to a stack U for some manifold U . (ii) A stack X is differentiable if there exists a morphism X ! X such that for any morphism U ! X, the resulting fibre product X X U is representable and the natural map between manifolds induced by the morphism X X U ! U is a surjective submersion. The morphism X ! X is said to be an atlas of X. Remark. In terms of algebraic geometry, a submersion is essentially a smooth map. Loosely speaking then, a representable morphism is a morphism with differentiable fibres. Example. (i) Let U ! W and V ! W be morphisms induced by submersions U ! W and V ! W (this implies in particular that U W V is again a manifold). As a consequence of the universal property of the (set-theoretic) fibre product, U W V is isomorphic to U W V . Since the induced map U W V ! V is clearly a surjective submersion, the stack U is differentiable. An atlas is provided by Id W U ! U . Any manifold can therefore be considered as a differentiable stack in a natural way. We sometimes simply write U for U if there is no risk of confusion. (ii) The morphism ! BG represented by  G 2 BG. / is an atlas for BG. Indeed, consider a morphism V ! BG associated with P ! V in BG.V /. Then . BG V /.X / D f.f W X ! ; g W X ! V; ˛ W X  G Š g  P g Š f.g W X ! V;  W X ! g  P / j pg  P B  D IdX g Š C 1 .X; P / D P .X /: (iii) Finally, consider an action W U  G ! G of the Lie group G on U . An atlas of ŒU=G, the so-called quotient stack of U and G, is provided by the morphism U ! ŒU=G corresponding to .U  G ! U; W U  G ! U /. Indeed, a calculation similar to (ii) shows that for a morphism V ! ŒU=G represented by .P ! V; u W P ! U /, one has U ŒU=G V Š P . Remark. Differentiable stacks form a full sub-2-category of the 2-category of CFGs over M consisting of differentiable stacks.

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Lie groupoids. product

Given an atlas x W X ! X, we can consider the (representable) fibre Iso.x/ WD X X X

together with its two canonical morphisms to X . Its inherent structure can be axiomatised as follows. Definition 2.4. (i) Let C be a category. A groupoid object in C or groupoid for short consists of two manifolds R and U and five structure maps, s W R ! U (the source map), t W R ! U (the target map), i W R ! R (the inverse map), 1 W U ! R (the unit map) and m W R U R D f.g; h/ 2 R  R j s.g/ D t .h/g ! R (the multiplication map). We usually write i.g/ D g 1 , 1.x/ D ex and m.g; h/ D gh. For any k, h and g in R these maps are required to satisfy • s.gh/ D s.h/, t .gh/ D t .g/, • .gh/k D g.hk/ whenever defined, • e t.g/ g D g D ges.g/ , • s.g 1 / D t .g/, t .g 1 / D s.g/, g 1 g D es.g/ , gg 1 D e t.g/ . (ii) A Lie groupoid is a groupoid object in M where s (and thus t ) is a submersion. A morphism between two Lie groupoids R Ã U and S Ã V is a differentiable functor which preserves the groupoid structure. More concretely, it is a pair of smooth maps ˆ W R ! S and W U ! V compatible with the structure maps, i.e. for all g; g 0 2 R and x 2 U we have .s.g// D s.ˆ.g//, .t .g// D t .ˆ.g//, ˆ.ex / D e.x/ and ˆ.gg 0 / D ˆ.g/ˆ.g 0 / whenever this makes sense. We denote this morphism by .ˆ; /. Remark. (i) The condition that s is a submersion implies in particular that R U R is again a manifold. (ii) If G is a small category, then G is a groupoid in the sense of Definition 2.1 (i) if and only if it is a groupoid object for the category of sets (with s.U ! V / D U , t.U ! V / D V , i taking a morphism to it inverse etc.). Schematically we can write a Lie groupoid as m

i

s

1

R U R ! R ! R Ã U ! R:

(2)

t

In general we simply write R Ã U for a Lie groupoid as given by (2). Definition 2.5. Let R Ã U be a Lie groupoid. (i) On U consider the equivalence relation x  y if and only if there exists g 2 R with s.g/ D x and t .g/ D y. The quotient U=  is called the space of orbits or coarse moduli space of the Lie groupoid. (ii) The isotropy group of x 2 U is the set Rx WD s 1 .x/ \ t 1 .x/ (this is indeed a group with respect to the natural group structure induced by m).

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Example. (i) Every manifold U defines the unit groupoid U à U with s, t , i and 1 the identity and p  p D p for p 2 U . The isotropy groups are Up D fpg and the coarse moduli space is U . More generally, consider a submersion X ! U . Then X U X defines a Lie groupoid with s and t the projections, 1 the diagonal, the inverse interchanging the two factors and multiplication sending .x; y/ and .y; z/ to .x; z/. (ii) Every Lie group G can be regarded as a Lie groupoid G à (with denoting the one point space), where m is usual multiplication, 1. / D eG and i the map taking a group element to its inverse. It follows that G D G while the coarse moduli space is . (iii) Every (left) G-space U gives the translation groupoid G  U à U . Here, s.g; u/ D u, t .g; u/ D gu, .g; hx/  .h; x/ D .gh; x/, i.g; x/ D .g 1 ; gx/ and 1.x/ D .eG ; x/. The isotropy group of x is the stabiliser under the action, i.e. the set of pairs .g; x/ such that gx D x. Further the coarse moduli space is just the space of orbits U=G. (iv) Let x W X ! X be an atlas of a differentiable stack. Then Iso.x/ à X defines a Lie groupoid. Indeed, Iso.x/ consists of triples .f W U ! X; g W U ! X; ' W x.f / Š x.g// and the canonical projections taking such a triple to f and g respectively define the source and target maps. Multiplication with .f 0 ; g 0 ; W x.f 0 / Š x.g 0 // is defined by .f; g 0 ; B ' W x.f / Š x.g 0 //. Since x induces a surjective submersion, Iso.x/ can be given a differentiable structure. Definition 2.6. (i) A Lie groupoid R à U is proper if the map .s; t / W R ! U  U is proper. (ii) A Lie groupoid R à U is étale if dim R D dim U , that is, s and t are local diffeomorphisms. Proper étale Lie groupoids arise from effective orbifolds as defined in Section 1. Indeed, let .U˛ ; G˛ ; f˛ / be an orbifold atlas for X . Put U D tU˛ and let R be the set of triples .x; y; f / such that x and y get mapped to the same point in X and f is a germ of a diffeomorphism mapping x to y. Then s.x; y; f / D x, t .x; y; f / D y, i.x; y; f / D .y; x; f 1 /, 1.x˛ / D .x˛ ; x˛ ; idU˛ / and .y; z; f /.x; y; g/ D .x; z; f B g/. Moreover, the sheaf topology on R turns s and t into local homeomorphisms which induce a differentiable structure on R for which s and t become local diffeomorphisms. Further, the resulting Lie groupoid R à U is proper by Proposition 5.29 in [MM]. Note that the isotropy group of a point x 2 U (in the sense of Section 1) is just Rx as given in Definition 2.5. Hence the isotropy groups are discrete and Proposition 5.20 in [MM] implies that R à U is also étale. Example. Take P 1 D C [ f1g and remove a disk D around 1. We obtain the kconehead encountered above by gluing in the cone D=Zk of angle 2=k, see Figure 1. The resulting space is still homeomorphic to P 1 . An orbifold atlas is given by .U0 D C; feg; f0 .z/ D Œz W 1/ and .U1 D C; Zk ; f1 .z/ D Œ1 W z k /. Indeed, z, w 2 U1 get mapped to the same point if and only if w D e 2l=k z for some l D 0; : : : ; k  1 so that e 2l=k induces the required germ of diffeomorphisms. On the other hand, if z 2 U0 and w 2 U1 get mapped to the same point, a germ is induced by f01 .w/ D w k .

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Figure 1. A cone of angle 2=k.

Morita equivalence. An atlas x W X ! X gives rise, as we have seen, to a Lie groupoid Iso.x/ Ã X . Conversely, one can associate with a Lie groupoid R Ã U a differentiable stack which carries an atlas giving back R Ã U up to isomorphism (see for instance [BX]). Of course, different Lie groupoids can give rise to isomorphic stacks in the same way two different atlases of a topological manifold can give rise to the same differentiable structure. This structure will be indeed the same if we can pass to a common refinement. We will formalise a similar concept for Lie groupoids now. Definition 2.7. (i)A Morita morphism is a Lie groupoid morphism .ˆ; / from R Ã U to S Ã V which satisfies the following two properties: • The diagram R

.s;t/



ˆ

 S

/U U

.s;t/

 /V V

is cartesian, i.e. R is isomorphic to the fibred product S V V .U  U /. • The map t B pr 1 W S s;V; U ! V sending .h; y/ to t .h/, is a surjective submersion. (ii) Two Lie groupoids R Ã U and S Ã V are Morita equivalent if there exists a third Lie groupoid T Ã W with Morita morphisms to R Ã U and S Ã V . Remark. (i) Here we followed the terminology of [BX]; [MM] speak of weak equivalences and weakly equivalent respectively. (ii) Morita equivalence defines an equivalence relation between Lie groupoids (see the remark after Proposition 5.12 in [MM]). Then we have (cf. Theorem 2.26 in [BX]) Proposition 2.8. Let X and Y be differentiable stacks which are associated with the Lie groupoids R Ã U and S Ã V . Then the stacks X and Y are equivalent if and only if R Ã U and S Ã V are Morita equivalent.

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We will sometimes abuse language and refer to a Lie groupoid associated with a stack as an atlas of the stack. Remark. One can turn the category Lie groupoids into a 2-category and establish a dictionary between differentiable stacks and Lie groupoids, see [BX]. Example. (i) Consider the stack U with atlas provided by the identity Id W U ! U . Then Iso.Id/ Ã U is the unit groupoid defined in the examples after Definition 2.5. (ii) Consider the quotient stack ŒU=G. The Lie groupoid Iso.U  G; / Ã U given by the atlas induced by .pr 1 W U  G ! U; W U  G ! U / is the translation groupoid G  U Ã U . Indeed, objects in U ŒU=G U.X / are determined by triples .f W X ! U; g W X ! U; ' W X ! G/ with '.x/ f .x/ D g.x/ (cf. the example after Definition 2.3 to see this). Sending this triple to .'; f / gets the isomorphism U ŒU=G U Š G  U . With the second projection as source map and group action as target map (sending .'; f / to ' f W X ! U ) we obtain the translation groupoid. In particular, taking U D we recover the Lie groupoid Iso.  G/ of BG coming from the natural atlas ! BG, which is just the Lie groupoid G Ã . In the sequel we shall be mainly interested in the case of quotient stacks. The definition of Morita equivalence takes an easier shape when applied to this special case. First, a morphism ŒU=G ! ŒV =H  consists of a morphism W U ! V and a group homomorphism W G ! H which are compatible in the following sense. If G and H denote the respective group actions of G and H on U and V , we require that .g G u/ D

.g/ H .u/

for all g 2 G and u 2 U . Then . ; / is a Morita equivalence if (M1) the diagram GU

.pr 2 ;G /



 H V

/U U 

.pr 2 ;H /

 /V V

is cartesian and (M2) the morphism H  U ! V;

.h; m/ 7! h H .m/;

is surjective. Remark. If both W U ,! V and to condition (M10 )

W G ,! H are inclusions, then (M1) is tantamount

8h 2 H and m 2 U W Œh m 2 U ) h 2 G :

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3 Symplectic toric DM stacks In this section we discuss the extension of the Lerman–Tolman theorem 1.6 to the stack setting. Deligne–Mumford stacks Definition 3.1. A Deligne–Mumford stack or DM stack for short is a differentiable stack which admits an atlas given by a proper étale Lie groupoid R Ã U . Its dimension is dim X D dim U D dim R. The dimension is indeed well defined, cf. Section 2.5 in [BX]. Example. (i) The unit Lie groupoid U Ã U is étale and proper. Hence every manifold considered as a stack is a DM stack. (ii) As discussed in Section 2, an effective orbifold in the sense of Section 1 gives rise to a proper étale Lie groupoid and thus to a DM stack. To define further geometric structure on a DM stack we take the viewpoint of [LM] and define objects with respect to a fixed (not necessarily étale) atlas R Ã U . To check independence of the atlas one can either show invariance under Morita equivalence or compatibility of these definitions with abstract stack theory. Symplectic DM stacks. For the definition of vector fields and differential forms on DM stacks we first generalise the concept of a Lie algebra associated with a Lie group to Lie groupoids. Definition 3.2. A Lie algebroid over a manifold M is a vector bundle a ! M with a Lie bracket on its space of sections C 1 .a/, together with a vector bundle morphism a W a ! TM called the anchor such that • the induced map C 1 .a/ ! X.M / between sections of A and vector fields on M is a Lie algebra morphism; • for all v; w 2 C 1 .a/ and f 2 C 1 .M /, the identity Œv; f w D f Œv; w C df .a.v//w holds. With a Lie groupoid R Ã U we can canonically associate a Lie algebroid r ! U as follows (cf. [MM] Section 6.1). For an arrow h W y ! x we can define a “left multiplication” Lh W t 1 .y/ ! t 1 .x/ by composition Lh .g/ WD hg. This lifts to the involutive vector subbundle ker ds ! R of TR ! R. Namely, given 2 .ker ds/g for some g W z ! y we define h WD dLh . / which lies in .ker ds/hg . A section X 2 C 1 .ker ds/ is invariant if X.hg/ D hX.g/. The invariant sections form a Lie subalgebra A of C 1 .ker ds/. As sections of A are determined by their restriction to the set of units, we get a linear isomorphism between A and the space of sections of the vector bundle r WD 1 ker ds ! U . In particular, r inherits a natural Lie algebra

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structure. We take a D dt W r ! T U as anchor map and call .r; a/ the associated Lie algebroid. Example. Consider a Lie group G as Lie groupoid via G Ã . The associated Lie algebroid g D 1 ker ds ! is a vector bundle whose Lie algebra structure is precisely the Lie algebra structure of left-invariant vector fields on G. We thus recover the usual Lie algebra of G. An arbitrary atlas of a DM stack is not necessarily étale as this property is not preserved under Morita equivalence. However, the following statement allows us to characterise the Lie groupoids representing a DM stack in terms of their associated Lie algebroids. Theorem 3.3 ([CM]). A Lie groupoid R Ã U is Morita equivalent to an étale groupoid if and only if the Lie algebroid associated with R Ã U has injective anchor. For a Lie groupoid R Ã U representing a DM stack we can therefore think of its associated Lie algebroid as a subbundle of T U . This makes DM stacks a convenient class to work with. In the sequel, we consider a DM stack X together with a fixed atlas R Ã U. We first define the space of k-forms on X by k .X/ WD f.˛1 ; ˛0 / 2 k .R/  k .U / j s  ˛0 D ˛1 D t  ˛0 g: We can regard k .X/ as k-forms annihilating r  T U . If r?  T  U denotes the annihilator of r, then ˛0 is a section of ƒk r? . Furthermore, ˛0 is invariant under the natural “action” of R on U , where g 2 R sends s.g/ to t .g/. Note that the exterior derivative d commutes with pullbacks. Hence the exterior derivative induces a welldefined map d W k .X/ ! kC1 .X/ sending .˛1 ; ˛0 / to .d˛1 ; d˛0 /. In particular, we can speak about closed forms, i.e. forms in the kernel of d . By Proposition 2.9 (ii) in [LM], the resulting de Rham complex  .X/ does not depend on the chosen atlas up to isomorphism. Analogously we define the space of vector fields X.X/. A vector field is a section of T U=r which is equivariant under the “action” of R on U . Concretely, call a pair .v1 ; v0 / in X.R/  X.U / compatible if ds.v1 / D v0 B s; dt .v1 / D v0 B t . Then we regard two pairs of compatible vector fields .v1 ; v0 / and .w1 ; w0 / as equivalent (denoted by ) if and only if they differ only by a compatible pair .u1 ; u0 / with u1 2 .ker ds C ker dt /. (Note that s is a surjective submersion, hence the relation ds.v1 / D v0 B s determines v1 up to sections of ker ds, and similarly for t .) We then define X.X/ WD f.v1 ; v0 / 2 X.R/  X.U / j ds.v1 / D v0 B s; dt .v1 / D v0 B t g=  : To keep notation simple we denote by .v1 ; v0 / both the compatible pair and the induced equivalence class. Again, up to isomorphism, X.X/ does not depend on the chosen atlas (Proposition 2.9 (i) in [LM]).

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Example. (i) For a manifold U take the associated Lie groupoid Iso.Id/ Ã U considered in Section 2. Then we recover the usual notion of a differential form and a vector field. (ii) If G acts freely and properly on U , let g] ! U denote the subbundle of fundamental vector fields in T U . Then ŒU=G Š U=G, whence k .ŒU=G/ D C 1 .ƒk g]? /G , the G-invariant k-forms which annihilate g] . Similarly, X.ŒU=G/ D C 1 .T U=g] /G , the G-invariant sections of T U=g] . From the previous definitions it follows that the contraction of a vector field v D .v1 ; v0 / with a form ˛ D .˛1 ; ˛0 /, ˛0 /

³

˛1 ; v0

³

˛ WD .v1

³

v ³

(where vj ˛j is the usual contraction k ! k1 ) is a well-defined operation k .X/ ! k1 .X/. A 2-form ! on a DM stack X is said to be non-degenerate if and only if contraction with ! induces a linear isomorphism X.X/ ! 1 .X/. Definition 3.4. A 2-form ! on a DM stack is called symplectic if it is non-degenerate and closed. A DM stack .X; !/ together with a symplectic form is called a symplectic DM stack. Example. In continuation of the previous example (ii) a symplectic form on ŒU=G Š U=G is a closed G-invariant 2-form on U whose kernel is precisely g] (this is the nondegeneracy condition). As an example, consider the action of a Lie group G  T d on C d with associated moment map G W C d ! g (see the example after Definition 1.1). Let 2 g be a regular value for G and assume that G acts freely on the embedded d  ] submanifold i W 1 G . / ,! C . Then the kernel of the closed 2-form ! D i !0 is g 1 so that ! descends to a symplectic form on G . /=G. This is the symplectic reduction principle (cf. for instance Section 23 in [CdS]) which underlies the construction after Theorem 3.8. Hamiltonian group actions. Next we wish to consider actions by a Lie group on a differentiable stack. Their definition is more subtle than in the case of manifolds for stacks are categories and thus group elements act as functors. However, the composition of two such functors may differ from the functor of the product of the corresponding group elements by a natural transformation. A precise definition of G-actions as used here is due to Romagny [R]. We will not give it here; instead, we rephrase G-actions on DM stacks in terms of Lie groupoids (see Proposition 1.5 in [R] and Proposition 3.2 in [LM]). Proposition 3.5. Suppose we have a G-action on a DM stack X for some Lie group G. Then there exists a G-atlas for X, that is, there exists a Lie groupoid R Ã U where G acts on both R and U freely and compatibly with the structure maps.

On complex and symplectic toric stacks

319

Remark. If in addition G acts properly on R and U , R=G and U=G are again manifolds. The Lie groupoid R=G Ã U=G is then an atlas for the differentiable stack X=G, the quotient of X by G. Of course, U=G Š U=G. Let R Ã U be a G-atlas of X. For v 2 g, let v ] D .v1] ; v0] / denote the induced fundamental vector field on R  U . Since the action of G commutes with the structure maps, v ] 2 X.X/.

³

Definition 3.6. A G-action on a symplectic DM stack .X; !/ is called hamiltonian if there is a G-atlas R à U with a G-equivariant g -valued function  D .1 ; 0 /, i.e.  2 0 .X/ ˝ g , such that   v ] ! D d h1 ; vi; d h0 ; vi for any v 2 g. Again we refer to  as the moment map of the action. Symplectic reduction. The next definition is also taken from [LM]. Definition 3.7. A symplectic toric DM stack is a symplectic DM stack .X; !/ with a hamiltonian action by a compact torus T such that • T acts effectively on the coarse moduli space for any given T -atlas R à U . • dim X D 2 dim T . Generalising the example after Definition 3.4 we get for any regular value 2 g of T d the symplectic toric DM stack Œ1 . /=T d . Indeed, Theorem 5.4 in [LM] Td gives the following. Theorem 3.8 (Lerman–Malkin [LM]). Let G  T be a closed subgroup and 1 !

! Ty ! T ! 1 an extension of the standard compact torus by a finite group . Let y denote the corresponding group in Ty and 2 g be a regular value for the moment G map Gy W C d ! g . Then the quotient stack   y y WD 1 . /=G C d == G y G y D T =G. is a symplectic toric DM stack acted on by the torus Ty =G Examples. One way of producing the data of Theorem 3.8 is to consider a labelled rational polytope   Rd  as in Theorem 1.6. Using the notational conventions of Section 1, we define a linear map ˇ W Rm ! Rd by ˇ.ej / D nj uj , where e1 ; : : : ; em is the standard basis of Rm . Since uj 2 N for j D 1; : : : ; m, the exact sequence 

ˇ

0 ! ker ˇ ! Rm ! Rd ! 0

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gives rise to the (additively written) exact sequence on the torus level N

ˇ N y WD ker ˇN ! T m Š .R=Z/m ! T d Š .R=Z/d ! 0; 0!G

that is,

y D fŒx D Œ.x1 ; : : : ; xm / 2 T m j x 2 Rm with ˇ.x/ 2 N g: G y Instead of ˇ we can also consider ˇ0 W Rm ! Note that ker ˇ is just the Lie algebra of G. d R defined by ˇ0 .ej / D uj which in the same way gives rise to a subgroup G  T m . Then there is a finite extension given by the split sequence nN y! 0! !G G!0

induced by the map nŒx N D Œ.n1 x1 ; : : : ; nm xm /. Finally, D  .1 =n1 ; : : : ; m =nm /, where the j are determined by (1), is a regular value for Gy by Theorem 8.1 in [LT]. Now Theorem 3.8 applies. Next we consider two concrete examples. The first one is induced by the moment polytope of P 2 (see Figure 2), but with two different non-trivial labellings. The resulting coarse moduli space has a singular divisor and thus codimension 1 singularities. The second example comes from a rational, non-smooth polytope with trivial labelling (see Figure 3). Here, the singularities have codimension 2. (i) The projective plane. We have the facets F1 , F2 and F3 (see Figure 2) which we label by .1; 1; 2/ and .2; 2; 2/. It follows that 1 D 2 D 0 and 3 D 1. The resulting maps ˇ1 and ˇ2 are given by ˇ1 W R3 e1 e2 e3

! 7 ! 7 ! 7 !

R2 ; 1  u1 ; 1  u2 ; 2  u3 ;

which we represent by the matrices   1 0 2 M1 D 0 1 2

ˇ2 W R3 e1 e2 e3

! 7 ! 7 ! 7 !

R2 ; 2  u1 ; 2  u2 ; 2  u3 ;

 and

M2 D

 2 0 2 : 0 2 2

(3)

Now ˇ1;2 .x1 ; x2 ; x3 / 2 Z2 if and only if x1;2 2x3 2 Z (for ˇ1 ) and 2.x1;2 x3 / 2 Z (for ˇ2 ). Since nN 1 Œx2 ; x2 ; x3  D Œx1 ; x2 ; 2x3  and nN 2 Œx2 ; x2 ; x3  D Œ2x1 ; 2x2 ; 2x3 , we get the exact sequences nN 1

y1 D fŒ2x; 2x; x j x 2 Rg ! G D .R=Z/3 ! 0 0 ! 1 Š Z2 ! G and y2 D fŒx C a; x C b; x j x 2 R; a; b 2 1 Zg 0 ! 2 Š Z2 ! G 2 nN 2

! G D .R=Z/3 ! 0:

On complex and symplectic toric stacks

F1

F3

321

u2 u1

F2

u3

Figure 2. The polytope and its normal fan of the projective plane P 2 .

y1 and G y2 act on C 3 via the In both cases we have 1;2 D f.0; 0; c/ j c 2 12 Zg. Then G inclusions (written multiplicatively) y1 ! T 3 ; S1 Š G t D Œ.2x; 2x; x/ 7! .t 2 ; t 2 ; t/; and y2 ! T 3 ; Z2  Z2  S 1 Š G   .Œa; Œb; t / D Œ.x C a; x C b; x/ 7! .1/2a t; .1/2b t; t : On the other hand, k .ker ˇk /, k D 1; 2, is spanned by .2; 2; 1/ and .1; 1; 1/ respectively so that Gy 1 .z0 ; z1 ; z2 / D 1 B .z0 ; z1 ; z3 / D 2jz0 j2 C 2jz1 j2 C jz2 j2 ; Gy 2 .z0 ; z1 ; z2 / D 2 B .z0 ; z1 ; z3 / D jz0 j2 C jz1 j2 C jz2 j2 : Hence 1 .k .// D 1 .2/ is diffeomorphic to S 5 and we obtain the toric symplectic y y Gk

Gk

y1;2 . The isotropy groups are trivial for .z0 ; z1 ; z2 / y1;2 D ŒS 5 =G DM stacks C 3 ==2 G 2 2 with jz0 j C jz1 j < c1;2 where c1 D 1 and c2 D 2, and otherwise isomorphic to Z2 . (ii) Weighted projective space. Here we take the polytope of Figure 3 with trivial labelling, i.e. ˇ.ei / D ui , so the associated matrix is   1 0 1 M D : (4) 0 1 2 Proceeding as above, we see that .ker ˇ/ is spanned by .1; 2; 1/ and y D fŒ.x; 2x; x/ j x 2 Rg G acts on C 3 via the inclusion y Š S 1 ! T 3; G

t 7! .t; t 2 ; t/:

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Now  D .0; 0; 2/, so 2 is a regular value for Gy .z0 ; z1 ; z2 / D jz0 j2 C 2jz1 j2 C jz2 j2 yD and 1 .2/ is diffeomorphic to S 5 . We obtain the toric symplectic stack C 3 ==2 G y G y – the 2-dimensional analogue of the 2-conehead. The only nontrivial isotropy ŒS 5 =G group is Z2 for the image of .0; 1; 0/.

F1

F3

u2 u1

F2 u3

Figure 3. The polytope and its normal fan of the weighted projective space P 2 .1; 1; 2/.

4 Complex toric DM stacks A (complex) toric variety is a normal variety which contains an algebraic torus T as an open dense subset and such that the action of T on itself extends to the whole variety. This definition was subsequently generalised by Iwanari [I] and Fantechi et al. [FMN] to (complex) toric DM stacks, that is, (separated) DM stacks together with a stacky DM torus as an open dense subset and such that its action extends to the stack (cf. Definition 3.1 in [loc. cit.]). Based upon this definition, Fantechi et al. carried out a classification of complex toric DM stacks. Prior to this, Borisov et al. [BCS] constructed complex toric DM stacks as quotients. It is this construction we want to outline in this section in the case of trivial generic stabiliser. Note that the construction of [BCS] via stacky fans gives stacks isomorphic to the toric DM stacks as considered in [FMN]. However, the isomorphism is not unique if there is a non-trivial generic stabiliser (cf. Theorem II and Remark 7.26 in [FMN]). Stacky fans. For a fan † we denote by †.i / the set of i -dimensional cones in †. In particular, †.1/ is the set of rays. Let m be the cardinality of †.1/. Definition 4.1. A stacky fan is a triple † D .†; N; ˇ/ where • N Š Zd is a lattice with dual lattice M D Hom.N; Z/; • † is a complete and simplicial fan in NQ D N ˝ Q, i.e. the union of all cones covers the whole NQ , and the generators of the rays of each cone are linearly independent; • ˇ W Zm ! N is a map such that if uj denotes the primitive generator of j 2 †.1/, then ˇ.ej / D nj  uj for some nj 2 N>0 . We think of ˇ as a choice of lattice generators for the rays in †.1/.

On complex and symplectic toric stacks

323

Remark. Instead of completeness it suffices to require that the rays of the fan † generate NQ (cf. the analogous construction of toric varieties, e.g. [C]). Geometrically this means that there are no torus factors, see for instance Section 5.1 in [CLS] or Remark 7.14 in [FMN]. Moreover, N can be any finitely generated abelian group. The case of toric varieties. A toric variety X D X.†/ with simplicial fan † and without torus factors can be written as a good geometric quotient m =H: X.†/ D C† m Here, C† and H are defined as follows. Consider C m as the direct product of copies of C D Spec kŒxj  for every ray j 2 †.1/. The monomials Y xj for  2 † x. / WD j 6D

generate the so-called irrelevant ideal I whose vanishing locus is Z (since † is assumed to be complete, it is actually enough to take only the monomials x. / associated with the top-dimensional cones  2 †.n/). Hence Z is simply a union of coordinate m hyperplanes. We set C† WD C m n Z. Let T D Hom.M; C  / be the algebraic torus of X. To obtain the group action we look at one of the most important exact sequences in toric geometry: 0 ! M !

m M

Z  Dj ! Cl.X / ! 0;

j D1

X w 7 ! hw; uj iDj ;

(5)

j

where Dj is the T -invariant divisor corresponding to the ray j , and Cl.X / is the divisor class group of X . We apply the functor HomZ .; C  / to this sequence. Although the Hom-functor is only left-exact, the sequence 1 ! Hom.Cl.X /; C  / ! .C  /m D Hom.Zm ; C  / ! T ! 1 m is still exact, for C  is divisible. Then H WD Hom.Cl.X /; C  / acts on C† via the  m natural inclusion into .C / . By Theorem 1.11 in Chapter 5 of [CLS] the quotient m =H is a geometric quotient and isomorphic to X.†/. C†

The generalisation to toric stacks. For a stacky fan † we proceed similarly and define the stack   m =H.ˇ/ : (6) X.†/ D C† m Here C† is constructed as before, but the definition of H.ˇ/ is more elaborate.

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Consider the map ˇ W Zm ! N . As remarked above, N could be any abelian group so that dualising ˇ destroys all torsion information. Although we are dealing here with a torsion-free N , we present the more general procedure used in [BCS]. So, instead of dualising ˇ directly we consider the mapping cone in the derived category of Z-modules, i.e. the exact triangle Zm

id

/ Zm

ˇ

Zm

ˇ

/N /N

Zm

 /N

id

/ Cone.ˇ/

/ Zm Œ1:

In the general situation that N is not free but a finitely generated abelian group, N needs to be replaced by its free resolution in the diagram above. Now we dualise to get .Zm / o O

id

.Zm /

ˇ

.Zm / o

ˇ

.Zm / o

M o M o

id

M Cone.ˇ/ o

.Zm / Œ1:

Rolling out this triangle by taking cohomology leads to the long cohomology sequence, but we are only interested in its end ˇ

ˇ_

H 0 Cone.ˇ/ ! M ! .Zm / ! H 1 Cone.ˇ/ ! Ext 1 .N; Z/ ! 0: Since N is free, Ext1 .N; Z/ vanishes, hence H 1 Cone.ˇ/ is just coker.ˇ  / and ˇ _ W .Zm / ! coker ˇ  is surjective. On the other hand, the fan † is complete so that ˇ W Zm ! N , though not necessarily surjective, has only finite cokernel. Hence H 0 Cone.ˇ/ D ker.ˇ  / D coker.ˇ/ D 0, which yields the exact sequence ˇ

ˇ_

0 ! M ! .Zm / ! coker.ˇ  / ! 0:

(7)

In essence, this is the sequence from (5). Therefore, applying the functor HomZ .; C  / gives an action of H.ˇ/ WD Hom.coker.ˇ  /; C  /: m on C† via the embedding Hom.ˇ _ ; C  / into Hom..Zm / ; C  /. This defines the stack X.†/ of (6).

325

On complex and symplectic toric stacks

Remark. (i) Explicitly, the map ˇ  is given by ˇ

M ! Hom.Zm ; Z/;

M !

  w 7! ej 7! hw; ˇ.ej /i D hw; nj  uj i ;

w 7!

m M

Z  Dj ;

jP D1

j hw; uj iDj :

Comparing this with the map in (5), we recover the classical toric case by taking for ˇ the map Œej 7! uj . (ii) For general N , the group H.ˇ/ is Hom.H 1 Cone.ˇ/ ; C  /. The case of torsion-free N corresponds precisely to complex toric DM orbifolds (cf. Lemma 7.15(2) in [FMN]). Open substacks. For toric varieties, the cones  of maximal dimension in the defining (complete and simplicial) fan † give open charts U of X.†/. For toric stacks Proposition 4.3 in [BCS] yields a similar statement. Let  be such a cone of dimension d . We can restrict the map ˇ W Zm ! N to ˇ W Zd ! N such that ˇ .N d / ˝ Q D  . Set N D im ˇ . This is a sublattice of N of finite order, i.e. N. / D N=N is a finite group. Then  D .; N; ˇ / defines an open substack X. / of X.†/. The important observation of Proposition 4.3 in [loc. cit.] is that X.†/ is locally the quotient by a finite group H.ˇ / Š N. /:   X. / D C d =H.ˇ / : Proposition 4.2. The quotient X. / D C d =H.ˇ / is isomorphic to X. /. Proof. Let ˇ W Zd ! N be given by ej 7! nj uj and define ˇ ;0 W Zd ! N; ej 7! uj . If W Zd ,! Zd is the obvious map such that ˇ D ˇ ;0 B , then there exists a big commutative diagram M Z =nj Z Š / coker ˇ  = coker ˇ  ;0 j OO OO M



ˇ



 ˇ;0

/ .Zd / O 

M

? / .Zd /

/ / coker ˇ  O ?  / / coker ˇ ;0 .

The inclusion in the lower right hand side corner and the isomorphism in the top row can be deduced from the snake lemma. We apply the functor Hom.; C  / to the exact sequence in the right column and obtain M Znj ! H.ˇ / ! H.ˇ ;0 / ! 0 0 ! j

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where the Znj denote the cyclic groups of ni -th roots of unity. It is well-known from geometric invariant theory that the morphism Spec A ! Spec AG is a good categorical quotient if G is a reductive group acting algebraically on the affine variety Spec A, whence C d =H.ˇ / D Spec CŒx1 ; : : : ; xd H.ˇ / : Using the previous exact sequence we conclude L  H.ˇ;0 / Š CŒx1 ; : : : ; xd H.ˇ;0 / CŒx1 ; : : : ; xd H.ˇ / D CŒx1 ; : : : ; xd  j Znj to obtain the isomorphism X. / Š X. /. Remark. Even though X. / is isomorphic to X. / as an affine variety, the torus actions are different. Furthermore, these spaces are also different when considered as orbifolds or stacks ŒC d =H.ˇ / and ŒC d =H.ˇ ;0 /: The action of H.ˇ ;0 / is free except on a closed subset of codimension at least 2, which becomes the singular locus of the quotient X. /. As soon as nj > 1, H.ˇ / does not act freely anymore. In particular, this action has Znj as isotropy group for any point in the divisor fxj D 0g of X. /. Examples. We revisit the examples from Section 3. (i) The projective plane. For the fan † of P 2 (see Figure 2) we consider again the maps ˇ1;2 : ˇ2 W Z3 ! N; ˇ1 W Z3 ! N; e1 7! 1  u1 ; e1 7! 2  u1 ; e2 7! 2  u2 ; e2 7! 1  u2 ; e3 7! 2  u3 ; e3 7! 2  u3 ;  we diagorepresented by the matrices M1;2 in (3). For the computation of coker ˇ1;2 nalise the transposes of M1;2 with an element in Gl.3; Z/ and get 1 0 1 0 1 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 2 0 2 0 @0 1 0A  @ 0 1 A D @0 1A ; @0 1 0A  @ 0 2 A D @0 2A : 2 2 1 2 2 0 0 1 1 1 2 2 0 0 _ Hence coker.ˇ1 / Š Z and coker.ˇ2 / Š Z2 ˚ Z2 ˚ Z. The maps ˇ1;2 are given by

ˇ1_ W Z3 e1 e2 e3

! coker.ˇ1 / 7 ! e1 C 2e3 7 ! e2 C 2e3 7 ! e3

Š $ $ $

Z; 2; 2; 1;

ˇ2_ W Z3 e1 e2 e3

! coker.ˇ2 / 7 ! e1 C e3 7 ! e2 C e3 7 ! e3

Š Z2 ˚ Z2 ˚ Z; N 0; N 1/; $ .1; N N 1/; $ .0; 1; N N 1/: $ .0; 0;

So the groups H1;2 D H.ˇ1;2 / act via T in the following way: H1 Š C  ! T ; H2 Š Z2  Z2  C  ! T ;

t 7! .t 2 ; t 2 ; t/; N

N t / 7! ..1/aN t; .1/b t; t /: .a; N b;

327

On complex and symplectic toric stacks

Putting everything together we obtain fromthe stacky fan †1;2 D .†; Z2 ; ˇ1;2 / the toric DM stack X.†1;2 / D C 3 n f0g=H1;2 . (ii) Weighted projective space. For the fan of P .1; 1; 2/ (see Figure 3) we take ˇ trivial as in Section 3. Diagonalising the transpose of M given in (4) yields 0 1 0 1 0 1 1 0 0 1 0 1 0 @0 1 0A  @ 0 1 A D @0 1A : 1 2 1 1 2 0 0 Hence coker.ˇ  / Š Z and the map ˇ _ is ˇ _ W Z3 e1 e2 e3

! coker.ˇ  / 7 ! e1 C e3 7 ! e2 C 2e3 7 ! e3

Š $ $ $

Z; 1; 2; 1:

So H acts via T by

H D C  ! T ; t 7! .t; t 2 ; t/;   and we therefore obtain X.†/ D C 3 n f0g=H .

5 Comparison of symplectic and complex toric DM stacks Consider the normal fan † of a given polytope and a choice of ray generators ˇ W Zm ! N , ej 7! nj uj . Our aim is to show that the differentiable stacks induced by .†; ˇ/ following the construction in Sections 3 and 4 are isomorphic, that is, they have Morita equivalent atlases. To apply the simplified criterion of Morita equivalence as given in Section 2, we first establish the inclusions m ker ˇN ,! H.ˇ/ and 1 † . / ,! C†

(where † is moment map induced by †, see below). Lemma 5.1. For ˇ W Zm ! N with finite cokernel, ker ˇN and Hom.coker ˇ  ; R=Z/ are naturally isomorphic. Proof. We apply the exact functor Hom.; R=Z/ to the sequence (7): 0o

N ˝ R=Z o

0o

NR =N o

ˇN

.R=Z/m o

Hom .coker ˇ  ; R=Z/ o

0

Rm =Zm o

ker ˇN o

0.

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Using the natural isomorphism C  D S 1  RC Š R=Z  R provided by the exponential map we get the first inclusion   ker ˇN D Hom coker ˇ  ; R=Z      Hom coker ˇ  ; R=Z  Hom coker ˇ  ; R Š H.ˇ/: „ ƒ‚ … μ CR

For the second inclusion, we apply as above  ˝ R to the sequence (7) and get _ ˇR

0 ! MR ! .Rm / ! coker ˇ  ˝ R ! 0: We compose the moment map 0 W C m ! .Rm / ;

.z1 ; : : : ; zm / 7! .jz1 j2 ; : : : ; jzm j2 /=2

_ with the map ˇR and obtain the moment map _ ˇR

0

† W C m ! .Rm / ! coker ˇ  ˝ R used for the reduction. Lemma 5.2. The map † does not depend on the “stacky” information, i.e. on the chosen nj 2 N in the definition of ˇ. In particular we obtain the same map for the trivial labelling nj D 1, j D 1; : : : ; m. Proof. The map ˇ _ is defined as the canonical projection .Zm / ! .Zm / = im ˇ  .  and Let ˇ0 be the map defined by ei 7! ui . After applying  ˝ R the images im ˇR  _ _ im ˇ0;R are equal, hence ˇR and ˇ0;R are equal. The map ˇ0 together with the fan † are just the data for the usual Cox construction m of the toric variety X.†/. As the definition of C† is also independent of ˇ we obtain 1 m the second inclusion † . /  C† , since this is already known for toric varieties, see Theorem 1.4 in Appendix 1 of [Gu]. The next step requires a closer look at the group action. We set  U D 1 † . /

«

 G D ker ˇN





/ V D Cm †

« / H D H.ˇ/:

As observed above we can write H D G  CR with CR D Hom.coker ˇ  ; R/. Let coker ˇ  D Zl ˚ T be an arbitrary splitting into the free and the torsion part. Since R is torsion-free, CR Š Hom.Zl ; R/ D Rl . Hence CR and its action on V do not depend on the coefficients nj in the definition of ˇ. In other words, all the stacky information of ˇ is already contained in G.

On complex and symplectic toric stacks

329

Theorem 5.3. Let .†; ˇ/ be a stacky fan. Then the two stacks  1    m † . /= ker ˇN and C† =H.ˇ/ are isomorphic. Proof. First we show that the inclusion of stacks ŒU=G ,! ŒV =H  satisfies (M10 ): 8h 2 H and u 2 U W Œh u 2 U ) h 2 G: Since H splits into H D G  CR we only need to test whether an h 2 CR with h u 2 U is necessarily zero. But this is independent of the specific coefficients nj in ˇ, so again we deduce the result from the already known case of toric varieties. That the inclusion of stacks also satisfies (M2), 

H  U ! V is surjective, follows from V D CR  U . This equality holds since all three ingredients are independent of the stacky information. Remark. Since the maps commute with the natural torus actions we actually have an m N equivariant isomorphism between Œ1 † . /= ker ˇ and ŒC† =H.ˇ/. This equivariant N isomorphism descends to a homeomorphism of the coarse moduli spaces 1 † . /= ker ˇ m and C† =H.ˇ/. Acknowledgments. We would like to thank Lars Petersen, David Ploog and the referee for valuable comments on the manuscript. Added in proof. After submission of the manuscript we became aware of Sakai’s article [S] which also treats the correspondence between complex and symplectic toric DM stacks.

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[Au]

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[BX]

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[CLS] D. Cox, J. Little, and H. Schenk, Toric varieties. Grad. Stud. Math. 124, Amer. Math. Soc., Providence, RI, 2011. [CM]

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I. Iwanari, The category of toric stacks. Compos. Math. 145 (2009), no. 3, 718–746.

[LM]

E. Lerman and A. Malkin, Hamiltonian group actions on symplectic Deligne-Mumford stacks and toric orbifolds. Adv. Math. 229 (2012), no. 2, 984–1000.

[LT]

E. Lerman and S. Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties. Trans. Amer. Math. Soc. 349 (1997), no. 10, 4201–4230.

[MM] I. Moerdijk and J. Mrˇcun, Introduction to foliations and Lie groupoids. Cambridge Stud. Adv. Math. 91, Cambridge University Press, Cambridge 2003. [R]

M. Romagny, Group actions on stacks and applications. Michigan Math. J. 53 (2005), no. 1, 209–236.

[S]

H. Sakai, The symplectic Deligne–Mumford stack associated to a stacky polytope. Results Math., to appear, Doi 10.1007/s00025-012-0240-3.

Andreas Hochenegger, Mathematisches Institut der Freien Universität Berlin, Arnimallee 3, 14195 Berlin, Germany E-mail: [email protected]

On complex and symplectic toric stacks Frederik Witt, Mathematisches Institut der Universität Münster, Einsteinstraße 62, 48149 Münster, Germany E-mail: [email protected]

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Contents 1 Introduction and main result . . . . . . . . . . . . . . . . 2 Jet theory . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Admissible vector fields . . . . . . . . . . . . . . . . . . . 4 Admissible higher-order infinitesimal deformations . . . . 5 Proof of Theorems 1.7 and 1.19 . . . . . . . . . . . . . . . 6 Example: Embeddings into complex-symplectic manifolds 7 Example: Deformation along a foliation . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction and main result 1.1 Introduction. Let f W Y ! X be the inclusion map of a compact reduced subspace of a complex manifold. We aim to deform the morphism f , keeping the complex spaces X and Y fixed.  For this purpose let us fix a first-order infinitesimal deformation of f , say  2 H 0 Y; f  TX – we refer to the earlier paper [KKL, Section 1] for a discussion of infinitesimal deformations, and for other notions used here. We ask for conditions to guarantee that  is effective. In other words, we ask for conditions that guarantee the existence of a disk   C, centered about 0, and a family of morphisms, F W   Y ! X;

.t; y/ 7! F t .y/;

the infinitesimal deformation induced by the family, such that Fˇ0 D f and such  that d ˇ 0  F ;0 WD dt tD0 F t 2 H Y; f TX , agrees with  . In case when Y is a manifold, the most general result in this direction is due to Horikawa.   Theorem 1.1 (Horikawa’s criterion, [Hor]). If H 1 Y; f  TX vanishes, then any firstorder infinitesimal deformation of f is effective.   Vanishing of the obstruction space H 1 Y; f  TX is a sufficient, but not a necessary condition for the existence of liftings. In settings where the geometry of the target manifold it is often possible to prove existence of liftings even  X is well-understood,  if H 1 Y; f  TX is large.  Clemens Jörder and Stefan Kebekus were supported in part by the DFG-Forschergruppe 790 “Classification of Algebraic Surfaces and Compact Complex Manifolds”.

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Earlier results. One situation where effectivity of infinitesimal deformations can sometimes be shown has been studied in [KKL]. The authors of [KKL] considered a coherent subsheaf F  TX of OX -modules, closed under Lie-bracket, and an infinitesimal deformation induced by F ,    2 H 0 Y; Image f  F ! f  TX : It was  shown in [KKL, Theorem  1.5] that  lifts to a family of morphisms if the space H 1 Y; Image f  F ! f  TX vanishes. Examples of sheaves F that appear this way include (singular) foliations, logarithmic tangent sheaves, or the tensor product of TX with the ideal sheaf of a subvariety. Result of this paper. This paper is concerned with the case where the infinitesimal deformation  is induced by a sheaf F  TX which is closed under Lie-bracket, but is not necessarily a sheaf of OX -modules. Examples are given by sheaves of Hamiltonian vector fields on complex-symplectic manifolds, or more generally sheaves of vector fields whose flows stabilize a given tensor. The main result, formulated in Theorems 1.7 and 1.19, generalizes and improves on [KKL, Theorem 1.5]. In case where X is complex-symplectic, and Y  X is a Lagrangian submanifold, this reproduces results of Ran, Voisin, Kawamata, and others. Aim and scope. Written for the IMPANGA Lecture Notes series, this paper aims at simplicity and clarity of argument. It does not strive to present the shortest proofs or most general results available. While everything said here can also be deduced from the abstract machinery of deformation theory, we argue in a rather elementary and geometric manner, constructing higher-order liftings of a given infinitesimal deformation using flow maps of carefully crafted time-dependent vector fields. 1.2 Main result. In order to formulate our result we start off with the necessary notation. Notation 1.2. If X is any complex manifold, denote the sheaf of locally constant functions on X by CX  OX . Throughout the present paper, we will frequently consider subsheaves F  TX , such as the sheaf of Hamiltonian vector fields on a complex-symplectic space, which are invariant under multiplication with constants, but not necessarily under multiplication with arbitrary regular functions. We call such F a sheaf of CX -modules. The main results of this paper consider the following setup. Setup 1.3. Let X be a complex manifold, and Y  X a reduced complex subspace of X, with inclusion map f W Y ! X . Further, let F  TX be a subsheaf of CX -modules, closed under Lie bracket.

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We aim to deform the inclusion map “along the subsheaf F ”. The following two notions help to make this precise. Definition 1.4 (Infinitesimal deformations locally induced by F ). In Setup 1.3, set   FY WD Image f 1 F ! f  TX  f  TX : We call FY thesheaf of infinitesimal deformations locally induced by F . Sections  2 H 0 Y; FY are called infinitesimal deformations of f which are locally induced by F . Definition 1.5 (Obstruction sheaves). In Setup 1.3, we call a subsheaf G  FY an obstruction sheaf for F if the Lie-bracket of any two vector fields in F which agree along the image f .Y / induces a section of G . More precisely, G  FY is called an obstruction sheaf for F if for any open set U  X and any two vector fields AE1 ; AE2 2 F .U / satisfying f  AE1 D f  AE2 , the preimage of the Lie-bracket is contained in G ,   f  ŒAE1 ; AE2  2 G f 1 .U / : Remark 1.6. Obstruction sheaves are generally not unique. Example 1.10 discusses a situation where there is more than one sheaf satisfying the requirements of Definition 1.5. The main result of our paper essentially says that Setup 1.3, any infinitesimal deformation of f which is locally induced by F is effective if there exists an obstruction sheaf whose first cohomology group vanishes. Theorem 1.7 (Existence of deformations). In Setup 1.3, assume that Y is compact   and assume that there exists an obstruction sheafG  FY for F such that H 1 Y; G D 0. Then any infinitesimal deformation  2 H 0 Y; FY locally induced by F is effective. In other words, there exists an open neighbourhood  of 0 2 C, and a family of morphisms F W   Y ! X; .t; y/ 7! F t .y/; the such that Fˇ0 D f and such  that  infinitesimal deformation induced by the family, d ˇ 0  F 2 H T F ;0 WD dt Y; f X , agrees with  . tD0 t Understanding that the formulation of Theorem 1.7 may sound a little technical, we end the present subsection with two examples. In Section 1.3 we will then see that the family of morphisms whose existence is guaranteed by Theorem 1.7 can often be chosen in a way that geometrically relates to the sheaf F . Example 1.8 (Complex-symplectic manifolds). Let .X; !/ be a complex-symplectic manifold and F  TX the sheaf of Hamiltonian vector fields. Let Y  X be any compact complex submanifold, with inclusion map f W Y ! X . We assume that Y is

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Kähler and consider the sheaf TY?  f  TX of vector fields which are perpendicular to Y , with respect to the symplectic form !. In other words, TY? is the sheaf associated to the presheaf ˇ  ˚ E df .B// Å D 0 for all BÅ 2 TY .U / ; U 7! AE 2 .f  TX /.U / ˇ .f  !/.A; where  U  Y runs over the open subsets of Y . We will show in Section 6 that H 0 Y; FY D H 0 Y; f  TX , and that TY? is an obstruction sheaf  in thissetting. Theorem 1.7 thus asserts that any infinitesimal deformation  2 H 0 X; f  TX always   lifts to a holomorphic family F of morphisms if the cohomology group H 1 Y; TY? vanishes. Remark 1.9 (Deformations of Lagrangian submanifolds). In the setting of Example 1.8, if Y   X is a Lagrangian submanifold, then TY? D TY . We obtain that vanishing of H 1 Y; TY , the tangent space to the Kuranishi-family of deformations of Y , is the only obstruction to lifting a given infinitesimal deformation. This partially reproduces results of Ran, Kawamata and Voisin on the unobstructedness of deformations of Lagrangian subvarieties, cf. [Ran], [Voi], [Kaw], [Kaw2] and the references there. Example 1.10 (Deformation along a foliation). Let X be a complex manifold and F  TX a (regular) foliation, i.e., a sub-vectorbundle of TX which is closed under Lie bracket. Again, let Y  X be any compact submanifold of X . Let T  Y be the set of points where the foliation is tangent to Y , T WD fy 2 Y j F jy  TY jy g: It is clear that FY D F jY . We will show in Section 7 that any of the two sheaves G1 WD FY

G2 WD FY ˝ JT     are obstruction sheaves in this setting. Thus, if either H1 Y; FY or H 1 Y; FY ˝JT vanishes, then any infinitesimal deformation  2 H 0 Y; F jY lifts to a holomorphic family of morphisms. and

1.3 Deformations along F . Although the sheaf F appears in the assumptions of Theorem 1.7, its conclusion seems to disregard F entirely, as the family of morphisms obtained in Theorem 1.7 need not be related to F in any way. However, in all the examples we have in mind, there is a way to construct a family of morphisms F W   Y ! X that relates to F geometrically. 1.3.1 Notation concerning higher-order infinitesimal deformations. For a precise formulation of this result, we need to discuss locally closed subspaces of the Douadyspace of all holomorphic maps Y ! X which parametrize deformations along F . The following notions concerning higher-order infinitesimal deformations of f will be used in the definition.

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Definition 1.11 (Higher-order infinitesimal deformations, cf. [KKL, Definition 2.12]). Let f W Y ! X be a morphism from a complex space Y to a complex manifold X . An n-th order infinitesimal deformation of f is a morphism fn W Spec CŒt =t nC1  Y ! X whose restriction to Y Š Spec C  Y agrees with f . The universal property of the Douady-space immediately yields another, equivalent, definition of higher-order infinitesimal deformations. Notation 1.12 (Douady-space of morphisms, cf. [CP, Section 2]). Let X , Y be complex spaces and assume  thatY is compact. We denote the Douady-space of morphisms from Y to X by Hom Y; X . Fact 1.13 (Higher-order deformations as morphisms to the Douady-space). In the setting of Definition 1.11, assume additionally that Y is compact. To give an n-th order infinitesimal deformation of f , it is then equivalent to give a morphism   fn W Spec CŒt =t nC1 ! Hom Y; X   whose closed point maps to the point Œf  2 Hom Y; X representing the morphism f . Alternatively, any n-th order deformation of f can also be seen as a section in the pull-back of the n-th order jet-bundle of X . Jet bundles and their fundamental properties are reviewed in Section 2.1 below. Fact 1.14 (Higher-order deformations as sections in Jet n , [KKL, Proposition 2.13]). In the setting of Definition 1.11, to give an n-th order infinitesimal deformation of f , it is equivalent to give a section fn W Y ! f  Jet n .X /. 1.3.2 Spaces of deformations along F . Sections in Jetn .X /, which appear in the description of higher-order deformations given in Fact 1.14, can be constructed using flows of (time-dependent) vector fields on X . We will recall in Section 2.3 that if U  X is any open set, if AE is any time-dependent vector field on U , and n any number, then the flow of AE induces a section  nE W U ! Jetn .U /. We call higher-order A infinitesimal deformations of f to be induced by time-dependent vector fields in F if they locally arise in this way. Definition 1.15 (Deformations induced by time-dependent vector fields in F ). In Setup 1.3, an n-th order deformation fn W Y ! f  Jetn .X / of f is said to be locally induced by time-dependent vector fields in F , if there exist a cover of Y by open subsets of X, say Y  [˛2A U˛ , and for any a 2 A time-dependent vector fields of the form AEa D

n X iD0

t i AEa;i ;

where AEa;i 2 F .U˛ /;

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such that fn jUa \Y D  nE jUa \Y , for all a 2 A. Aa

Definition 1.16 (Spaces of deformations along F ). In Setup 1.3, assume additionally that Y is compact. A locally closed analytic subspace H  Hom.Y; X / containing Œf  is called space of deformations along F , if for any infinitesimal deformation fn of f that is locally induced by time-dependent vector fields in F , the corresponding morphism Spec CŒt =t nC1 ! Hom.Y; X / factors through H . Example 1.17 (Complex-symplectic manifolds). In the setting of Example 1.8, where .X; !/ is a complex-symplectic manifold, set H WD fg 2 Hom.Y; X / j g  ! D f  !g  Hom.Y; X /: Since (time-dependent) Hamiltonian vector fields preserve the symplectic form !, it is clear that H is a space of deformations along F . Example 1.18 (Foliated manifolds). In the setting of Example 1.10, where F  TX is a foliation, the existence of a space of deformations along F with very good properties has been shown in [KKL, Corollary 5.6]. 1.3.3 Existence of deformations along F . In cases where a space of deformations along F exists, the deformation family constructed in Theorem 1.7 can be chosen to factor via that space. This complements and strengthens Theorem 1.7 in our special situation. Theorem 1.19 (Existence of deformations along F , strengthening of Theorem 1.7). In the setting  of Theorem 1.7, assume in addition that there exists a space H  Hom Y; X of deformations along F . Then there exists a family F W   Y ! X such that F satisfiesall properties stated in Theorem 1.7, and such that the associated  map F W  ! Hom Y; X factors through H . 1.4 Outline of this paper, acknowledgements 1.4.1 Outline. Section 2 summarizes fundamental facts concerning jet bundles associated with a complex manifold that will be needed in the sequel. Subsections 2.1 and 2.2 review the notions of jet bundles and time-dependent vector fields, respectively. The sections in the jet bundles arising from flow maps associated with time-dependent vector fields are discussed in the subsequent Subsection 2.3. Section 3 is the technical core of the paper. In Setup 1.3 the choice of an obstruction sheaf leads to the notion of admissible time-dependent vector fields on X . The rather technical definition of these time-dependent vector fields is justified by the properties of the jets induced by them, as it is formulated in Subsection 3.1 and proven in the remainder of Section 3.

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The local definition and properties of admissible time-dependent vector fields and the induced jets are globalized in Section 4, yielding the notion of admissible higherorder infinitesimal deformations of the inclusion f W Y ! X . This leads us to a lifting criterion for infinitesimal higher-order deformations of f based on the cohomology vanishing assumption that appears in the statement of Theorem 1.7. Once the result in Section 4 is established, the actual proof of Theorem 1.7 and Theorem 1.19 is a short argument that is outlined in Section 5. The paper concludes with a detailed review of the two examples mentioned so far, namely the deformation on complex-symplectic manifolds and on foliated manifolds in Sections 6 and 7, respectively. 1.4.2 Acknowledgements. A first version of the main results appeared in the diploma thesis of Clemens Jörder, [Jö], supervised by Stefan Kebekus. Work on the project was initiated by discussions between Stefan Kebekus and Jaroslaw A. Wisniewski that took place during the 2009 MSRI program in algebraic geometry. Both authors would also like to thank Jun-Muk Hwang for numerous discussions on the subject. This paper was written as a contribution for the proceedings of the 2010 IMPANGA summer school. The authors thank the organizers of that event.

2 Jet theory The proof of Theorem 1.7 uses the convenient language of jet bundles. Sections 2.1 and 2.2 summarize basic facts and definitions about jet bundles and time-dependent vector fields, respectively. Section 2.3 contains an important formula concerning the jets induced by flow maps of time-dependent vector fields. A more detailed introduction to jets is found in [KKL, Section 2] and the references quoted there. 2.1 Jet bundles. Jet bundles generalize the notion of tangent bundles. If X is any complex manifold, an n-th order jet on X is an equivalence class of curve germs, where two germs are considered equivalent if they agree to n-th order. A precise definition is given as follows. Definition 2.1 (Jets on complex manifolds). Let X be any complex manifold and x 2 X a point. An n-th order jet at x 2 X is a morphism  W Spec CŒt =t nC1 ! X of complex spaces which maps the closed point of Spec CŒt =t nC1 to x. Definition 2.2 (Jet bundle). If X is a complex manifold, define the n-th order jet bundle of X as   Jet n .X / WD Hom Spec CŒt =t nC1 ; X : As a set, Jet n .X / equals the set of n-th order jets on X .

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Recall from Fact 1.14 that higher-order infinitesimal deformations fn W Spec CŒt =t nC1  Y ! X of f can be considered as X -morphisms fn W Y ! Jetn .X /. A detailed description of the structure of jet bundles is therefore important. Fact 2.3 (Affine bundle structure, cf. [KKL, Fact 2.8.3]). Let X be a manifold. Then the following facts hold true for any natural number n  0: 1. The complex space Jetn .X / is a manifold. The obvious forgetful morphism n;m W Jet n .X / ! Jetm .X / is holomorphic for all n  m  0. There are natural isomorphisms Jet0 .X / Š X and Jet1 .X / Š TX . 2. The morphism nC1;n W JetnC1 .X / ! Jetn .X / has the structure of an affine bundle. The associated vector bundle of translations on Jet n .X / is the pullback of the tangent bundle. Notation 2.4. In the setting of Fact 2.3, if ;  2 JetnC1 .X / are two jets whose n-th order parts agree, nC1;n . / D nC1;n . /, the affine bundle structure mentioned in Item 2 allows to express the difference between  and  as a tangent vector vE 2 TX jx . In this context, we write vE D   . The following descriptions of jets is immediate from the universal property of the Hom space. Fact 2.5 (Jet bundles in deformation theory). Let f W Y ! X be a holomorphic map between a compact complex space Y and a complex manifold X . To give an n-th order jet at Œf  2 Hom.Y; X /, it is equivalent to give a holomorphic section  n W Y ! f  Jet n .X / WD Jetn .X / X Y . 2.2 Time-dependent vector fields. We follow the standard approach familiar from the theory of ordinary differential equations and define a time-dependent vector field on a complex manifold X as a vector field on the product of X and a “time axis”. 2.2.1 Notation. The following notation concerning time-dependent vector fields on X and on the Cartesian product X  C will be used throughout this paper. Notation 2.6 (Cartesian product). Let X be a complex manifold. Consider the product X  C. Let t be the standard coordinate on C, with associated vector field @t@ 2  H 0 X  C; TXC . The projection from X  C to the first factor is denoted by pX W X  C ! X . Finally, let j t W X ! X  C; x 7! .x; t / be the inclusion map. Notation 2.7 (Time-dependent vector fields). Let X be a complex manifold. A timedependent vector field AE on X is a vector field AE on the product X  C contained in the subspace     H 0 X  C; pX .TX /  H 0 X  C; TXC :

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For any time t 2 C, the restriction of AE to time t is denoted by AEt WD j t . / 2 H 0 .X; TX / Definition 2.8 (Vector field with constant flow in time). If AE is any time-dependent vector field on a complex manifold X , set   E WD @ C AE 2 H 0 X  C; TXC : D.A/ @t E the vector field with constant flow in time associated with A. E We call D.A/ 2.2.2 Calculus involving time-dependent vector fields. For later reference, we state without proof a formula involving time-dependent vector fields and their Lie brackets. The formula is easily checked by a direct computation in local coordinates. E BÅ 2 p  .TX / be any two Lemma 2.9 (Calculus of time-dependent vector fields). Let A; X time-dependent vector fields on X , and let n 2 N be any number. Using Notation 2.6, the following equations hold. E t n  B Å D Œt n  A; E B Å D t n  ŒA; E B Å ŒA;   tn n n1 t t Å E Å Å LD.A/ E nŠ  B D nŠ  ŒA; B C .n1/Š  B C

(2.9.1) tn

nŠ



@ Å B @t

(2.9.2)

2.3 Jets induced by time-dependent vector fields. If X is a complex manifold,   x 2 X a point and AE 2 H 0 X  C; pX .TX / a time-dependent vector field on X , then the local flow of AE through x induces a curve germ x at x, uniquely determined by the properties x .0/ D x and x0 .t / D AEt .x .t // for all t . We denote the associated jets as follows.  Definition 2.10 (Jets induced by time-dependent vector fields). Let AE 2 H 0 X   C; pX .TX / be a time-dependent vector field on a complex manifold X . We denote by  nE W X ! Jetn .X / A

the holomorphic section which assigns to each point x 2 X the n-th order jet at x 2 X E associated with the A-integral curve through x. The following is the key observation of this paper and of the previous paper [KKL]. E BÅ on X In its simplest form, Fact 2.11 considers two time-dependent vector fields A, whose associated n-th order jets  nE and  nÅ agree at a point x 2 X . It gives a formula A

B

for the difference between .n C 1/-st order jets  nC1 .x/ and  nC1 .x/ which, using E Å A

B

the affine bundle structure of JetnC1 .X / ! Jetn .X /, can be identified with a tangent

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.x/   nC1 .x/ 2 TX jx . This formula allows for explicit computations of vector  nC1 Å B AE obstruction cocycles relevant when trying to lift infinitesimal deformations from n-th to .n C 1/-st order. We refer to [KKL, Section 3] for a more detailed discussion. Fact 2.11 (Difference formula, cf. [Jö, Corollary 2.5.3], [KKL, Theorem 4.3]). Let   AE1 ; : : : ; AEn ; BÅ 2 H 0 X  C; pX .TX / be time-dependent vector fields on a complex manifold X. Let x 2 X be any point such that the induced n-th order jets agree at x,  nE .x/ D  nE .x/ D    D  nÅ .x/: A1

A2

B

Using Fact 2.3 and Notation 2.4 to identify the difference between the .n C 1/-st order .x/ and  nC1 .x/ with a tangent vector in TX jx , the difference is given as jets  nC1 E Å An

B

  Å .x/ 2 TX jx ;  nC1 .x/   nC1 .x/ D LD.AE / B    B LD.AEn / D.B/ 0 Å E B

An

1

where LD.AE / denotes Lie-derivative with respect to the vector field D.AEi / with coni stant flow in time.

3 Admissible vector fields We consider the following setup throughout the present section. Setup 3.1. Let X be a complex manifold, equipped with a Lie-closed subsheaf F  TX of CX -modules. Let Y  X be a reduced complex subspace of X with inclusion map f W Y ! X, and let further G  FY be an obstruction sheaf for F , in the sense of Definition 1.5. The aim of this section is to define and discuss admissible vector fields. These are time-dependent vector fields whose induced jets are particularly well-behaved when used to deform the inclusion map f W Y ! X . Since the deformation of f will be defined locally on Y , we do not assume compactness of Y in this section. To begin, we fix the notion of a time-dependent vector field in F , see Definition 1.15. Notation 3.2 (Time-dependent vector fields in F ). A time-dependent vector field AE 2 H 0 .X  C; pX .TX // is said to be a time-dependent vector field in F , if it can be expressed as a finite sum n X AE D t i AEi iD1

where AEi 2 H 0 .X; F / is a time-independent vector field in F , and t is the standard coordinate on C.

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  Definition 3.3 (Admissible vector field). In Setup 3.1, let AE 2 H 0 X  C; pX .TX / be a time-dependent vector field in F , and let n  1 be any natural number. The vector field AE is said to be n-admissible for the obstruction sheaf G , if for any natural number 1  m  n, the restriction to Y of Lm E AE at time t D 0 is a section of the obstruction D.A/ sheaf. In other words, if  m  ˇ   L E AE 0 ˇY 2 H 0 Y; G : D.A/

 Remark 3.4 (Admissible fields and derivative in time direction). If AE 2 H 0 X  C; pX .TX / is any time-dependent vector field on X , then @ E E E LD.A/ E A D LAC E E @ A D L @ A D  LAE @t D  LD.A/ @t

@t

More generally, we have Lm E AE D  Lm E D.A/  ˇ  D.A/  m L E @t@ 0 ˇY 2 H 0 Y; G for all 1  m  n.

@ . @t

@ : @t

If AE is n-admissible for G , then

D.A/

Remark 3.5 (Time-independent vector fields in F are admissible). A time-independent section of F is n-admissible for arbitrary n and arbitrary obstruction sheaf, when considered as time-dependent vector field. In other words, any field     AE 2 H 0 X; F  H 0 X  C; pX1 .F / is n-admissible for any G . 3.1 Jets induced by admissible fields. The geometric meaning of Definition 3.3 is perhaps not obvious. However, the usefulness of the concept will immediately become clear once we look at jets induced by admissible vector fields. The following two propositions, which form the technical core of this paper, summarize the main features. Proofs are given in Subsections 3.2–3.5 below. Proposition 3.6 (Extension of jets from n-th to .n C 1/-st order). In Setup 3.1, let   AE 2 H 0 X  C; pX .TX / be an n-admissible vector field for G with n  1. If    Å 2 H 0 X; F  H 0 X; TX / is any time-independent vector field in F whose    Å Y 2 H 0 Y; G , then there exists a restriction to Y lies in the obstruction sheaf, j   time-dependent vector field BÅ 2 H 0 X  C; pX .TX / such that the following holds true. 1. The time-dependent vector field BÅ is .n C 1/-admissible for G . 2. The n-th order jets induced by AE and BÅ agree on Y , ˇ ˇ  nEˇY D  nÅ ˇY : A

B

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Å 3. The difference between the induced .n C 1/-st order jets is given by , ˇ ˇ ˇ Åˇ :  nÅ ˇY   nEˇY D  Y B

A

Proposition 3.7 (Differences of jets induced by admissible vector fields). In Setup 3.1,   E BÅ 2 H 0 X  C; p  .TX / be two time-dependent let n  1 be any number and A; X vector fields, both of them n-admissible for the obstruction sheaf G . If the induced n-th order jets agree on Y , ˇ ˇ  nEˇY D  nÅ ˇY ; B

A

then the difference between the .nC1/-st order deformations of f lies in the obstruction sheaf G . In other words, ˇ ˇ   ˇ   nC1 ˇ 2 H 0 Y; G :  nC1 Y Y Å E B

A

The proofs of Propositions 3.6 and 3.7 are quite elementary, but somewhat lengthy and tedious. The reader interested in gaining an overview of the argumentation is advised to skip Sections 3.2–3.5 on first reading and continue with Section 4 on page 349, where Propositions 3.6 and 3.7 are used to lift first-order infinitesimal deformations of f to arbitrary order. 3.2 Preparation for the proof of Proposition 3.6. The proof of Proposition 3.6 relies on the following computational lemma. Å be any time-independent vector field Lemma 3.8. In the setup of Proposition 3.6, let E in F and consider the time-dependent vector field in F , t nC1 Å tn Å C E: BÅ WD AE C  nŠ .n C 1/Š

(3.8.1)

Then the following equalities hold up to higher-order terms of t , for all 1  m  n, Lm

Å D.B/

Å Lm .B/

E D.A/

E C .A/

t nm Å  .nm/Š

E CE Å C nŒA; E  Å LnC1Å BÅ LnC1E .A/ D.B/

D.A/

mod .t nmC1 /;

(3.8.2)

mod .t /:

(3.8.3)

Lemma 3.8 easily follows from a direct but rather tedious computation. Details are found in the preprint version of this paper, available on the arXiv. 3.3 Proof of Proposition 3.6. Consider the time-dependent vector field tn Å t nC1 Å BÅ WD AE C  C E; nŠ .n C 1/Š

  Å WD nŒA; E  Å  LnC1 AE : where E 0 E D.A/

(3.8.4)

Å Y is a section of the obstruction sheaf G , Equation (3.8.2) immediately Since j Å Equation (3.8.3) shows that BÅ is in implies that BÅ is n-admissible. By choice of E, fact .n C 1/-admissible. This already shows Property (1) claimed in Proposition 3.6.

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Proof of Proposition 3.6 (2). To show Property (2), we will prove by induction that ˇ ˇ ˇ D  m ˇ ; for all 1  m  n: m (3.8.5) E Y Å Y A

B

We start the induction with the case where m D 1. In this case Equation (3.8.5) E and D.B/ Å agree along Y at time t D 0. That, simply asserts that the vector fields D.A/ Å however, is clear by choice of B. For the inductive step, assume that Equation (3.8.5) was shown for a certain number 1  m < n. It will then follow from Fact 2.11 that the difference between the .mC1/-st order jets is given as ˇ ˇ   ˇ ˇ   mC1 ˇ D Lm D.B/ Å ˇˇ :  mC1 (3.8.6) E Y Y 0 Å E B

But since Lm

E D.A/

D.A/

A



tn Å  nŠ

Y

nC1

t Å C .nC1/Š E E D.A/  n  Å C t nC1 E Å D Lm E tnŠ  .nC1/Š

Å D Lm D.B/

E C D.A/

 by (3.8.4)

D.A/



t nm Å  .nm/Š

mod .t nmC1 /

by (2.9.2);

it is clear that the difference (3.8.6) vanishes as required. This finishes the proof of Property (2) claimed in Proposition 3.6. Proof of Proposition 3.6 (3). Fact 2.11 and Property (2) together imply that the difference between the .n C 1/-st order jets is given by ˇ ˇ   ˇ ˇ   nC1 ˇ D Ln Å ˇˇ : D. B/  nC1 E Y Y 0 Å E B

A

D.A/

Y

As in the proof of Property (2) above we obtain that Ln

E D.A/

Å  Å mod .t /; D.B/

finishing the proof of Proposition 3.6. 3.4 Preparation for the proof of Proposition 3.7. The proof of Proposition 3.7 makes use of two computational lemmas concerning Lie derivatives that are formulated and proved in the current Section 3.4. The actual proof of Proposition 3.7 is given in Section 3.5. To start, recall the classical Jacobi identity, formulated in terms of Lie derivatives. Å 2 H 0 .X; TM / be vector fields on a comRemark 3.9 (Jacobi identity). Let XÅ; YÅ ; Z plex manifold M . Written in terms of Lie derivatives rather than Lie brackets, the Jacobi identity asserts that Å Å Å LŒX; Å B LY Å Z  LY Å B LX Å Z: ÅY Å  Z D LX

(3.9.1)

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Lemma 3.10. In the setup of Proposition 3.7, define Å E D.B/ Å D L E D.B/ Å2 WD ŒD.A/; R D.A/ and inductively Åm WD ŒD.A/; E R Åm1  D Lm1 D.B/ Å for m > 2. R E D.A/

 0



Å 2 H X  C; TXC is any vector field on the product X  C and m  2 any If Z Å can be expressed as a linear combination of terms TÅ of the form number, then LRÅm Z   Å ; TÅ D LFÅ B    B LFÅm Z 1

E or equal to D.B/. Å where all FÅi are either equal to D.A/ Proof. We prove Lemma 3.10 by induction on m. If m D 2, then the Jacobi identity (3.9.1) asserts that ÅDL E Å Å Å LRÅ2 Z Å Z D LD.A/ E B LD.B/ Å Z  LD.B/ Å B LD.A/ E Z; ŒD.A/;D.B/ showing the claim in case m D 2. For m > 2, the Jacobi identity gives Å D L E Åm1 Z Å D L E B L Åm1 Z Å  L Åm1 B L E Z; Å LRÅm Z ŒD.A/;R  D.A/ D.A/ R R showing the claim inductively. Lemma 3.11. In the setup of Proposition 3.7, if FÅ1 ; : : : ; FÅn is a sequence of timeE or equal to D.B/, Å dependent vector fields such that all FÅi are either equal to D.A/ then    ˇˇ   LFÅ B    B LFÅn @t@ (3.11.1) ˇ 2 H 0 Y; G : 1

0 Y

E or all FÅi are equal to D.B/, Å then the statement follows Proof. If all FÅi are equal to D.A/ Å from Remark 3.4. We can thus assume without loss of generality that FÅn D D.B/, E and that at least one of the FÅi , for i < n, is equal to D.A/. As a first step in the proof of Lemma 3.11, we show the following claim. Claim 3.11.2. Let 1  i  n  2 be any number, and consider the vector field1   TÅ WD LFÅ B    B LFÅ B LŒFÅ ;FÅ  B LFÅ B    B LFÅn @t@ 1 i 1 i i C1 i C2   D LFÅ B    B LFÅ B LFÅ B LFÅ B LFÅ B    B LFÅn @t@ (3.11.3) 1 i 1 i i C1 i C2 @  LFÅ B    B LFÅ B LFÅ B LFÅ B LFÅ B    B LFÅn @t : 1

i 1

i C1

i

i C2

ˇ   Then TÅ0 ˇY 2 H 0 Y; G . 1

The equality in (3.11.3) is again Jacobi’s identity in the form of Remark 3.9.

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Using a somewhat tedious inductive argument, which we leave to the reader, one verifies that the vector field TÅ can be expressed as follows: X TÅ WD ŒPÅ˛ ; TÅ ˛  ˛ a subsequence of .1; 2; : : : ; i  1/

  where the subsequence ˛ is written as ˛ D ˛.1/; : : : ; ˛.k/ and the complementary   subsequence is denoted by ˛ xD ˛ x.1/; : : : ; ˛ x.i  1  k/ , and where   B LFÅ FÅiC1 PÅ˛ WD LFÅ B    B LFÅ ˛.1/

˛.k/

i

and TÅ ˛ WD LFÅ

˛ x .1/

B    B LFÅ

˛ x .l/

B LFÅ

i C2

B    B LFÅn

@ @t

:

There are two things to note in this setting. 1. Since F is closed under Lie-bracket, totime t D 0 of the vector  the  restriction  field TÅ ˛ is a section of F , that is, TÅ ˛ 2 H 0 X; F , for all subsequences ˛.  0



0

2. The vector field .PÅ˛ /0 2 H X; F is the restriction to time t D 0 of an iterated Lie bracket of at most .n  1/ time-dependent vector fields in F , all inducing the same .n  1/-st order jets on Y  X . Fact 2.11 therefore implies that .PÅ˛ /0 vanishes on Y , again for all subsequences ˛. Consequently, the restriction of TÅ0 to Y satisfies X X   ŒPÅ0˛ ; TÅ0˛  D ŒPÅ0˛ C TÅ0˛ ; TÅ0˛  2 H 0 Y; G ; TÅ0 jY D ˛

˛

by Definition 1.5. This ends the proof of Claim 3.11.2. Claim 3.11.2 asserts that Equation (3.11.1) holds if and only if it holds after permuting the operators LFÅ and LFÅ . In other words, Equation (3.11.1) holds if and i i C1 only if    ˇˇ   LFÅ B    B LFÅ B LFÅ B    B LFÅn @t@ ˇ 2 H 0 Y; G : 1

i C1

i

0 Y

Using the classical “bubble-sort” algorithm, we can therefore sort the FÅi and assume without loss of generality that there exists a number k such that FÅ1 ; FÅ2 ; : : : ; FÅk are E whereas FÅkC1 ; : : : ; FÅn are all equal to D.B/. Å In this situation an all equal to D.A/, argument similar to, but easier than the proof of Claim 3.11.2 then shows the following. Claim 3.11.4. If SÅ denotes the following vector field,   @ SÅ WD LFÅ B    B LFÅn @t@  LD.B/ Å B LFÅ B    B LFÅn @t ;  0

1



then SÅ0 jY 2 H Y; G .

2

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C. Jörder and S. Kebekus

The proof of Claim 3.11.4 is left to the reader. Claim 3.11.4 allows to replace E by D.B/. Å Applying the ‘Sorting Claim 3.11.2’ and the ‘Replacement FÅ1 D D.A/ Claim 3.11.4’ exactly k times, this reduces the Lemma 3.11 to the case where FÅ1 D Å where the lemma is known to be true, thus finishing the proof of    D FÅn D D.B/, Lemma 3.11. 3.5 Proof of Proposition 3.7. By Fact 2.11, the time-independent vector field  Å WD Ln 

E D.A/



Å D.B/

0

  2 H 0 X; TX

and  nC1 at time t D 0 describes the difference between the .n C 1/-st order jets  nC1 Å AE B Å Y is a section of the obstruction along Y  X. We need to show that the restriction j sheaf G . For simplicity of argument, we discuss the cases n D 1 and n > 1 separately in the next two subsections. Å and of D.A/ E and D.B/ Å to Proof in case n D 1. If n D 1, expand the definition of  obtain ˇ   ˇ     ˇ  ˇ  ˇ Å ˇ D L E BÅ ˇ C L @ BÅ ˇ C L E @ ˇ : Å ˇ D L E D.B/  Y A A @t 0 Y D.A/ 0 Y 0 Y 0 Y @t „ ƒ‚ … „ ƒ‚ … 2G by Rem. 3.4

2G by Rem. 3.4

To conclude, it thus suffices to show that  ˇ  ˇ    LAE BÅ 0 ˇY D LAE BÅ0 ˇY 2 H 0 Y; G : 0

This, however, follows from the assumption that AE and BÅ be admissible vector fields,   so that AE0 ; BÅ0 2 H 0 X; F , and from Definition 1.5 of obstruction sheaf. This proves Proposition 3.7 in case n D 1. Proof in case n > 1. Consider the time-dependent vector field Å WD Ln1 D.B/: Å R E D.A/

Å0 jY D 0. As one Fact 2.11 and the assumed equality of n-th order jets imply that R consequence, we obtain that  ˇ  LRÅ AE0 ˇY D LRÅ 0

E

0 CA0

ˇ   AE0 ˇY 2 H 0 Y; G ;

according to Definition 1.5, where obstruction sheaves were introduced. The equation ˇ  ˇ    ˇ   ˇ  ˇ Å ˇ D  L Å D.A/ E ˇ D  L Å AE0 ˇ  L Å @ ˇ Åˇ D L E R  Y Y R R0 R @t 0 Y D.A/ 0 Y 0 Y „ ƒ‚ … 2G

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Deformation along subsheaves, II

thus asserts that to prove Proposition 3.7, it suffices to show that   ˇ   LRÅ @t@ 0 ˇY 2 H 0 Y; G :

(3.11.5)

To this end, recall from Lemma 3.10 that LRÅ @t@ can be expressed as a linear combination of terms TÅ of the form   TÅ D LFÅ B    B LFÅn @t@ 1

E D.B/g Å for 1  i  n. Lemma 3.11 then asserts that every one of with FÅi 2 fD.A/;   these terms is contained in H 0 Y; G when restricted at time t D 0 to Y , thus finishing the proof of Proposition 3.7 in case n > 1.

4 Admissible higher-order infinitesimal deformations In this section, we generalize the notion of admissibility to jets of arbitrary order. We employ Proposition 3.6 and 3.7 to show that – under a suitable cohomology vanishing assumption – first-order infinitesimal deformations that are locally induced by admissible vector fields can always be lifted to admissible infinitesimal deformations of arbitrary order. We maintain the assumptions spelled out in Setup 3.1 on page 342. Definition 4.1 (Admissible higher-order infinitesimal deformations). In Setup 3.1, let n  1. A section  n W Y ! Jetn .X / over f is said to be admissible for the S obstruction sheaf G , if there exists a cover of Y  X by open subsets, say Y  j 2J Xjı , and time-dependent vector fields AEj on Xjı for any j 2 J such that the following two conditions hold true for every j 2 J . 1. The vector field AEj is n-admissible for the obstruction sheaf G jYiı , where Yjı WD Y \ Xjı . ˇ ˇ 2. The restriction of  n to Yjı is induced by AEj . In other words,  n ˇY ı D  nE ˇY ı . j

Aj

j

Remark 4.2. Recall from Remark 3.5 that time-independent vector fields in  F are always admissible. In Setup 3.1, any infinitesimal deformation  2 H 0 Y; FY locally induced by F is therefore admissible for the obstruction sheaf G in the sense of Definition 4.1 above. Proposition 4.3   (Lifting admissible sections to arbitrary order). In Setup 3.1, suppose that H 1 Y; G D 0. Then any section  n W Y ! Jetn .X / over f that is admissible for G admits a lift to a section  nC1 W Y ! JetnC1 .X / that is likewise admissible for G . S Proof. Fix an open cover Y  j 2J Xjı and n-admissible vector fields AEj as in Definition 4.1. We write Yjı WD Xjı \ Y , Xjık WD Xjı \ Xkı and Yjık WD Yjı \ Ykı for j; k 2 J .

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ˇ ˇ Since  nE ˇY ı D  nE ˇY ı for j; k 2 J by Item (2) in Assumptions 4.1, we may Aj

jk

Ak

jk

calculate the difference between .n C 1/-st order jets. By Assumption (1) and by Proposition 3.7, this difference is described by a section in the obstruction sheaf G , that is, ˇ ˇ ˇ ı   nC1 ˇ ı 2 G .Y ı /: (4.3.1) CÅj k WD  nC1 jk Y Y E E Aj

jk

Ak

jk

Å ˇ The general properties of affine bundles imply that the family S .Cj k /j k is a Cech 1cocycle of the sheaf G with respect to the open cover Y D j Yjı . The cohomology ˇ vanishing assumption ensures that .CÅj k /j k is a Cech 1-coboundary, that is, that there ı exist sections CÅj 2 G .Y / such that j

ˇ ˇ CÅj k D CÅk ˇY ı  CÅj ˇY ı 2 G .Yjık /:

(4.3.2)

Taking (4.3.1) and (4.3.2) together, the sections ˇ ˇ ı C CÅj W Y ı ! JetnC1 .X /;  nC1 i Y E

(4.3.3)

jk

Aj

jk

j

with j running over the index set J , glue, giving a section  nC1 W Y ! Jet nC1 .X / over  n . Refining the open cover, Equation (4.3.3) and Proposition 3.6 allow to assume that Å Cj is induced by a vector field on Xjı which is .nC1/-admissible for G jYjı , as required.

5 Proof of Theorems 1.7 and 1.19 We maintain assumptions and notation of Theorems 1.7 and 1.19, where X is a complex manifold, F  TX a Lie-closed subsheaf of CX -modules, where Y  X is a reduced, compact complex subspace  with inclusion map f W Y ! X , and G an obstruction sheaf for F . Let  2 H 0 Y; FY be deformation of f that is locally  an infinitesimal  induced by F , and assume that H 1 Y; G D 0. 5.1 Proof of the Theorem 1.7. By Proposition 4.3,  1 W Y ! Jet1 .X /  the section  0 corresponding to the first order deformation  2 H Y; FY of f inductively admits lifts to sections  n W Y ! Jetn .X / for any natural number n. The family . n /n2N corresponds to a formal curve Spec CŒŒt  ! Hom.Y; X /

(5.0.4)

at Œf  2 Hom.Y; X / whose associated Zariski tangent vector equals  . The existence of a holomorphic curve at Œf  2 Hom.Y; X / with derivative  then follows from a classical result of Michael Artin, [Art, Theorem 1.2].

Deformation along subsheaves, II

351

5.2 Proof of Theorem 1.19. If H  Hom.Y; X / is a subspace of deformations along F , then the formal curve (5.0.4) factors through H , Spec CŒŒt  ! H  Hom.Y; X /: In particular, the holomorphic curve at Œf  2 Hom.Y; X / given by [Art] can be required to lie in H .

6 Example: Embeddings into complex-symplectic manifolds In this section, we apply Theorem 1.7 to embedding morphisms into complex-symplectic manifolds. The following assumptions will be maintained throughout the present section. Setup 6.1. Let .X; !/ be a complex-symplectic manifold, let F  TX be the subsheaf of Hamiltonian vector fields, and f W Y ! X the inclusion of a compact submanifold. 6.1 Infinitesimal deformations locally induced by Hamiltonian vector fields. We start off with a discussion of the sheaf of infinitesimal deformations locally induced by F . Perhaps somewhat surprisingly, it will turn out that all infinitesimal deformations of f are locally induced by F , as long as Y is either a curve, surface, or a Kähler manifold.   Notation 6.2. If U  Y is any open set, and AE 2 f  TX .U /, consider the associated   E / 2 f  1 and the associated form E WD .df /. E/ 2 1 . section AE WD f  ! .A; X Y A A Proposition 6.3 (Infinitesimal deformations locally induced by Hamiltonians). In   Setup 6.1, let AE 2 H 0 Y; f  TX be an infinitesimal deformation of f . If AE is closed, then AE is locally induced by F . Proof. The question being local on Y , we can assume that there are coordinates x1 ; : : : ; xn ; y1 ; : : : ; ym on X such that the submanifold Y is given as Y D fx1 D    D xn D 0g. If AE is closed, it can locally be written as AE D dg, where g 2 OY is a suitable holomorphic function. The section AE is then written as AE D dg C

n X

gi .y1 ; : : : ; ym /  dxi

iD1

where again gi 2 OY are suitable holomorphic functions. Consider the function G WD g C

n X

gi .y1 ; : : : ; ym /xi ;

iD1

defined on a neighborhood of Y in X. To finish the proof, observe that the image of E its associated Hamiltonian vector field Hamilton.G/ in f  TX agrees with A.

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Corollary 6.4 (All infinitesimal deformations are locally induced by Hamiltonians). If Y is Kähler or if dim Y < 3, then any infinitesimal deformation of f is locally induced by F . In other words,     H 0 Y; Image FY ! f  TX D H 0 Y; f  TX : Proof. The proof follows immediately from Proposition 6.3 and from the fact that any holomorphic 1-form on a compact curve, surface or Kähler manifold is automatically closed, cf. [BHPV, Chapter IV.2] and [Voi2, Theorem 8.28]. 6.2 Obstruction sheaf in the symplectic setup. Next, we show that in the symplectic setting, there exists an obstruction sheaf that can be understood geometrically. We recall the notation of vector fields that are perpendicular to Y , already introduced in Example 1.8. Notation 6.5. Let TY?  f  TX be the subsheaf of sections which are perpendicular to Y , i.e., the sheaf associated to the presheaf ˇ E B/ Å D 0 for all BÅ 2 TY .U /g U 7! fAE 2 f  .TX /.U / ˇ .f  !/.A; where U  Y runs over the open subsets. The sheaf TY? is a subbundle of f  TX . Its rank equals codimX Y . We aim to show that TY? is an obstruction sheaf, in the sense of Definition 1.5. To start, we need to show that TY?  FY . In order to prove this, apply the bundle 1 isomorphism TX Š X induced by ! to both sides of the inclusion. To prove TY?  1 FY we thus need to show that sections of the kernel of df W f  X ! 1Y can locally be extended to closed forms, defined on a neighborhood of Y . In complete analogy to the proof of Proposition 6.3, this follows from a short calculation in local coordinates which we leave to the reader. The following proposition then shows the remaining property required for TY? to be an obstruction sheaf. Proposition 6.6 (Lie brackets of Hamiltonian vector fields). Let U  X be any open Å 2 F .U / be two Hamiltonian vector fields on subset, Y ı WD Y \ U , and let FÅ; G ı Å is perpendicular to Y . In other U that agree along Y . Then the Lie bracket ŒFÅ; G ?  Å Å ı words, f ŒF ; G 2 TY .Y /. Proof. The statement of Proposition 6.6 being local on Y , it suffices to show that   Å jY ı D 0 (6.6.1) ! VÅ ; ŒFÅ; G   for any vector field VÅ 2 H 0 U; TX which is tangent to Y ı , i.e., which satisfies VÅ jy 2 TY jy  TX jy

for all y 2 Y ı :

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Deformation along subsheaves, II

Å Y ı D 0 implies that Given any such VÅ , the equality .FÅ  G/j LFÅGÅ .VÅ /jY ı D 0:

(6.6.2)

Equality (6.6.1) then follows with   Å jY ı 0 D LFÅGÅ !.VÅ ; G/

Å Y ı C !.VÅ ; L Å Å G/j Å Yı D !.LFÅGÅ VÅ ; G/j F G Å Yı D !.VÅ ; ŒFÅ; G/j

Å Yı D 0 since .FÅ  G/j Å preserve ! since FÅ and G by (6.6.2)

This finishes the proof of Proposition 6.6. Corollary 6.7 (Obstruction sheaf in the symplectic setup). The sheaf TY?  FY is an obstruction sheaf for the sheaf F  TX of Hamiltonian vector fields. 6.3 A space of deformations along F . As a last step in the discussion of symplectic spaces, we aim to identify a space of deformations along F . Since Hamiltonian vector fields preserve the symplectic form by definition, the following space is a candidate. Definition 6.8 (Space of morphisms with prescribed pull-back of !). Consider subspace Hom! .Y; X / WD fg 2 Hom.Y; X / j g  .!/ D f  .!/g  Hom.Y; X / with its natural structure as a (not necessarily reduced) complex space. The following is now an elementary consequence of the fact that Hamiltonian vector fields preserve !. Fact 6.9 (Hom! .Y; X / is a space of deformations along F , cf. [Jö, Proposition 3.1.10]). If n is any number and fn any n-th order infinitesimal deformation of f that is locally induced by time-dependent Hamiltonian vector fields, then the corresponding morphism fn W Spec CŒt =t nC1 ! Hom.Y; X / factors via Hom! .Y; X /. In particular, Hom! .Y; X /  Hom.Y; X / is a space of deformations along F . 6.4 Summary of results in the symplectic case. Putting Corollaries 6.4, 6.7, Fact 6.9 and the main results of this paper, Theorems 1.7 and 1.19 together, the following corollary summarizes our results in the symplectic setting. Corollary 6.10. Let f W Y ! X be the embedding of a compact curve, surface of Kähler submanifold into a complex-symplectic manifold .X; !/. Moreover assume

354

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    that H 1 Y; TY? D 0. Then any infinitesimal deformation  2 H 0 Y; f  TX is effective. More is true: There exists a family F W   Y ! X of morphisms such that the infinitesimal deformation induced by F agrees with  , i.e., F ;0 D  , and such that F t .!/ D f  .!/ for any t 2 . 6.5 Generalisations. Many of the results contained in this section carry over  to1the case where X is not necessarily symplectic, but carries a two-tensor ! 2 H 0 X; X ˝ 1 X , which need not be alternating, symmetric, or non-degenerate. Details are found in [Jö].

7 Example: Deformation along a foliation This subsection is concerned with the case when X carries a regular foliation, that is, a subbundle F  TX closed under Lie bracket. We fix the following setup for the present section. Setup 7.1. Let F  TX be a regular foliation on a complex manifold X , and let f W Y ! X be the inclusion of a compact submanifold Y  X . 7.1 Infinitesimal deformations locally induced by the foliation. Since F is a subbundle of TX , it is clear that in the context of Setup 7.1, the sheaf of infinitesimal deformations locally induced by F is FY D f  F . The space of infinitesimal deformations locally induced by F is then H 0 Y; f  F . 7.2 Obstruction sheaf in the foliated setup. Of course, the restricted subbundle FY  f  .TX / is an obstruction sheaf for the foliation F . If the leaves intersect the submanifold f .Y / transversely, then FY is in fact the only possible obstruction sheaf. However, if the set (7.1.1) T WD fy 2 Y j F jy  TY jy g of points where the foliation is tangent to Y is non-empty, then the following proposition asserts that the proper subsheaf G WD JT ˝ FY  FY is an obstruction sheaf as well. Proposition 7.2. In Setup 7.1, let U  X be any open subset, set Y ı WD Y \ U and E BÅ 2 F .U / are two vector fields in F that agree along Y ı , then T ı WD T \ U . If A; the Lie bracket vanishes along T ı ,   E B Å jT ı D 0 2 FY .T ı /: f  ŒA;

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Deformation along subsheaves, II

Proof. We consider vector fields as derivations acting on the structure sheaf OX . From   E B:g Å this point of view, we need to show ŒA; .x/ D 0 for any point x 2 T ı and any germ of function g 2 OX;x . We know that     E ; E B:g Å D ŒA; E BÅ  A:g E D A: E .BÅ  A/:g E E A:g ŒA;  .BÅ  A/: „ ƒ‚ … „ ƒ‚ … DWa

DWb

E Y ı D Bj Å Y ı . Since AE 2 F is tangent to The terms a and b vanish along Y ı because Aj Y at all points x 2 T ı , the assertion follows. Corollary 7.3 (Obstruction sheaf in the case of a foliation). In Setup 7.1, any sheaf G satisfying JT ˝ F jY  G  F jY is an obstruction sheaf for the foliation F . 7.3 A space of deformations along F . A space of deformations along F has been constructed in [KKL]. The following notation is useful in the description of its main property. Notation 7.4 (Velocity vector field for families of morphisms, cf. [KKL, Section 1.B]). Let F W   Y ! X be a family of morphisms such that F0 D f . Given a point y 2 Y , we can consider the curve Fy W  ! X;

t 7! F .t; y/:

Given t0 2  and taking derivatives in t for all y at time t D t0 , this gives a section   F ;t0 2 H 0 Y; .F t0 / TX ; called velocity vector field at time t0 . Fact 7.5 (Space of deformations along a foliation, cf. proof of [KKL, Corollary 5.6]). In Setup 7.1, there exists a space of deformations along F , denoted HomF .Y; X /  Hom.Y; X /; with the following additional property. If F W  ! HomF .Y; X / is any holomorphic curve germ, with associated family of morphisms F W   Y ! X , then the velocity vector fields are in the pull-back of the foliation F , for all times t 2 . In other words, F ;t 2 H 0 Y; F t F for all t 2 . The actual statement of [KKL, Corollary 5.6] only implies that infinitesimal deformations induced by time-independent vector fields in F factor through HomF .Y; X /. However, the proof in loc. cit. can be generalized with minimal changes to the case of time-dependent vector fields, as required in Definition 1.16.

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7.4 Summary of results in the case of a foliated manifold. Using Corollary 7.3 and Fact 7.5, the following corollary summarizes our results in the case of a foliated manifolds. Corollary 7.6. In Setup 7.1, let G be any subsheaf satisfying JT ˝ FY  G  F jY ;

  where T  Y is the space defined in (7.1.1) above. Assume that H 1 Y; G D 0.   Then any infinitesimal deformation  2 H 0 Y; F jY is effective. More is true: There exists a family F W   Y ! X of morphisms such that F0 D f , F ;0 D  and, such that the velocity vector fields F ;t are contained in H 0 Y; F t F , for all t 2 . Remark 7.7. It might be worth noting that   to obtain the conclusions of Corollary 7.6, 1 it suffices to prove vanishing H Y; G D 0 for a single obstruction sheaf G . Since the obstruction sheaf is often not uniquely defined, this gives extra leeway which might be useful in applications.

References [Art]

M. Artin, On the solutions of analytic equations. Invent. Math. 5 (1968), 277–291.

[BHPV] W. P. Barth, K. Hulek, C. A. M. Peters, and A. Van de Ven, Compact complex surfaces. Second ed., Ergeb. Math. Grenzgeb. (3) 4, Springer-Verlag, Berlin 2004. [CP]

F. Campana and T. Peternell, Cycle spaces. In Several complex variables VII, Encyclopaedia Math. Sci. 74, Springer-Verlag, Berlin 1994, 319–349.

[Hor]

E. Horikawa, On deformations of holomorphic maps. I. J. Math. Soc. Japan 25 (1973), 372–396

[Jö]

C. Jörder, Deformation of morphisms. Diploma thesis, University of Freiburg, 2010;

[Kaw]

Y. Kawamata, Unobstructed deformations. A remark on a paper of Z. Ran: “Deformations of manifolds with torsion or negative canonical bundle” [J. Algebraic Geom. 1 (1992), no. 2, 279–291]. J. Algebraic Geom. 1 (1992), no. 2, 183–190.

www.freidok.uni-freiburg.de/volltexte/7897http://www.freidok.uni-freiburg.de/volltexte/7897.

[Kaw2] Y. Kawamata, Erratum on: “Unobstructed deformations. A remark on a paper of Z. Ran: ‘Deformations of manifolds with torsion or negative canonical bundle’” [J. Algebraic Geom. 1 (1992), no. 2, 183–190]. J. Algebraic Geom. 6 (1997), no. 4, 803–804. [KKL]

S. Kebekus, S. Kousidis, and D. Lohmann, Deformations along subsheaves. L’Enseign. Math. (2) 56 (2010), no. 3–4, 287–313.

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Z. Ran, Lifting of cohomology and unobstructedness of certain holomorphic maps. Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 1, 113–117.

[Voi]

C. Voisin, Sur la stabilité des sous-variétés lagrangiennes des variétés symplectiques holomorphes. In Complex projective geometry (Trieste, 1989/Bergen, 1989), London Math. Soc. Lecture Note Ser. 179, Cambridge University Press, Cambridge 1992, 294–303.

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C. Voisin, Hodge theory and complex algebraic geometry. I. English ed., translated from the French by Leila Schneps, Cambridge Stud. Adv. Math. 76, Cambridge University Press, Cambridge 2007.

Clemens Jörder, Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstraße 1, 79104 Freiburg, Germany E-mail: [email protected] Stefan Kebekus, Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstraße 1, 79104 Freiburg, Germany E-mail: [email protected]

Some degenerations of G2 and Calabi–Yau varieties Michał Kapustka

1 Introduction We shall denote by G2 the adjoint variety of the complex simple Lie group G2 . In geometric terms this is a subvariety of the Grassmannian G.5; V / consisting of 5spaces isotropic with respect to a chosen non-degenerate 4-form ! on a 7-dimensional vector space V . In this V4 context the word non-degenerate stands for 4-forms contained in the open orbit in V of the natural action of Gl.7/. It is known (see [1]) that this open orbit is the complement of a hypersurface of degree 7. The hypersurface is the closure of the set of 4-forms which can be decomposed into the sum of 3 simple forms. The expected number of simple forms needed to decompose a general 4-form is also 3, meaning that our case is defective. In fact this is the only known example (together with the dual .k; n/ D .3; 7/) with 3  k  n  3 in which a general k-form in an n-dimensional space cannot be decomposed into the sum of an expected number of y2 of 5-spaces simple forms. A natural question comes to mind. What is the variety G isotropic with respect to a generic 4-form from the hypersurface of degree 7? From the above point of view it is a variety which is not expected to exist. We prove that the y2 is linearly isomorphic to the closure of the image of P 5 by Plücker embedding of G y2 the map defined by quadrics containing a fixed twisted cubic. We check also that G is singular along a plane and appears as a flat deformation of G2 . Next, we study varieties obtained by degenerating the twisted cubic to a reducible cubic. All of them appear to be flat deformations of G2 . However only one of them appears to be a linear section of G.5; V /. It corresponds to the variety of 5-spaces isotropic with respect to a 4-form from the tangential variety to the Grassmannian G.4; 7/. The two other degenerations corresponding to configurations of lines give rise to toric degenerations of G2 . The variety G2 as a spherical variety is proved in [7] to have such degenerations, but for G2 the constructed degeneration is not a Gorenstein Fano variety. In the context of applications, mainly for purposes of mirror symmetry of Calabi–Yau manifolds, it is important that these degenerations lead to Gorenstein toric Fano varieties. Our two toric varieties are both Gorenstein and Fano, they admit respectively 3 and 4 singular strata of codimension 3 and degree 1. Hence the varieties obtained by intersecting these toric 5-folds with a quadric and a hyperplane have six and eight nodes respectively. The small resolutions of these nodes are Calabi– Yau threefolds which are complete intersections in smooth toric Fano 5-folds and are connected by conifold transition to the Borcea Calabi–Yau threefolds of degree 36, which are sections of G2 by a hyperplane and a quadric, and will be denoted X36 . This is the setting for the methods developed in [3] to work and provide a partially conjectural construction of mirror. Note that in [5] the authors found a Gorenstein

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toric Fano fourfold whose hyperplane section is a nodal Calabi–Yau threefold admitting a smoothing which has the same Hodge numbers, degree, and degree of the second Chern class as X36 . It follows by a theorem of Wall that it is diffeomorphic to it and by connectedness of the Hilbert scheme is also a flat deformation of it. However, a priori the two varieties can be in different components of the Hilbert scheme, hence do not give rise to a properly understood conifold transition. In this case it is not clear what is the connection between the mirrors of these varieties. y2 are also used in the paper for the construction of The geometric properties of G another type of geometric transitions. A pair of geometric transitions joining X36 and the complete intersection of a quadric and a quartic in P 5 . The first is a conifold transition involving a small contraction of two nodes the second a geometric transition involving a primitive contraction of type III. In the last section we consider a different application of the considered constructions. We apply it to the study of polarized K3 surfaces of genus 10. By the Mukai linear section theorem (see [14]) we know that a generic polarized K3 surface of genus 10 appears as a complete linear section of G2 . A classification of the non-general cases has been presented in [10]. The classification is however made using descriptions in scrolls, which is not completely precise in a few special cases. We use our construction to clarify one special case in this classification. This is the case of polarized K3 surfaces .S; L/ of genus 10 having a g51 (i.e. a smooth representative of L admits a g51 ). In particular we prove that a smooth linear section of G2 does not admit a g51 . Then, y2 has a g 1 and that K3 we prove that each smooth 2-dimensional linear section of G 5 surfaces appearing in this way form a component of the moduli space of such surfaces. More precisely we get the following. Proposition 1.1. Let .S; L/ be a polarized K 3 surface of genus 10 such that L admits exactly one g51 , then .S; L/ is a proper linear section of one of the four considered degenerations of G2 . Proposition 1.2. If .S; L/ is a polarized K 3 surface of genus 10 such that L admits a g51 induced by an elliptic curve, and S has Picard number 2, then .S; L/ is a proper y2 . linear section of G The methods used throughout the paper are elementary and rely highly on direct computations in chosen coordinates including the use of Macaulay2 and Magma.

2 The variety G2 In this section we recall a basic description of the variety G2 using equations. Lemma 2.1. The variety G2 appears as a 5-dimensional section of the Grassmannian G.2; 7/ with seven hyperplanes (non complete intersection). It parametrizes the set of V 2-forms fŒv1 ^ v2  2 G.2; V / j v1 ^ v2 ^ ! D 0 2 6 V g, where V is a 7-dimensional vector space and ! a non-degenerate 4-form on it.

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By [1] we can choose ! D x1 ^ x2 ^ x3 ^ x7 C x4 ^ x5 ^ x6 ^ x7 C x2 ^ x3 ^ x5 ^ x6 C x1 ^ x3 ^ x4 ^Vx6 C x1 ^ x2 ^ x4 ^ x5 . The variety G2 is then described in its linear span W  P . 2 V / with coordinates .a; : : : ; n/ by 4  4 Pfaffians of the matrix: 0 1 0 f e g h i a B f 0 d j k l b C B C B e d 0 m n g  k c C B C B g j m 0 c b d C B C: B h k C n c 0 a e C B @ i l g C k b a 0 f A a b c d e f 0

y2 3 The variety G V From [1] there is a hypersurface of degree 7 in 4 V parameterizing 4-forms which may be written as a sum of three pure forms. The generic element of this hypersurface corresponds to a generic degenerate 4-form !0 . After a suitable change of coordinates we may assume (see [1]) that !0 D x1 ^ x2 ^ x3 ^ x7 C x4 ^ x5 ^ x6 ^ x7 C x2 ^ y2 D fŒv1 ^ v2  2 x3 ^ x5 ^ x6 C x1 ^ x3 ^ x4 ^Vx6 . Let us consider the variety G G.2; V / j v1 ^ v2 ^ !0 D 0 2 6 V g. Analogously as in the non-degenerate case it is described in it’s linear span by 4  4 Pfaffians of a matrix of the form 0 1 0 0 e g h i a B 0 0 d j g l b C B C B e d 0 m n k c C B C B g j m 0 0 b d C B C: B h g n C 0 0 a e C B @ i l k b a 0 f A a b c d e f 0 y2 contains a smooth Fano fourfold Directly from the equations we observe that G F described in the space .b; c; d; f; j; l; m; k/ by the 4  4 Pfaffians of the matrix 0 1 0 d j l b B d 0 m k c C B C B j m 0 b d C B C: @ l k b 0 f A b c d f 0 Remark 3.1. In fact we see directly a second such Fano fourfold F 0 isomorphic to y2 with the space F and meeting F in a plane. It is analogously the intersection of G .e; h; i; a; n; c; k; f /. We shall see that there is in fact a one parameter family of such Fano fourfolds any two intersecting in the plane .c; k; f /.

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y2 from the plane spanned by .c; k; f / Observation 3.2. The image of the projection of G 1 5 is a hyperplane section of P  P . y2 to P 10 with coordinates .a; b; d; e; g; h; i; j; l; m; n/ Proof. The projection maps G Observe that the equations of the projection involve 2  2 minors of the matrix   e g h i a n ; d j g l b m y2 and do not involve c; k; f . It follows as these equations appear in the description of G that the image is contained in a hyperplane section P of a P 1  P 5 . Next we check that the map is an isomorphism over the open subset given by g D 1 of P . y2 is a conic. Proposition 3.3. The Hilbert scheme of projective 3-spaces contained in G y2 . Moreover the union of these 3-spaces is a divisor D of degree 8 in G Proof. We start by proving the following lemmas. Lemma 3.4. Let a plane P have four points of intersection with a G.5; V /, such that they span this plane. Then P \ G.5; V / is a conic parameterizing all 5-spaces containing a 3-space W . Proof. The proof follows from [15], as three points in G.2; V / always lie in a G.2; A/ for some subspace A of dimension 6. y if and only if Lemma 3.5. A projective 3-space …  G.2; 7/ is contained in G there exists a vector u in V and a 4-space v1 ^ v2 ^ v3 ^ v4 2 G.4; 7/ such that u ^ !0 D u ^ v1 ^ v2 ^ v3 ^ v4 and … is generated by u ^ v1 , u ^ v2 , u ^ v3 , u ^ v4 . Proof. To prove the if part we observe that our conditions imply u ^ vi ^ !0 D 0 for i D 1; : : : ; 4. Let us pass to the proof of the only if part. Observe first that any projective 3-space contained in G.2; V / is spanned by four points of the form u ^ v1 ,u ^ v2 ,u ^ v3 ,u ^ v4 . By our assumption on !0 the form u ^ !0 ¤ 0, and it is killed by the vectors u; v1 ; : : : ; v4 , hence equals u ^ v1 ^ v2 ^ v3 ^ v4 . Now, it follows from Lemma 3.5 that the set of projective 3-spaces contained in y is parametrized by those Œv 2 P .V / for which v ^ !0 2 G.5; 7/. The form !0 G may be written as the sum of three simple forms corresponding to three subspaces P1 , P2 , P3 of dimension 4 in V , each two meeting in a line and no three having a nontrivial intersection. Hence the form v ^ !0 may be written as the sum of three simple forms corresponding to three subspaces of dimension 5 each spanned by v and one of the spaces Pi . By lemma 3.4 the sum of these three 5-forms may be a simple form only if they all contain a common 3-space. But this may happen only if v lies in the space spanned by the lines Pi \ Pj . Now it is enough to see that the condition v ^ !0 is simple, corresponds for the chosen coordinate space to ..v ^ !0 / /2 D 0

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and perform a straightforward computation to see that it induces a quadratic equation on the coefficients of v 2 spanfP1 \ P2 ; P1 \ P3 ; P2 \ P3 g. y2 with fg D h D In coordinates the constructed divisor is the intersection of G D 0g. The latter defines on G.5; 7/ the set of lines intersecting the distinguished plane. We compute in Macaulay2 its degree. Remark 3.6. From the above proof it follows that the form !0 defines a conic Q in P .V / by Q D fŒv 2 P .V / j v ^ !0 2 G.5; 7/g. Observe that any secant line y2 . Indeed let v1 ; v2 2 V be two vectors such that of this conic is an element of G Œv1  ; Œv2  2 Q. Then vi ^ !0 defines a 5 space …i  V for i D 1; 2. Consider now the product v1 ^ v2 ^ !0 . If it is not zero it defines a hyperplane in V . It follows that dim.…1 \ …2 / D 4 and !0 can then be written in the form !0 D u1 ^ u2 ^ u3 ^ u4 C v1 ^ v2 ^ ˛. According to [1] this decomposition corresponds to a non general degenerate form !0 giving us a contradiction. y2 is a P 3 of lines passing through The proof implies also that each P 3 contained in G a chosen v in the conic and contained in the projective 4-space corresponding to v ^!0 . Remark 3.7. For any three points v1 , v2 , v3 lying on the distinguished conic Q, there exists a decomposition of !0 into the sum of 3 simple forms ˛1 , ˛2 , ˛3 such that v1 ^ .!0  ˛1 / D v3 ^ .!0  ˛2 / D v3 ^ .!0  ˛3 / D 0. In other words for any triple of points on the conic there is a decomposition with corresponding 4-spaces P1 , P2 , P3 such that .v1 ; v2 ; v3 / D .P1 \ P2 ; P1 \ P3 ; P2 \ P3 /. y2 has a 5-dimensional family of presentations into Remark 3.8. A three form defining G the sum of three simple forms corresponding to three subspaces P1 , P2 , P3 , however all these presentations induce the same space spanfP1 \ P2 ; P1 \ P3 ; P2 \ P3 g. This y consisting of lines contained in a space corresponds to the only projective plane in G y consist of lines passing through a projective plane. All other planes contained in G point and contained in a 3-space. y2 from F is a birational map onto P 5 whose Proposition 3.9. The projection of G inverse is the map ' defined by the system of quadrics in P 5 containing a common twisted cubic. Proof. Observe that the considered projection from F decomposes into a projection from the plane spanned by c, k, f and the canonical projection from P 1  P 5 onto P 5 . The latter restricted to P is the blow down of P 1  P 3 . It follows that the map is an isomorphism between the open set given by g D 1 and its image in P 5 . Let us write down explicitly the inverse map. Let .x; y; z; t; u; v/ be a coordinate system in P 5 . Consider a twisted cubic curve given by u D 0, v D 0 and the minors of the matrix   x y z : t x y Let L be the system of quadrics containing the twisted cubic. Choose the coordinates .a; : : : ; n/ of H 0 .L/ in the following way: .a; : : : ; n/ D .uy; vy; ytx 2; vx; ux; y 2 xz; uv; u2; uz; v 2; xyCzt; vz; vt; ut /:

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We easily check that the corresponding map is well defined and inverse to the projection y2 with substituted coordinates. by writing down the matrix defining G 0 1 0 0 ux uv u2 uz uy B 0 C 0 vx v 2 uv vz vy B C 2 C B ux vx 0 vt ut xy C zt yt  x C B B uv v 2 vt 0 0 vy vx C B 2 C B u C uv ut 0 0 uy ux B C @ uz vz xy  zt vy uy 0 y 2  xz A 0 uy vy yt C x 2 vx ux y 2 C xz

Remark 3.10. The images of the 4-dimensional projective spaces containing the twisted cubic form a pencil of smooth Fano fourfolds each two meeting in the plane which is the image of the P 3 spanned by the twisted cubic. The statement follows from the fact that we can change coordinates in P 5 and hence we can assume that any two chosen Fano fourfolds obtained in this way are F and F 0 in Remark 3.1. y2 is a plane. Lemma 3.11. The singular locus of G Proof. To see that the distinguished plane is singular it is enough to observe that each line secant to the distinguished conic C is the common element of two projective 3y2 . These are the spaces of lines corresponding to the points of spaces contained in G intersections of the secant line with C . By the same argument it follows also that the divisor D 0 is singular in the plane. To check smoothness outside let us perform the following argument. Clearly the system j2H  Ej on the blow up of P 5 in the twisted cubic separates points and tangent directions outside the pre-image transform of the P 3 spanned by the twisted cubic. It remains to study the image of the exceptional divisor, which is D 0 . Now observe that for any F1 and F2 in the pencil of Fano fourfolds y2 described in Remark 3.10 there is a hyperplane in P 13 whose intersection with G 0 y decomposes in F1 , F2 and D . It follows that the singularities of G2 may occur only in the singularities of D 0 and in the base points of the pencil. We hence need only to prove that D 0 is smooth outside the distinguished plane. This follows directly from the description of the complement of the plane in D 0 as a vector bundle over the product of the twisted cubic with P 1 . Remark 3.12. Observe that the map induced on P 5 contracts only the secant lines of the twisted cubic to distinct points of the distinguished P 2 . y2 by 2 hypersurfaces is nodal. Remark 3.13. A generic codimension 2 section of G We check this by taking a codimension 2 linear section and looking at its singularity. y is a flat deformation of G. Lemma 3.14. The variety G

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Proof. We observe that both varieties arise as linear sections of G.2; V / by some P 10 . Moreover we easily find an algebraic family with those as fibers. Indeed consider the family parameterized by t 2 C of varieties given in P 13 by the 4  4 Pfaffians of the matrices: 0 B B B B B B B B @

0 tf f 0 e d g j h k i l a b

e d 0 m n gCk c

g j m 0 c b d

h g  t k n tc 0 a e

i l k b a 0 f

a b c d e f 0

1 C C C C C: C C C A

For each t 2 C the equations describe the variety of isotropic 5-spaces with respect to the form ! t D x1 ^ x2 ^ x3 ^ x7 C x4 ^ x5 ^ x6 ^ x7 C x2 ^ x3 ^ x5 ^ x6 C x1 ^ x3 ^ x4 ^ x6 C tx1 ^ x2 ^ x4 ^ x5 . The latter is a nondegenerate fourform for t ¤ 0. It follows that for t ¤ 0 the corresponding fiber of the family is isomorphic to G2 and y2 . for t D 0 it is equal to G The assertion then follows from the equality of their Hilbert polynomials, which we compute using MACAULAY 2.

4 Further degenerations y2 by considering degenerations of the twisted Observe that one can further degenerate G 5 cubic C in P . In particular the twisted cubic can degenerate to one of the following: • the curve C0 which is the sum of a smooth conic and a line intersecting it in a point; • a chain C1 of three lines spanning a P 3 ; • a curve C2 consisting of three lines passing through a common point and spanning a P 3. Let us consider the three cases separately. Let us start with the conic and the line. In this case we can assume that the ideal of C0 is given in P 5 by fu D 0; v D 0g and the minors of the matrix   x y z : t x 0 Then the image of P 5 by the system of quadrics containing C0 can also be written as a section of G.2; 7/ consisting of 2-forms killed by the 4-form !1 D x1 ^ x2 ^ x3 ^ x7 C x2 ^ x3 ^ x5 ^ x6 C x1 ^ x3 ^ x4 ^ x6 . To find the deformation family

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y2 corresponding to twisted cubics we consider the family of varieties isomorphic to G given by fu D 0; v D 0g and the minors of the matrix   x y z : t x y We conclude comparing Hilbert polynomials. V Remark 4.1. The forms !0 and !1 represent the only two orbits of forms in 3 .V / whose corresponding isotropic varieties are flat degenerations of G2 . To prove it we use the representatives of all nine orbits contained in [1] and check one by one the invariants of varieties they define using Macaulay2. In all other cases the dimension of the isotropic variety is higher. In the case of a chain of lines the situation is a bit different. Proposition 4.2. The variety G2 admits a degeneration over a disc to a Gorenstein toric Fano 5-fold whose only singularities are three conifold singularities in codimension 3 toric strata of degree 1. y2 is a degeneration of G2 over a disc it is enough to prove that G y2 admits Proof. As G 5 such a degeneration. We know that the latter is the image of P by the map defined by the system of quadrics containing a twisted cubic C . Let us choose a coordinate system .x; y; z; t; u; v/ such that C is given in P 5 by fu D 0; v D 0g and the minors of the matrix   x y z ; t x y then choose the chain of lines C1 to be defined by fu D 0; v D 0g and the minors of the matrix   0 y z : t x 0 Let T be the variety in P 13 defined as the closure of the image of P 5 by the system of quadrics containing C0 . It is an anti-canonically embedded toric variety with corresponding dual reflexive polytope: . 0 0 1 0 0 / . 0 0 0 1 0 / . 1 1 1 1 1 / . 0 0 0 0 1 / . 1 0 0 0 0 / . 0 1 0 0 0 / . 1 1 1 0 1 / . 0 0 1 0 1 / . 1 1 0 1 1 /

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We check using Magma that the singular locus of this polytope has three conifold singularities along codimension 3 toric strata of degree 1. Consider the family of quadrics parameterized by  containing the curves C defined by fu D 0; v D 0g and the minors of the matrix   x y z : t x y For each  ¤ 0 the equations of the image of P 5 by the quadrics agree with the minors of the matrix: 0 0 0 e g h i B 0 0 d j g l B B e d 0 m n k B B g j m 0 0 b B B h g n 0 0 a B @ i l k b a 0 a b c d e f

corresponding system of a b c d e f 0

1 C C C C C; C C C A

in the coordinates .a; : : : ; n/ D .uy; vy; ytx 2; vx; ux; y 2 xz; uv; u2; uz; v 2; xyCzt; vz; vt; ut /: The latter define a variety isomorphic to Gy2 for each . It is easy to check that this family degenerates to T when  tends to 0. By comparing Hilbert polynomials we y2 , hence of G2 . obtain that it is a flat degeneration of G In the case of the twisted cubic degenerating to three lines meeting in a point we obtain a Gorenstein toric Fano 5-fold with 4 singular strata of codimension 3 and degree 1 which is a flat deformation of G2 . The corresponding dual reflexive polytope is: . 1 1 1 1 1 / . 0 0 1 0 0 / . 0 0 0 1 0 / . 0 0 0 0 1 / . 1 0 0 0 0 / . 0 1 0 0 0 / . 1 1 1 1 0 / . 1 1 1 0 1 / . 1 1 0 1 1 / . 2 2 1 1 1 / 4.1 Application to mirror symmetry. One of the methods of computing mirrors to Calabi–Yau threefolds is to find their degenerations to complete intersections in Gorenstein toric Fano varieties. Let us present the method, contained in [3], in our

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context. We aim to use the constructed toric degeneration to compute the mirror of the Calabi–Yau threefold X36 . As the construction is still partially conjectural we omit details in what follows. Consider the degeneration of G2 to T . We have, X36 is a generic intersection of G2 with a hyperplane and a quadric. On the other hand when we intersect T with a generic hyperplane and a generic quadric we get a Calabi–Yau threefold Yy with six nodes. It follows that Yy is a flat degeneration of X36 . Moreover Yy admits a small resolution of singularities, which is also a complete intersection in a toric variety. We shall denote it by Y . The variety Y is a smooth Calabi–Yau threefold connected to X by a conifold transition. Due to results of [2] the variety Y has a mirror family Y  with generic element denoted by Y  . The latter is found explicitly as a complete intersection in a toric variety obtained from the description of Y by the method of nef partitions. The authors in [2] prove that there is in fact a canonical isomorphism between H 1;1 .Y / and H 1;2 .Y  /. Let us consider the one parameter subfamily X  of the family Y  corresponding to the subspace of H 1;2 consisting of elements associated by the above isomorphism to the pullbacks of Cartier divisors from Yy . The delicate part of this mirror construction of X36 is to prove that a generic Calabi– Yau threefold from the subfamily X  has six nodes satisfying an appropriate number of relations. This is only a conjecture ([3, Conjecture 6.1.2]) which we are still unable to solve, also in this case. Assume that the conjecture is true. We then obtain a construction of the mirror family of X36 as a family of small resolutions of the elements of the considered subfamily.

5 A geometric bi-transition In this section we construct two geometric transitions between Calabi–Yau threefolds based on the map from Proposition 3.9. Let us consider a generic section X of Gy2 by a hyperplane and a quadric. Observe that X has exactly two nodes and admits a smoothing to a Borcea Calabi–Yau threefold X36 of degree 36. Observe moreover that X contains a system of smooth K3 surfaces each two intersecting in exactly the two nodes. Namely these are the intersections of the pencil of F with the quadric and the hyperplane. Blowing up any of them is a resolution of singularities of X . Let us consider the second resolution i.e. the one with the exceptional lines flopped. It is a Calabi–Yau threefold Z with a fibration by K3 surfaces of genus 6 and generic Picard number 1. Observe moreover that according z 5 of P 5 in the twisted to Proposition 3.9 the map ' 1 factors through the blow up P cubic C . Let E be the exceptional divisor of the blow up and H the pullback of the hyperplane from P 5 . In this context Z is the intersection of two generic divisors of type j2H  Ej and j4H  2Ej respectively. Lemma 5.1. The Picard number .Z/ D 2 Proof. We follow the idea of [11]. Observe that both systems j2H  Ej and j4H  2Ej z 5 . On P z 5 both divisors contract the proper transform are base point free and big on P

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of P 3 to P 2 . It follows by [17, Theorem 6] that the Picard group of Z is isomorphic z 5 which is of rank 2. to the Picard group of P Moreover Z contains a divisor D 0 fibered by conics. In one hand D 0 is the proper transform of the divisor D from Proposition 3.3 by the considered resolution of singularities, on the other hand D 0 is the intersection of Z with the exceptional divisor E. It follows that D 0 is contracted to a twisted cubic in P 5 by the blowing down of E and the contraction is primitive by Lemma 5.1. It follows that Z is connected by a conifold transition involving a primitive contraction of 2 lines with X , and by a geometric transition involving a type III primitive contraction with the complete intersection Y2;4  P 5 . Remark 5.2. We can look also from the other direction. Let C be a twisted cubic, Q2 a generic quadric containing it, and Q4 a generic quartic singular along it. Then the intersection Q2 \ Q4 contains the double twisted cubic and two lines secant to it. Taking the map defined by the system of quadrics containing C the singular cubic is blown up and the two secant lines are contracted to two nodes.

6 Polarized K3 surfaces genus 10 with a g51 In this section we investigate polarized K3 surfaces of genus 10 which appear as sections of the varieties studied in this paper. Proposition 6.1. A polarized K 3 surface .S; L/, which is a proper linear section of a G2 does not admit a g51 . Proof. Let us first prove the following lemma: Lemma 6.2. Let p1 ; : : : ; p5 be five points on G.2; V / of which no two lie on a line in G.2; V / and no three lie on a conic in G.2; V / and such that they span a 3-space P . Then fp1 ; : : : ; p5 g  G.2; W /  G.2; V / for some 5-dimensional subspace W of V . Proof. Let p1 ; : : : ; p5 correspond to planes U1 ; : : : ; U5  V . By Lemma 3.4 we may assume that no four of these points lie on a plane. Assume without loss of generality that p1 ; : : : ; p4 span the 3-space. If dim.U1 CU2 CU3 CU4 / D 6 the assertion follows from [15, Lemma 2.3]. We need to exclude the case U1 C U2 C U3 C U4 D V . In this case (possibly changing the choice of p1 ; : : : ; p4 from the set fp1 ; : : : ; p5 g) we may choose a basis in one of the two following ways fv1 ; : : : ; v7 g such that v1 ; v2 2 U1 , v3 ; v4 2 U2 , v5 ; v6 2 U3 , and either v7 ; v1 C v3 C v5 2 U4 or v7 ; v1 C v3 2 U4 . Each point of P is then represented by a bi-vector w D av1 ^ v2 C bv3 ^ v4 C cv5 ^ v6 C dv7 ^ .v1 C v3 C v5 /; or w D av1 ^ v2 C bv3 ^ v4 C cv5 ^ v6 C dv7 ^ .v1 C v3 /;

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for some a; b; c; d 2 C. By simple calculation we have w 2 D 0 if and only if exactly one of the a, b, c, d is nonzero, which gives a contradiction with the existence of p5 as in the assumption. Now assume that L has a g51 . It follows from [15, (2.7)] that it is given by five points on L spanning a P 3 . By Lemma 6.2 these points are contained in a section of G2 with a G.2; 5/. We conclude by [13, Lemma 3.3] as five isolated points cannot be a linear section of a cubic scroll by a P 3 . Proposition 6.3. Every smooth polarized surface .S; L/ which appears as a complete y2 is a K 3 surface with a g 1 . linear section of G 5 y2 is a flat deformation of G2 it’s smooth complete linear section of diProof. As G mension 2 are K3 surfaces of genus 10. Moreover each of these surfaces contains an elliptic curve of degree 5 which is a section of the Fano fourfold F . Let us consider the converse. Let .S; L/ be polarized K3 surface of genus 10, such that L admits a g51 induced by and elliptic curve E and do not admit a g41 . By the theorem of Green and Lazarsfeld [9] this is the case for instance when L admits a g51 and does not admit neither a g41 nor a g72 . We have E:L D 5 and E 2 D 0 hence h0 .O.L/jE / D 5 and h0 .O.L  E/jE / D 5. It follows from the standard exact sequence that h0 .O.L  E//  6 and h0 .O.L  2E//  1. We claim that jL  Ej is base point free: Indeed, denote by D its moving part and  its fixed part. Clearly jD  Ej is effective as jL  2Ej is. Observe that D cannot be of the form kE 0 with E 0 an elliptic curve, because as D  E is effective we would have E 0 D E hence k  3 which would contradict h0 .O.L  E//  6. Hence we may assume that D is a smooth irreducible curve and h1 .O.D// D 0. By Riemann–Roch we have 4 C D 2 D 2h0 .O.D// D 2h0 .O.D C /  4 C .D C /2 and analogously 4 D 4 C E 2 D 2h0 .O.E// D 2h0 .O.E C /  4 C .E C /2 ; because jD  Ej being effective implies  is also the fixed part of jE C j. It follows that L: D .D C E C /:  0, which contradicts ampleness of L. It follows from the claim that jLEj is big, nef, base point free and h0 .O.LE// D 6. Observe that jL  Ej is not hyper-elliptic. Indeed, first since .L  E/:E D 5 it cannot be a double genus 2 curve. Assume now that there exists an elliptic curve E 0 such that E 0 :.L  E/ D 2 then L:E  4 because jL  2Ej being effective implies .L  2E/:E 0  0. This contradicts the nonexistence of g41 on L. Hence jLEj defines a birational morphism to a surface of degree 8 in P 5 . Observe moreover that the image of an element in jL  2Ej is a curve of degree 3 spanning a P 3 . The latter follows from the fact that by the standard exact sequence 0 ! O.E/ ! O.L  E/ ! O .L  E/ ! 0

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for  2 jL  2Ej we have h0 ..L  E/j / D 4. Next we have two possibilities: 1. The system jL  Ej is trigonal, then the image of 'jLE j .S / is contained in a cubic threefold scroll. The latter is either the Segre embedding of P 1  P 2 or a cone over a cubic rational normal scroll surface. 2. The surface 'jLE j .S / is a complete intersection of three quadrics. Moreover for the image C D 'jLE j ./ we have the following possibilities: • Either C is a twisted cubic, • or C is the union of a conic and a line, • or C is the union of three lines. Consider now the composition of 'jLE j with the birational map given by quadrics in P 5 containing C . It is given by a subsystem of jLj D j2.L  E/  .L  2E/j. Moreover in every case above .S / spans a P 10 , because in each case the space of quadrics containing 'jLE j .S / is 3-dimensional. It follows that is defined on S by the complete linear system jLj. Finally .S; L/ is either a proper linear section of one of the three considered degenerations of G2 or a divisor in the blow up of a cubic scroll in a cubic curve. In particular we have the following. Proposition 6.4. Let .S; L/ be a polarized K 3 surface of genus 10 such that L admits a g51 induced by an elliptic curve E but no g41 . If moreover jL  Ej is not trigonal, then .S; L/ is a proper linear section of one of the four considered degenerations of G2 . Remark 6.5. The system jL  Ej is trigonal on S if and only if there exists an elliptic curve E 0 on S such that one of the following holds: 1. L:E 0 D 6 and E:E 0 D 3, 2. L:E 0 D 5 and E:E 0 D 2. Now observe that in both cases we obtain a second g51 on L. In the first case it is given by the restriction of E 0 and in the second we get at least a g72 by restricting jLE E 0 j, the latter gives rise to a g51 by composing the map with a projection from the singular point of the image by the g72 (there is a singular point by Noether’s genus formula). We can now easily prove Proposition 1.1. Proof. Proof of Proposition 1.1 Indeed the existence of exactly one g51 excludes both the existence of a g72 and of a g41 , hence the g51 is induced by an elliptic pencil jEj on S. Moreover by Remark 6.5 we see that jL  Ej is then trigonal.

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Proposition 1.2 follows directly from Proposition 6.4 and the fact that in the more degenerate case we clearly get a higher Picard number due to the decomposition of C . Remark 6.6. The K3 surfaces obtained as sections of considered varieties fit to the case g D 10, c D 3, D 2 D 0, and scroll of type .2; 1; 1; 1; 1/ from [10] (Observe that there is a misprint in the table, because H 0 .L  2D/ should be 1 in this case). The y2 embedding in the scroll corresponds to the induced embedding in the projection of G from the distinguished plane. Acknowledgements. I would like to thank K. Ranestad, J. Buczy´nski and G.Kapustka for their help. I acknowledge also the referee for useful comments. The project was partially supported by SNF, No 200020-119437/1 and by MNSiW, N N201 414539.

References [1] H. Abo, G. Ottaviani, and Ch. Peterson, Non-defectivity of Grassmannians of planes. J. Algebraic Geom. 21 (2012), no. 1, 1–20. [2] V. V. Batyrev and L. A. Borisov, On Calabi–Yau complete intersections in toric varieties In Higher-dimensional complex varieties (Trento, 1994), de Gruyter, Berlin 1996, 37–65. [3] V. V. Batyrev, I. Ciocan-Fontanine, B. Kim, and D. van Straten, Conifold transitions and mirror symmetry for Calabi–Yau complete intersections in Grassmannians. Nuclear Phys. B 514 (1998), no. 3, 640–666. [4] V. V. Batyrev, I. Ciocan-Fontanine, B. Kim, and D. van Straten, Mirror symmetry and toric degenerations of partial flag manifolds. Acta Math. 184 (2000), no. 1, 1–39. [5] V. V. Batyrev and M. Kreuzer, Constructing new Calabi–Yau 3-folds and their mirrors via conifold transitions. Adv. Theor. Math. Phys. 14 (2010), no. 3, 879–898. [6] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language. J. Symbolic Comput. 24 (1997), 235–265. [7] M. Brion and V. Alexeev, Toric degenerations of spherical varieties. Selecta Math. (N.S.) 10 (2004), 453–478. [8] D. R. Grayson and M. E. Stillman, Macaulay 2, a software system for research in algebraic geometry. Available at www.math.uiuc.edu/Macaulay2/http://www.math.uiuc.edu/Macaulay2/ [9] M. Green and R. Lazarsfeld, Special divisors on curves on a K3 surface. Invent. Math. 89 (1987), 357–370. [10] T. Johnsen and A. L. Knutsen, K3 projective models in scrolls. Lecture Notes in Math. 1842, Springer-Verlag, Berlin 2004. [11] G. Kapustka, Primitive contractions of Calabi–Yau threefolds II. J. London Math. Soc. (2) 79 (2009), no. 1, 259–271. [12] M. Kapustka, Geometric transitions between Calabi–Yau threefolds related to KustinMiller unprojections. J. Geom. Phys. 61 (2011), no. 8, 1309–1318.

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[13] M. Kapustka and K. Ranestad, Vector bundles on Fano varieties of genus 10. Preprint, arXiv:1005.5528v2 [math.AG]. [14] S. Mukai, Curves, K3 surfaces and Fano 3-folds of genus  10. In Algebraic geometry and commutative algebra in honor of Masayoshi Nagata, Kinokuniya, Tokyo 1988, 357–377. [15] S. Mukai, Curves and Grassmannians. In Algebraic geometry and related topics (Inchon, 1992), Conf. Proc. Lecture Notes Algebraic Geom. I, International Press, Cambridge, Mass., 1993, 19–40. [16] S. Mukai, Non-abelian Brill-Noether theory and Fano 3-folds. Sugaku Exp. 14 (2001), no. 2, 125–153. [17] G. V. Ravindra and V. Srinivas, The Grothendieck-Lefschetz theorem for normal projective varieties. J. Algebraic Geom. 15 (2006), 563–590. Michał Kapustka, Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland; and Institute of Mathematics, Jagiellonian University of Kraków, ul.Łojasiewicza 6, 30-348 Kraków, Poland E-mail: [email protected]

Notes on Kebekus’ lectures on differential forms on singular spaces Mateusz Michałek The present article contains the notes from three lectures whose aim is a discussion of hyperbolicity of moduli and related topics.

1 Lecture one 1.1 Classical theory. Let us first recall basic facts that are the motivation for further constructions. Fact 1.1. Let X be a smooth, projective variety of dimension n over C. There exist 1 n and the dualizing sheaf !X D X . The following the sheaf of differential forms X theorems hold: • For any locally free sheaf F on X we have H i .X; F / Š H ni .X; F  ˝ !X /_ (Serre duality). • For any ample invertible sheaf L on X we have H i .X; L ˝ !X / D 0 for all i > 0 (Kodaira vanishing). 1.2 Three constructions for singular varieties. From now on we assume that X is a normal, possibly singular, variety. There exist several constructions of sheaves of differential forms that in general give different results. 1 be the sheaf of Kähler differentials. Construction 1.1 (Kähler differentials). Let X 1 On U D Spec A the sections X .U / form an OX .U /-module generated by formal symbols df for f 2 A that satisfy the relations d.f C g/ D df C dg, d.fg/ D g.df / C f .dg/ and dc D 0 for c a constant.

This first construction has got many advantages. First of all it is very natural. Moreover, given a morphism of two varieties f W X ! Y one can pull-back differential forms from Y to X . 1 does not have to be locally free. Moreover, it It also has a few disadvantages: X does not have to be reflexive and may even have torsion. For a precise criterion when the sheaf of Kähler differentials on a cone over a smooth projective variety hasV torsion n 1 see [GRo]. One does not have a Harder–Narasimhan theory. For X WD n X neither Serre duality nor Kodaira vanishing holds. The motivation for the second construction is Serre duality. For a projective scheme over a field one always has the Grothendieck dualizing sheaf !X . This means that for

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any coherent sheaf F we have an isomorphism Hom.F; !X / Š H n .X; F /_ . Moreover, if X is Cohen–Macaulay then Serre duality applies. Unfortunately, in case of a n singular variety, the sheaves !X and X can be different. Let us now describe the sections of !X . The intuition is that these are differential forms defined away from the singularities. More precisely let Z be the singular locus of X and let U D X n Z. We have got i W U ! X the natural inclusion. One has !X D i .nU /. As the sections of !U do not depend on a subset of codimension 2, neither do the sections of !X . We call this property the second Riemann extension property, or using the notation of [B, p. 128] we say that !X is normal. The sheaf !X is also torsion free so using the characterization [Har80, Proposition 1.6] it is reflexive. Œp as i .pU /. Construction 1.2 (Reflexive differentials). We define X

One of the advantages of this construction is that we get a reflexive and in particular torsion free sheaf. As a consequence we have got a Harder–Narasimhan filtration. As already mentioned Serre duality applies if X is Cohen–Macaulay. One can also have results on the positivity on moduli spaces – for precise results see Theorem 3.13 and Œp p  and .X / are both reflexive sheaves that the discussion after it. Let us note that X agree on U . As the codimension of the singular locus it at least 2, they must be equal. Œp the vanishing theorems do not hold and in general one However for the sheaf X cannot define the pull-back. The following theorem will be the motivation for the last construction. Theorem 1.2 (Grauert–Riemenschneider [GR]). Let  W Xz ! X be a resolution of singularities of the variety X . Let ! zX WD  .!Xz /. Then the following holds: 1. The sheaf ! zX is a subsheaf of the Grothendieck dualizing sheaf !X and the inclusion does not depend on the resolution. 2. For any ample invertible sheaf L and any integer q > 0 we have H q .X; ! zX ˝ L/ D 0:



Construction 1.3. Let us choose any resolution of singularities  W Xz ! X . We z p WD  p . define  X z X

z p is also torsion free. The push-forward of a torsion free sheaf has no torsion, so  X Moreover, by the above theorem one obtains results on vanishing. In this case one can also define the pull-back properly. Unfortunately the results on Serre duality do not apply. 1.3 Comparison of the three constructions. The advantages and disadvantages of each construction are summed up in the following table.

Notes on Kebekus’ lectures on differential forms on singular spaces

Type

Reflexive

Pull-back

No torsion

Duality

Vanishing

p X Œp X zp  X

✕

✓

✕

✕

✕

✓

✕

✓

✓

✕

✕

✓

✓

✕

✓

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To sum up one might paraphrase Grauert: “We need to make a choice when defining differential forms on singular spaces.” The reason for this, is that depending on what property we are interested in, we might have to consider different sheaves. If we need the results on vanishing the last construction is the most appropriate, but if we want to make advantage of Serre duality we should consider the second construction. Of course one would be interested in having just one sheaf that would satisfy both. This motivates an interesting question: when the last two constructions partially coincide, that is for which X we have !X D ! zX . To answer it let us consider a resolution of singularities  W Xz ! X . We assume that the -exceptional locus E is an snc1 divisor that is mapped onto the singular locus of X. From the definition, we know that the sections of ! zX are differential forms on Xz . z We also know that X n E is isomorphic to X n Z, where Z is the singular locus of X. This means that sections of !X on an open set U are differential forms on  1 .U / n E. zX agree if and only if each differential form defined We see that definitions of !X and ! on  1 .U / n E extends to a differential form on  1 .U /. Under the assumption that X is Gorenstein both definitions agree iff X is canonical2 . zp Main aim. Our aim will be to address the same problem for p-forms, that is when  X Œp and X coincide. Under mild assumptions on singularities that are always satisfied when dealing with the minimal model program, we will see that any p-form defined away from the exceptional divisor, extends on it. Let us now state precisely the theorem – all necessary definitions are given in Lecture two. Theorem 1.3 (Greb, Kebekus, Kovács, Peternell, Theorem 1.5 [GKKP]). Let .X; D/ be a log canonical pair. Let P  X be the non-klt locus and let  W Xz ! X be a log z be the largest reduced divisor contained in  1 .P /. resolution of singularities. Let D z is reflexive and equal to Œp .log D/. Then for any integer p the sheaf  pz .log D/ X X Equivalently any p-form defined on the smooth locus of an open set U  X can be extended to a p-form on any resolution of singularities.  For values of p that are small with respect to the dimension of singular locus stronger results are known. The reader is advised to consult [SvS] and [F]. 1

simple normal crossing The formal definition of canonical singularities will appear in the next section. As we will see it follows from definition that X is canonical iff on the resolution of singularities any n-form defined outside the exceptional locus extends on it. 2

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2 Lecture two We will start by recalling well-known facts on the Minimal Model Program and logarithmic sheaves. 2.1 Minimal Model Program. First we will make a short review of the classical results about the Minimal Models for surfaces. Given a smooth algebraic surface X we may blow-down all 1-curves. We obtain a map  W X ! X such that depending on the Kodaira dimension .X / of X either of the following holds. 1. If .X/  0: The canonical divisor KX is nef and defines a fibration KX W X ! Z such that dim Z D .X / D .X /. This is a situation of the Kodaira fiber space. 2. If .X/ < 0: There exists a fibration m W X ! Z such that KX is ample on fibers and .Z/ C 1 D .X /, where  is the Picard number. This is the Mori fiber space. The aim of the Minimal Model Program would be to extend this result to higher dimensions in the following way: Dream 2.1. Let X be a projective manifold. Then we have a birational map  W X Ü X such that X is normal, Q-factorial, with sufficiently mild singularities. Moreover, 1 does not contract any divisor and one of the following holds: 1. If .X/  0: The divisor KX is nef and defines a map X ! Z, such that dim Z D .KX / D .KX /. 2. If .X/ < 0: There exists a morphism m W X ! Z such that .Z/ C 1 D .X / and KX is ample on fibers. It turns out that in case of higher dimensions one should work with pairs .X; D/, where D is a divisor. Such a setting often allows to make inductive arguments, by passing to the divisor and hence decreasing the dimension. Let us now remind definitions concerning types of singularities. All the definitions and much more information can be found in [KM] and [KMM]. Let X be an algebraic variety with an effective Q-divisor D. Let f W Y ! X be a resolution of singularities z be the strict transform of D. and let D Definition 2.2 (Log resolution). We assume that the exceptional locus E of f is a divisor that is mapped onto the singular locus of X . We say that f is a log resolution z is an snc divisor. of the pair .X; D/ if E C D

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Let us fix canonical divisors KY and KX respectively on Y and X , such that f .KY / D KX . We assume that KX C D is Q-Cartier so that its pull-back is well defined. For a given resolution of singularities f we define ai such that z D f  .KX C D/ C KY C D

X

a i Ei ;

where Ei are the irreducible exceptional divisors of f . As D is a Q-divisor we may P write it as D D bi Di , where Di are irreducible and bi 2 Q. We call bi the coefficients of D. The following definitions are crucial for the Minimal Model Program [KM, Definition 2.34, Theorem 2.44]. Definition 2.3 (Canonical, lc, dlt, klt). We say that the pair .X; D/ is • canonical if there exists a log resolution f such that ai  0 for all i , • lc (log canonical) if all the coefficients of D are less or equal to 1 and there exists a log resolution f such that ai  1 for all i , • dlt (divisorial log terminal) if all the coefficients of D are less or equal to 1 and there exists a log resolution f such that ai > 1 for all i, • klt (Kawamata log terminal) if all the coefficients of D are strictly smaller then 1 and there exists a log resolution f such that ai > 1. Moreover, in the cases lc and klt one can write “for any log resolution” instead of “there exists a log resolution” [KMM, Lemma 0.2.12]. For the dimension equal to three the Dream 2.1 comes true in the following setting. Theorem 2.4. Let .X; D/ be a dlt pair where dim X D 3. Then we have a birational map  W X Ü X such that .X ; D / is a dlt pair. Moreover, one of the following holds: 1. The divisor KX C D is nef and defines a map X ! Z, such that dim Z D .KX C D/ D .KX C D /. 2. There exists a morphism m W X ! Z such that .Z/C1 D .X / and KX C D is ample on fibers.  More information on the Minimal Model Program can be found in [KM] and [KMM]. For the recent developments in this area the reader is advised to consult [BCHM], where the case of a klt pair and a big divisor is treated. One also hopes that results similar to Theorem 2.4 can be established in higher dimensions.

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2.2 Logarithmic sheaves. Now, we will review basic facts about logarithmic sheaves. First let us consider the following motivation. Let X be a compact manifold and D a smooth divisor. Let U D X n D. By T we denote the tangent sheaf. Given a vector field s 2 H 0 .U; TU / one may ask when it extends to X and stabilizes D. If both of these conditions are satisfied then we obtain a flow on X that stabilizes D. This will be a motivation to define TX . log D/. Its sections will correspond to vector fields whose flow stabilizes D. The formal definition is as follows. Definition 2.5 (TX . log D/). Let X be a projective manifold, D an snc divisor. Let U be any open set on which D is defined by an equation U . We set TX . log D/.U / D f@ 2 Der.OX .U // W @U 2 D .U /g; where D is the ideal sheaf of D and Der.OX .U // is the module of deriviations. One can see that the above definition on affine pieces gives a locally free sheaf on X that is a subsheaf of the tangent bundle. By dualizing the inclusion TX . log D/  TX 1 1 we get X  TX . log D/ DW X .log D/. Fact 2.6. For a projective manifold X and an snc divisor D one has the following 1 description of X .log D/: 1 1 2 .log D/ D fı 2 X ˝ OX .D/ W d ı 2 X ˝ OX .D/g: X

Fact 2.7. The following equality holds: n ^

1 X .log D/ D OX .KX C D/:

For X0 non compact manifold, we may compactify it to a variety X in such a way that D D X n X0 is an snc divisor. Fact 2.8 (Kodaira–Iitaka dimension for non compact varieties). The Kodaira–Iitaka diV 1 mension . n X .log D// does not depend on X and hence it is a birational invariant of X0 . We denote it .X0 /. Proof. This follows from the fact stated in [I, p. 326]. For an integer m > 0 one can consider birational invariant Pxm .X0 / of X0 called logarithmic m-genus. There are positive real numbers ˛, ˇ > 0 such that for m sufficiently large ˛m.X0 /  Pxm .X0 /  ˇm.X0 / : Hence the Kodaira–Iitaka dimension is indeed a birational invariant. Let us now consider a manifold X and a smooth codimension one subvariety D. Fact 2.9. One has got the following exact sequence called “residue sequence”: 1 1 0 ! X ! X .log D/ ! OD ! 0:

It can be generalized for higher differentials.

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Fact 2.10 (Property 2.3 b, [EV92]). Let X be a smooth variety and D a reduced, irreducible divisor. There is the following exact sequence: p p1 p 0 ! X ! 0: ! X .log D/ ! D

Let us assume that .X; D/ is a dlt pair. In this case reflexive differentials  .log D/ D .p .log D// have got a lot of nice properties, similar with logarithmic Kähler differentials on smooth spaces with an snc divisor. For example we get the following exact sequence. Œp

Fact 2.11. Let .X; D/ be a dlt pair. There exists an exact sequence Œp Œp p1 ! X .log D/ ! D ! 0: 0 ! X

For a much more general analogue of the sequence for manifolds the reader is advised to consult [GKKP, Theorem 11.7]. There and in [KK08a] one can find more results on reflexive differentials on dlt pairs.

3 Lecture three In this lecture we will focus on applications of theorems on reflexive differentials. 3.1 Pull-back of reflexive differentials. First let us note that the pull-back of logarithmic differential sheaves is closely related to our main aim: the extension of the differential form from the smooth locus onto the resolution of singularities. Let  W Xz ! X be a resolution of singularities. Suppose that we have a pull-back Œp map d  W   .X / ! Œp D pz . The last equality holds, because Xz is smooth. A z X

X

Œp is a differential form on the smooth locus. Hence the existence of d  section of X precisely guarantees that all such sections extend to differential forms on Xz . This is one of the reasons why the following theorem is crucial.

Theorem 3.1 (Theorem 4.3 [GKKP]). Let f W X ! Y be a morphism of normal varieties. Suppose that Y is klt and the image of X is not contained in the singular Œp locus. Then there exists the pull-back morphism df W f  .Œp  Y / ! X . Remark 3.2. The statement of [GKKP, Theorem 4.3] is much stronger. The proof presented there uses the main theorem of that paper [GKKP, Theorem 1.5] on the extension of reflexive differentials. Hence the three conditions 1. existence of the pull-back map for reflexive differentials, 2. extension of differential forms from the smooth locus onto the resolution of singularities, z from the Construction 1.3 3. reflexivity of the sheaf  pz .log D/ X are closely related. We will now present some applications of Theorem 3.1.

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3.2 Bogomolov–Sommese vanishing. Using Theorem 3.1 one can obtain an extension of Bogomolov–Sommese vanishing. Let us first recall the theorem. Theorem 3.3 (Bogomolov–Sommese vanishing, cf. Corollary 6.9 [EV92]). Let X be p a projective manifold, D an snc divisor and A  X .log D/ an invertible subsheaf. Then the Kodaira–Iitaka dimension .A/ is not greater then p.  Here the definition of the Kodaira–Iitaka dimension for a reflexive sheaf is as follows. Definition 3.4 (Kodaira–Iitaka dimension of a sheaf). Let X be a normal projective variety and A a reflexive sheaf of rank one. If h0 .X; .A˝n / / D 0 for all n 2 N, then we say that A has Kodaira–Iitaka dimension .A/ WD 1. Otherwise, set M WD fn 2 N W h0 .X; .A˝n / / > 0g, recall that the restriction of A to the smooth locus of X is locally free and consider the natural rational mapping n W X Ü P .H 0 .X; .A˝n / /_ / for each n 2 M: The Kodaira–Iitaka dimension of A is then defined as .A/ WD max dim n .X /: n2M

The Bogomolov–Sommese vanishing can be generalized to log canonical pairs as follows. Theorem 3.5 (Bogomolov–Sommese vanishing for lc pairs, Theorem 7.2 [GKKP]). Œp Let .X; D/ be an lc pair, where X is projective. If A  X .log D/ is a Q-Cartier reflexive subsheaf of rank one, then .A/  p. Proof in a simple case. We assume that D D 0 and X is klt. Let us consider a Cartier Œp divisor A  X . Let  W Xz ! X be a resolution of singularities with an exceptional divisor E. The pull-back   .A/ is a Cartier divisor on Xz . By Theorem 3.1 it is a subsheaf of Œp D pz . Using the standard Bogomolov–Sommese vanishing theorem z X X we obtain .  .A//  p. As  is surjective, we have .A/ D .  .A// and so .A/  p. 3.3 Lipman–Zariski conjecture. Here we will present the application of reflexive differentials to prove a special case of the Lipman–Zariski conjecture. Conjecture 3.6 (Lipman–Zariski conjecture). Let X be a variety such that the tangent sheaf TX is locally free. Then X is smooth. We give a proof of an interesting special case. Theorem 3.7 (Lipman–Zariski Conjecture for klt spaces, Theorem 6.1 [GKKP]). Let X be a klt space such that the tangent sheaf TX is locally free. Then X is smooth.

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Proof. Suppose that X is not smooth. As the question is local, we may assume that TX is free of rank n. Let 1 ; : : : ; n be global sections of TX that generate the tangent sheaf. We consider a resolution of singularities  W Xz ! X called the functorial resolution [Kol07, Theorems 3.35 and 3.45]. Let E be the exceptional divisor. As the singular locus of X is invariant with respect to any automorphism and due to the fact that we have chosen a functorial resolution, we may apply [GKK10, Corollary 4.7]. We see that  .TXz . log E// is reflexive. Hence we can lift the sections i to i0 2 H 0 .Xz ; TXz . log E//  H 0 .Xz ; TXz /: The smooth locus of X is isomorphic to Xz n E. As the sections i were independent, also the sections i0 must be independent on Xz n E. This means that we can find differential forms 1 ; : : : ; n 2 H 0 .Xz n E; 1z / that are a dual basis to i0 on Xz n E. X On this set we have i .j0 / D ıij 1Xz nE ; where ıij is the Kronecker delta and 1Xz nE is a constant function equal to 1. Using the extension theorem 1.3 we can extend each i z 1 /. Of course the equality 0 . 0 / D ıij 1 z holds on Xz . Let q be a to i0 2 H 0 .X; X i j z X smooth point of the divisor E. As i0 were in H 0 .Xz ; TXz . log E// the tangent vectors i0 .q/ must in fact be tangent to E 3 . In particular they have to be linearly dependent. This contradicts the equality i .j0 .q// D ıij . 3.4 Hyperbolicity of moduli. We will now present applications to moduli problems – for more details the reader is advised to consult [K]. First let us state the part of Shavarevich conjecture dealing with hyperbolicity. It was proved by Parshin and Arakelov [Par68], [Ara71]. Theorem 3.8 (Shavarevich hyperbolicity, [Par68], [Ara71]). Let f B W X B ! Y B be a smooth, complex, projective family of curves of genus g > 1 over a smooth quasiprojective base curve Y B . If Y B is isomorphic to one of the following varieties: • the projective line P 1 , • the affine line A1 , • the affine line minus one point C  , or • an elliptic curve, then any two fibers of f B are necessarily isomorphic.



In other words any map form Y B as above to the moduli stack of algebraic curves Mg is constant. If Y B is higher dimensional, then the map to the moduli stack must contract all curves isomorphic to those mentioned in the theorem. The straightforward generalization of Theorem 3.8 to families of higher dimension does not hold. A counterexample for surfaces over a curve is presented in [K, 2.1]. 3

The intuition that the flow corresponding to a section of T . log D/ stabilizes D might be helpful.

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However families of minimal surfaces [Mig95], [OV01] and families of canonically polarized manifolds4 of any dimension are well studied [Kov00], [VZ03]. In particular the following theorem holds. Theorem 3.9 (Hyperbolicity for families of canonically polarized varieties). Let f W X B ! Y B be a smooth, complex, projective family of canonically polarized varieties of arbitrary dimension, over a smooth quasi-projective base curve Y B . Then the conclusion of Shafarevich hyperbolicity, Theorem 3.8 holds.  Now, we will focus on generalizations of Theorem 3.8. First we define the variation of a family of canonically polarized complex manifolds. Definition 3.10 (Variation of the family, isotriviality). Let X B ! Y B be a projective family of canonically polarized complex manifolds over an irreducible base Y B . This defines a map from Y B to the moduli scheme5 . The variation of f B , denoted by Var.f B /, is defined as the dimension of the image of Y B in the moduli space. We have Var.f B / D 0 iff all fibers of f B are isomorphic; in this case, the family B f is called “isotrivial”. A general definition of the variation of a family can be found in [K, 2.5]. In [Vie01] Viehweg proposed the following generalization of the Shafarevich Hyperbolicity Theorem 3.8. Conjecture 3.11 (Viehweg’s conjecture). Let f B W X B ! Y B be a smooth projective family of canonically polarized varieties, over a quasi-projective manifold Y B . If Var.f B / D dim Y B then .Y B / D dim Y B , i.e. Y B is of log general type. Remark 3.12. In fact this conjecture was formulated by Viehweg in bigger generality – the details can be found in [Vie01]. The following result can be found in [KK08b] and implies Viehweg’s conjecture for dim Y B  3. Theorem 3.13 (Relation between the moduli map and the MMP, [KK08b], Theorem 1.2). Let f B W X B ! Y B be a smooth projective family of canonically polarized varieties, over a quasi-projective manifold Y B of dimension dim Y B  3. Let Y be a smooth compactification of Y B such that D WD Y n Y B is an snc divisor. Then any run of the minimal model program on the pair .Y; D/ will terminate in a Kodaira or Mori fiber space whose fibration factors the moduli map birationally. We will give an idea of a proof in a very simple case, highlighting the relation with extension of differential forms. However first let us present the statement of the theorem in detail. 4 5

A variety is canonically polarized iff the canonical bundle is ample. For the existence of coarse moduli scheme in this case see [Vie95].

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3.5 Discussion of Theorem 3.13. Let M be the coarse moduli space for polarized manifolds and let B W Y B ! M be the moduli map associated with the family f B . This defines a rational map W Y Ü M. Let  W Y Ü Y be the map obtained by running the minimal model program 2.4 and let Y ! Z be the associated Kodaira or Mori fiber space. Theorem 3.13 asserts the existence of a map Z Ü M, such that the following diagram commutes: XB  YB



 Y _ _ _/ Y    M o_ _ _ Z .

If .Y B /  0, then the minimal model program terminates in a Kodaira fiber space. In this case dim Z D .Y B /. In the special case when .Y B / D 0, then f B is isotrivial. If .Y B / D 1 then the minimal model program terminates in a Mori fiber space. In this case dim Z < dim Y B and the moduli map is not generically finite. It follows that if Y B is not of log general type, then the map to the moduli space is not generically finite. 3.6 Idea of the proof of Theorem 3.13 in a very simple case. We will now present one of the arguments used in the proof of Theorem 3.13. We focus on the easiest case .Y B / D 1 and Var.f B / D dim Y B . We will only show that the Picard number .Y / ¤ 1. This would show that there is a nontrivial fiber space structure on Y , which can be used for inductive arguments. Assume to the contrary that .Y / D 1. The main ingredient is the following theorem of Viehweg and Zuo [VZ02, 1.4(i)]: Theorem 3.14 (Existence of pluri-differentials). Let f B W X B ! Y B be a smooth projective family of canonically polarized complex manifolds, over a smooth complex quasi-projective base. Assume that the family is not isotrivial and fix a smooth projective compactification Y of Y B such that D WD Y n Y B is an snc divisor. Then there exist a number m > 0 and an invertible sheaf A  Symm 1Y .log D/ whose Kodaira–Iitaka dimension is at least the variation of the family, i.e. .A/  Var.f B /.  Remark 3.15. It turns out that the use of reflexive differentials allows to extend this result to the singular case. This way we obtain a reflexive, rank one sheaf A0  ˝m  ..Œ1 / such that .A0 /  Var.f B /. Y .log D // Let us remind that we are working under the hypothesis: .Y B / D 1, Var.f B / D dim Y B , .Y / D 1 and we want to obtain a contradiction. Let C  Y be a general complete intersection curve. As we have supposed that .Y / D 1 and .Y B / D 1,

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we know that .KY C D / is ample. As C avoids the singular locus of Y the sheaf 1 Œ1 Y .log D /jC equals Y .log D /jC and is locally free on C . We obtain c1 .Œ1 Y .log D /jC / < 0; so

/ < 0: c1 .1Y .log D /˝m jC

Since the map to the moduli space induced by f B is non constant, the variation Var.f B / is positive, so the Kodaira–Iitaka dimension .A0 / > 0. Note that as C is smooth and A0 is a reflexive, rank one sheaf, then A0 jC is invertible. One obtains c1 .A0 jC / > 0. Let B  Œ1 Y .log D / be the maximal destabilizing subsheaf with respect to C . We have seen that ..1Y .log D //˝m / contains a subsheaf that is positive with respect to C . On the other hand, since we are working in characteristic zero, .B ˝m / is the maximal destabilizing subsheaf of ..1Y .log D //˝m / , so it also has a positive degree. This implies that B has a positive degree with respect to C . Let r be the rank of B, that is strictly smaller then dim Y , as c1 .BjC / > 0, so B ¤ Œ1 Y .log D /. As Y is Q-factorial, for some k > 0 we have a Cartier divisor L D .det B/˝k / . The Picard number of Y is equal to one and B is positive with respect to C , so the divisor L has to be ample. In particular the Kodaira dimension of det B  Œr Y .log D / is equal to the dimension of Y what contradicts the generalized Bogomolov–Sommese vanishing 3.5 and therefore ends the proof. Acknowledgements. The author would like to thank Stefan Kebekus very much for valuable suggestions and detailed explanations of his interesting lectures. He also thanks Adrian Langer for useful arguments after Remark 3.15.

References [Ara71] S. J. Arakelov, Families of algebraic curves with fixed degeneracies. Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1269–1293; English transl. Math. USSR-Izv. 5 (1971), 1277–1302. [B]

W. Barth, Some properties of stable rank-2 vector bundles on P n . Math. Ann. 226 (1977), 125–150.

[BCHM] C. Birkar, P. Cascini, C. D. Hacon, and J. Mckernan, Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23 (2010), no. 2, 405–468. [EV92]

H. Esnault and E. Viehweg, Lectures on vanishing theorems. DMV Seminar 20, Birkhäuser Verlag, Basel 1992.

[F]

H. Flenner, Extendability of differential forms on nonisolated singularities. Invent. Math. 94 (1988), no. 2, 317–326.

[GR]

H. Grauert and O. Riemenschneider, Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen. Invent. Math. 11 (1970), 263–292.

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[GRo]

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D. Greb and S. Rollenske, Torsion and cotorsion in the sheaf of Kähler differentials on some mild singularities. Math. Res. Lett. 18 (2011), no. 6, 1259–1269.

[GKK10] D. Greb, S. Kebekus, and S. J. Kovács, Extension theorems for differential forms, and Bogomolov–Sommese vanishing on log canonical varieties. Compos. Math. 146 (2010), 193–219. [GKKP] D. Greb, S. Kebekus, S. Kovács, and T. Peternell, Differential forms on log canonical spaces. Publ. Math. Inst. Hautes Études Sci. 114 (2011), 87–169. [Har80] R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), no. 2, 121–176. [I]

S. Iitaka, Algebraic geometry. An introduction to birational geometry of algebraic varieties. Grad. Texts in Math. 76, Springer-Verlag, New York 1982.

[KMM] Y. Kawamata, K. Matsuda, and K. Matsuki, Introduction to the minimal model problem. In Algebraic geometry (Sendai, 1985), Adv. Stud. Pure Math. 10, North-Holland, Amsterdam 1987, 283–360. [K]

S. Kebekus, Differential forms on singular spaces, the minimal model program, and hyperbolicity of moduli stacks. In Handbook of moduli, in honour of David Mumford, International Press, to appear; preprint arXiv:1107.4239v2 [math.AG].

[KK08a] S. Kebekus and S. Kovács, Families of varieties of general type over compact bases. Adv. Math. 218 (2008), no. 3, 649–652. [KK08b] S. Kebekus and S. Kovács, The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties. Duke Math. J. 155 (2010), no. 1, 1–33. [Kol07]

J. Kollár, Lectures on resolution of singularities. Ann. of Math. Stud. 166, Princeton University Press, Princeton, NJ, 2007.

[KM]

J. Kollár and S. Mori, Birational geometry of algebraic varieties. Cambridge Tracts in Math. 134, Cambridge University Press, Cambridge 1998.

[Kov00] S. Kovács, Algebraic hyperbolicity of fine moduli spaces. J. Algebraic Geom. 9 (2000), no. 1, 165–174. [Mig95] L. Migliorini, A smooth family of minimal surfaces of general type over a curve of genus at most one is trivial. J. Algebraic Geom. 4 (1995), no. 2, 353–361. [OV01]

K. Oguiso and E. Viehweg, On the isotriviality of families of elliptic surfaces. J. Algebraic Geom. 10 (2001), no. 3, 569–598.

[Par68]

A. N. Parshin, Algebraic curves over function fields. Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1191–1219 (in Russian).

[SvS]

J. Steenbrink and D. van Straten, Extendability of holomorphic differential forms near isolated hypersurface singularities. Abh. Math. Sem. Univ. Hamburg 55 (1985), 97–110.

[Vie95]

E. Viehweg, Quasi-projective moduli for polarized manifolds. Ergeb. Math. Grenzgeb. (3) 30, Springer-Verlag, Berlin 1995.

[Vie01]

E. Viehweg, Positivity of direct image sheaves and applications to families of higher dimensional manifolds. In School on vanishing theorems and effective results in algebraic geometry (Trieste, 2000), ICTP Lecture Notes 6, Abdus Salam International Centre for Theoretical Physics, Trieste 2001, 249–284.

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[VZ02]

E. Viehweg and K. Zuo, Base spaces of non-isotrivial families of smooth minimal models. In Complex geometry (Göttingen, 2000), Springer-Verlag, Berlin 2002, 279–328.

[VZ03]

E. Viehweg and K. Zuo, On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds. Duke Math. J. 118 (2003), no. 1, 103–150.

Mateusz Michałek, Faculty of Mathematics and Computer Science, Jagiellonian University, 30-387 Kraków, Poland E-mail: [email protected]

Lecture notes on K3 and Enriques surfaces Notes by Sławomir Rams Shigeru Mukai

Algebraic varieties are the subject of study of algebraic geometers. The place of K3 and Enriques surfaces among them can be depicted in the following way: A L G E B R A I C V A R I E T I E S

Kodaira dimension  D 0

 f algebraic varieties of dimension  3 g

3 Calabi–Yau varieties holomorphic symplectic varieties

 f algebraic surfaces g

3 K3 surfaces: KX  0; q D 0 Enriques surfaces: 2KS  0; KS 6 0; q D 0

 f algebraic curves g

3 elliptic curves (genus 1)

Recall that both K3 and Enriques surfaces belong to the class of algebraic varieties with Kodaira dimension 0. The former satisfy KX  0, whereas the latter fulfill the conditions 2KS  0 and KS 6 0, where K is the canonical class (and q WD h1 .O/ is the irregularity). They can be seen as 2-dimensional analogues of elliptic curves. Moreover, these two kinds of surfaces are closely related to each other. For each Enriques surface S there exists a K3 surface X and a fixed-point-free involution " W X ! X such that S D X="; namely every Enriques surface is a quotient of K3 surface by a fixed-point-free involution. 

Supported in part by the JSPS Grant-in-Aid for Scientific Research (B) 22340007.

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Notation. In what follows we will keep the notation X and S to denote a K3 and an Enriques surface respectively, which are connected by the relation described above. Acknowledgement. In these lectures the author gave a short survey of [12] and [10, Appendix] on the automorphisms of K3 surfaces, and made an interim report of a work in progress toward similar results for Enriques surfaces. This note, except for §9, is an output from the TEXnotes taken by Prof. S. Rams, to whom he is very grateful. The author would like to thank Dr. H. Ohashi for his useful comments. Finally the author would like to thank the organizers for their hospitality.

1 Enriques surfaces and the Mathieu group M12 The main subject of our considerations will be around the following Conjecture 1.1. For a finite group G the following conditions are equivalent:1 [A] G has an M-semi-symplectic action on an Enriques surface, and [B] G is a subgroup of one of Gi , where i D 1; 2; 3; 4 and the data concerning the groups Gi are collected in the following table. Decomposition type of C t 

Group

Order

Root type

G1

S5

120

.1 C 5 C 6/

C

.2 C 10/

A1 C A9

G2

.Z=3/˚2 Ì D8

72

.1 C 2 C 9/

C

.6 C 6/

A5 C A5

G3

Q8 Ì S3

48

.1 C 3 C 8/

C

.4 C 8/

A3 C A7

G4

A6

360

.1 C 1 C 10/

C

.6 C 6/

A5 C A5

In the table above S5 (resp. A6 ) stands for the symmetric (resp. the alternating) group, Q8 is the quaternion group of order 8 and D8 is the dihedral group of order 8. Other necessary definitions will be explained in further sections of the paper. In particular we will sketch the proof of Theorem 1.2. The groups Gi , i D 1; 2; 4, have M-semi-symplectic actions on Enriques surfaces.

2 Mathieu groups Below we collect basic definitions and properties of Mathieu groups that we will use in the sequel. Recall that if we put  WD f1; : : : ; 24g then we have the inclusion 2 D F224  G ol; 1

See the footnote in §7 and a new conjecture in §9.

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where G ol stands for the (extended binary) Golay code. For a vector/word C WD .c1 ; : : : ; c24 / 2 F224 one puts jC j WD #fj W cj ¤ 0g and calls jC j the weight of the word/vector C . The Golay code is a 12-dimensional vector subspace of F224 with the weight enumerator X t jC j D 1 C 759  t 8 C 2576  t 12 C 759  t 16 C t 24 : C 2G ol

Every vector/word C corresponds to a subset of  via 2 D F224 . Obviously the unique word C such that jC j D 24 corresponds to the set , whereas 1 in the above equality stands for the empty set. The words of weight 8 (resp. 12) are called (special) octads (resp. (special) dodecads). The Golay code is characterized by the Steiner property St.5; 8I 24/, i.e., for every subset Q   that consists of five elements, there exists a unique octad that contains Q. In particular, the number of octads is exactly 24 58 D 759 5

as above. Let us fix a dodecad C  . We have the following definitions. Definition 2.1. M24 M23 M12 M11

WD fg W g is an even permutation of  such that g.G ol/ D G olg; WD stabilizer of the transitive action M24 Õ ; WD fg 2 M24 W g.C / D C g; WD stabilizer of the transitive action M12 Õ C :

Observe that  WD  n C is also a dodecad. Thus by definition, M12 acts simultaneously on two dodecads C and  . The action of M11 on  is transitive, while the action of M11 on C acquires a fixed point. Thus the two actions are not equivalent. Let G be a finite group. We consider the following condition: [C] The group G is embeddable into M11  M12 in such a way that G decomposes the dodecad C into at least 3 orbits and decomposes  into at least 2 orbits. In our proof of Theorem 1.2, we will use the following Fact 2.2. Condition [B] () Condition [C]:

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3 K3 surfaces and M24 We start this section with the following definition: Definition 3.1. a) A surface X is called a K3 surface if it is a compact complex manifold of dimension 2 with q D 0 and with a nowhere vanishing holomorphic 2-form !. b) We say that an automorphism g 2 Aut.X / is symplectic (resp. anti-symplectic) if g  ! D ! (resp. g  ! D !). We have Theorem 3.2 ([12, Theorem 0.3]). For a finite group G the following conditions are equivalent: [A] G has a symplectic action on some K3 surface, [B] G is embeddable into M23  M24 in such a way that G decomposes  into at least 5 orbits. As a result of the above theorem one obtains a complete classification of maximal finite groups acting symplectically on a K3 surface (see [12, Theorem 0.6]). It should be pointed out that the groups G1 , G2 , G3 , G4 appear in the list of 11 possible groups in [12]. Recall the following example from [12]: Example 3.3. We consider the Fermat quartic X4 in P 3 : x 4 C y 4 C t 4 C z 4 D 0: By adjunction formula and Lefschetz theorem it is a K3 surface. The nowhere vanishing holomorphic form on X4 can be obtained as the residue of a rational 3-form on P 3 :   d.x=t / ^ d.y=t/ ^ d.z=t / P3 ! D ResX4 : .x=t /4 C .y=t /4 C .z=t /4 C 1 One can see immediately that X4 has many symmetries. The automorphism group of X4 as a projective variety can be easily written down: Aut.X4  P 3 / D .Z=4/3 Ì S4 ; where the action of the group .Z=4/3 (resp. the symmetric group S4 ) is induced by the multiplication of the coordinates by a primitive 4-th root of unity (resp. the permutation of coordinates). It can be checked that the action of .Z=4/3 is not symplectic. More precisely, the image of the natural homomorphism Aut.X4  P 3 / ! Aut.C!/ ' C 

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393

is a cyclic group of order 4. We put F384 to denote its kernel. By definition jF384 j D .43  4Š/=4 D 384 and this group acts symplectically on X4 . In this particular case one obtains the decomposition into orbits  D q5iD1 i of the type 24 D 1 C 1 C 2 C 4 C 16; so that the first four orbits form a special octad. We have the isomorphism: F384 D fg 2 M24 W g.i / D i for all i D 1; : : : ; 5g: Basic Observation 3.4. Let g ¤ id be a symplectic automorphism of a K3 surface of order n < 1. Then we have 2  n  8, the set Fix.g/ of fixed points is finite and the following equality holds j Fix.g/j D

24 DW .n/: n  …pjn .1 C p1 /

Remark ([12, Observation (0.2)]). Let g 0 2 M23  M24 be an element of order n. Then the following equality holds j Fix.g 0 /j D .n/: Namely, the number of fixed points of g Õ X and g 0 Õ  coincide. Let S D X=" be an Enriques surface. For each g 2 Aut.S / there exists a lift gQ 2 Aut.X/. (There are two lifts of g. The other is g".) Definition 3.5. An automorphism g 2 Aut.S / is semi-symplectic if the lift gQ is either a symplectic or an anti-symplectic automorphism of the K3 surface X . It should be pointed out that the natural analogue of Basic Observation 3.4 does not always hold for semi-symplectic automorphisms of Enriques surfaces. In particular, the set Fix.g/ is not finite in general when the order of g is even.

4 K3 and Enriques surfaces There are many projective models of K3 surfaces – the quartics in P 3 among them. Let us recall the Enriques’ description of Enriques surfaces as polarized degree-6 surfaces: Denote by Sx a sextic surface mildly singular along 6 edges of the tetrahedron x  y  z  t D 0:

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## #  LL LL ##  LL ##  LL#  # LLL   ## LLLL  LL  # LL ##  LL   LL #  LL  #    L # LL  ## LL  c  ?? L c c  # ??  ## c c c c c c c c  cLLLL ??  LL c  L ?  c c c c c ##   c c c ?c??c ##    ?? ## ??     # ?   ?? ##  ??  #  ?? ##  ??   ?? ## ?? ##  ??#  ?#  ##??   # ??  ## ???  # 

x Then .S; OS .1// will be a (polarized) EnConsider the normalization  W S ! S. riques surface of degree 6 as we see. Observe that Sx is given by the equation Sx W

q.x; y; z; t /xyzt C .ay 2 z 2 t 2 C bx 2 z 2 t 2 C cx 2 y 2 t 2 C dx 2 y 2 z 2 / D 0: q: quadratic 10 monomials

4 monomials

In this way we have 14 monomials, and the 4-dimensional torus .C  /4 of diagonal matrices acts on the space of equations. So the family of surfaces has dimension 10. We have KS    .K Sx /  .the edges of the tetrahedron/ and KSx  2(plane section), that is, the canonical divisors of the normalisation are cut out by quadrics passing through the edges of the tetrahedron. This implies 2KS  0 and KS 6 0 (cf. [7, Chapter 4, §6]). More explicitly and directly, we have Proposition 4.1. S is an Enriques surface and the covering K3 surface X is a divisor of tridegree .2; 2; 2/ in P 1 P 1 P 1 which is invariant under the involution " W .u; v; w/ ! .u; v; w/; where u, v, w are inhomogeneous coordinates of three P 1 -factors.

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Proof. Sx is the image of X W q.1; vw; uw; uv/ C .au2 v 2 w 2 C bu2 C cv 2 C dw 2 / D 0 of tridegree .2; 2; 2/ in P 1 P 1 P 1 by the morphism associated with the linear system spanned by h1; vw; uw; uvi  H 0 .P 1 P 1 P 1 ; OP 1 P 1 P 1 .1; 1; 1//. Remark 4.2. By the standard Cremona transformation of P 3 , Sx is transformed to the sextic   1 1 1 1 0 x S Wq ; ; ; x 2 y 2 z 2 t 2 C .ax 2 C by 2 C cz 2 C dt 2 /xyzt D 0 x y z t of a similar kind. This is the image of the same K3 surface X by another morphism associated with hu; v; w; uvwi. So this gives the same Enriques surface S after the normalization but the polarization (of degree 6) differs from that of Sx by the 2-torsion of Pic S. Remark 4.3 (Connection with Farkas’ lecture). Under the Segre embedding X is a surface of degree 12: X  P 1 P 1 P 1 ,! P 7 : Its hyperplane section is a degree-12 canonical curve C12  P 6 of genus 7. In this way by considering the hyperplane sections of the Enriques sextic we obtain genus-4 Prym canonical curves D6 ! P 2 . Going back to automorphisms, we have the following: Let G Õ S D X=" be an action of a finite group G on an Enriques surface and let ! be a nowhere vanishing holomorphic 2-form on the K3 cover X . Then the action of G is semi-symplectic if z the pull-back of G, preserves the set f!; ! D " !g  H 0 .2 / (see and only if G, S Definition 3.5). Definition 4.4. Let G Õ S D X=" be a semi-symplectic action of a finite group G on an Enriques surface. We say that the action is M-semi-symplectic if ´ 4 for every g 2 G of order 2, 4, top .Fix.g// D 2 for every g 2 G of order 8 (cf. §9). In this definition, “M” alludes to the Mathieu group, in this case M12 , as in the following considerations. Remark 4.5. Some data concerning g 0 2 M11 are collected in the following table:

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Order of g 0

Permutation type

# of fixed points on C

1

1

12

2

.2/4

4

3

3

.3/

3

4

.4/2

4

5

.5/2

2

6

.6/.3/.2/

1

8

.8/.2/

2

From this table and the Lefschetz formula, we can see that a semi-symplectic action on an Enriques surface is M-semi-symplectic if and only if top .Fix.g// D Trace.g  Õ H .S; Q// D #.Fix.g 0 //; where g 0 2 M11 has the same order as g. Example 4.6. Let X be the quartic  P 3 with 15 nodes given in P 4 by the following equations: x C y C z C u C v D 0; 1 1 1 1 1 C C C C D 0: x y z u v Since each biregular involution of P 3 has two lines of fixed points, its restriction to a quartic has eight fixed points. Therefore, in order to obtain fixed point free involutions one has to study rational maps. Here we consider the birational involution   1 1 1 1 1 " W .x; y; z; u; v/ ! ; ; ; ; : x y z u v One can check that it defines a fixed point free involution on the quartic in question. This is a special case of the involution  in the table in §8. As the quotient one obtains an Enriques surface with root type E6 C A4 . The action of the symmetric group S5 on X defines a semi-symplectic S5 -action on S . We claim that the action of S5 is not M-semi-symplectic. Indeed, consider the automorphism g.12/ induced by the transposition .12/ 2 S5 . Then one can see that Fix.g.12/ / D .plane quartic [ 8 points/=" ;

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397

which implies that top .Fix.g.12/ // D 2 ¤ 4: On the other hand, the action of the alternative group A5 is M-semi-symplectic since, for the automorphism g.12/.34/ corresponding to the permutation .12/.34/ 2 A5 , one obtains Fix.g.12/.34/ / D .elliptic curve [ 8 points/=" ; so top .Fix.g.12/.34/ // D 4. Example 4.7. Let X be the complete intersection of three diagonal quadrics x12 C x32 C x52 D x22 C x42 C x62 ; x12 C x42 D x22 C x52 D x32 C x62 in P 5 . As is shown in [12, §2], this K3 surface has a symplectic action of one of the eleven maximal groups H192 D 24 D12 . The involution " W .x1 ; x2 ; x3 ; x4 ; x5 ; x6 / 7! .x1 ; x2 ; x3 ; x4 ; x5 ; x6 / defines a fixed point free involution and H192 acts semi-symplectically on the Enriques surface S D X=". But this action is not M-semi-symplectic. Recall that a symplectic automorphism of a K3 surface of order two is called a Nikulin involution. Such an involution has exactly 8 fixed points. The Example 4.6 shows that an analogue of the above result (i.e. top .Fix.g// D 4) does not always hold for semi-symplectic automorphisms of Enriques surfaces. This is the reason of Definition 4.4.

5 Root systems of Enriques surfaces In Conjecture 1.1 we stated the following correspondence between finite groups Gi , i D 1; 2; 3; 4 which may act semi-symplectically on an Enriques surface S and the root system of S : G1 G2 G3 G4

! A1 C A 9 ; ! A5 C A 5 ; ! A3 C A 7 ; ! A5 C A 5 :

In this section we explain what a root system of an Enriques surface is.2 2

Though defined differently, the root system of an Enriques surface was first introduced by Nikulin [16].

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The universal K3-cover of an Enriques surface X ! S induces the homomorphism of homology groups: 

H2 .X; Z/

/ H2 .S; Z/

.Z22 ; intersect. prod./

Z10 ˚ Z=2Z.

We define the rank-12 lattice H2 .S; Z! / WD ker ŒH2 .X; Z/ ! H2 .S; Z/ with the bilinear form := 12  .intersection product/. This bilinear form in question is Z-valued ([5]). Moreover, the resulting lattice is isomorphic to I2;10 – the unique odd unimodular lattice of signature .2; 10/, so in an appropriate basis ˛1 ; : : : ; ˛12 the intersection form on H2 .S; Z! / is given by the diagonal matrix: 0 1 ::: 1 0 B 0 1 C B C B C 1 B C: B :: C : : @ : A : 1

We define the period 3 of S as Z



Z ˛1

!; : : : ;

˛12

!

2 C 12 :

This vector is uniquely determined up to constant multiplication and up to the action of the orthogonal group O.2; 10I Z/ of the lattice I2;10 . The kernel of Z ! H2 .S; Z / ! C; ˛ 7! !; ˛

coincides with the kernel of the push-forward Pic X ! Pic S. This sublattice is denoted by Pic! S and called the twisted Picard lattice. By the Riemann–Roch theorem, Pic! S does not contain .1/-elements. Pic! S is an odd lattice if and only if the pullback homomorphism Br.S / ! Br.X / of the Brauer groups is zero by Beauville [4, Corollary 5.7, Lemma 5.9]. The root system R of S is the sublattice generated by .2/-elements in the twisted Picard lattice Pic! S. If C is a smooth rational curve on S, then its pull-back to X is a disjoint union of two smooth rational curves CC and C . Hence the difference 3 In the literature this period of an Enriques surface was first considered by Allcock [1] for arithmetic reason.

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399

ŒCC   ŒC  is a root of R. Roughly speaking, one has the following correspondences for a twisted 2-cycle ˛ 2 H2 .S; Z! /: ˛ 2 kerŒH2 .S; Z! / ! C 2

˛ 2 ker and .˛ / D 2

! ˛ is algebraic, ! ˛ produces a curve isomorphic to P 1 on S:

z  R of The root system R of an Enriques surface (together with an overlattice R 1 index 1 or 2) describes the configuration of P ’s on S.

6 Leech lattice and K3 surfaces In order to sketch the proof of Theorem 3.2 we recall the definition of Leech lattice below. We maintain the notation of §1, in particular  D f1; : : : ; 24g and G ol stands for the Golay code. We consider the free Z-module M Z D Zei with inner product .; / such that ei2 D 2 and .ei ; ej / D 0 for i ¤ j; i2

and define the Leech lattice ƒ  Q  to be a lattice that is commensurable with Z :4 ´ ƒ WD

1 2

P

ai ei W

μ i) all coordinates ai are even or all are odd, ii) ¹i W ai  k.mod 4/º 2 G ol for all k D 0; 1; 2; 3; : P iii) i ai  4a1 .mod 8/

The restriction of the inner product on Q  to ƒ is even (i.e. for each v 2 ƒ one has v 2 2 2Z), positive-definite and unimodular. Moreover, it has no roots: .v 2 /  4 for every v 2 ƒ; v ¤ 0: By definition, the Mathieu group M24 acts on ƒ isometrically. For a subgroup G  M24 we define the invariant sublattice and the anti-invariant one by ƒG WD fv 2 ƒ j g:v D v

for all g 2 Gg and

ƒG WD .ƒG /?  ƒ;

respectively. Recall that for a K3-surface X we have the lattice H2 .X; Z/ D Z22 with the bilinear form given by intersection numbers. After those preparations we are in position to sketch the proof of the implication (Condition [B] ) Condition [A]) in Theorem 3.2 (the proof appeared in [10, Appendix]): Sketch of the proof. We assume that G is embeddable into M23  M24 in such a way that G decomposes  into at least 5 orbits. It implies that the group G acts on the Leech lattice, and we obtain the anti-invariant sublattice ƒG . 4 For computational purpose the Niemeier lattice of type .A1 /24 is more convenient as is used in [10, Appendix].

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The sublattice ƒG has no roots and is definite. One can show that the assumption on the number of orbits yields that rank.ƒG /  19 D 24  5: The latter combined with a computation of the discriminant group and discriminant form yields a primitive embedding (recall that for a K3 surface, the signature of the intersection product on H2 .X; Z/ is .3; 19/): ƒG .1/ ,! H2 .K3; Z/: Using the Torelli theorem for K3 surfaces ([2]) one shows that G has a symplectic action on a K3 surface.

7 Finite M-semi-symplectic actions on Enriques surfaces The situation for Enriques surfaces is similar to the one for K3 surfaces except for the fact that we have to add the condition M-semi-symplectic. Recall that by Fact 2.2 the Conjecture 1.1 reads Conjecture 1.1. For a finite group G the following conditions are equivalent: [A] G has an M-semi-symplectic action on an Enriques surface; [C] G is embeddable into M11  M12 in such a way that G decomposes C (resp.  ) into at least 3 (resp. 2) orbits. Now we sketch the proof of Theorem 1.2. The details will be published elsewhere. Consider a decomposition of  into a pair of complementary (special) dodecads:  D C q  : By assumption, we have an action of the group G D G1 ; G2 or G4 : G  M11 Õ C

.resp.  /:

Now, we consider the sublattices of the Leech lattice ƒ: ƒ˙ WD ƒ \ Q ˙  Q : Obviously their (orthogonal) sum is a sublattice of the Leech lattice ƒ  ƒC C ƒ : ˙ If we put ƒ˙ G WD ƒG \ ƒ , then  ƒG  ƒC G C ƒG ;

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401

and the fact that C (resp.  ) consists of at least 3 (resp. 2) orbits implies that rank.ƒC G /  9 D 12  3

and

rank.ƒ G /  10 D 12  2:

The above inequalities and a computation of discriminant forms enable us to obtain primitive embeddings5 : 1 ƒC G . 2 /

isometric

1 ƒ G . 2 /

,!

,!

H2 .Enriques surf.; Z/ signature .1; 9/; H2 .Enriques surf.; Z! / signature .2; 10/:

Recall that, by definition, H2 .Enriques surf.; Z! / is the kernel of the map 

H2 .K3 surf.; Z/ ! H2 .Enriques surf.; Z/ signature .3; 19/ signature .1; 9/ induced by the universal K3 cover  W X ! S . Then, using the Torelli Theorem for Enriques surfaces ([8], [3], [15], [2] and [1]), one shows that G has a semi-symplectic action on an Enriques surface. The fact that the action in question is M-semi-symplectic results almost immediately from the definition. Remark 7.1. The Enriques surface S with an M-semi-symplectic action of G1 D S5 constructed above is is an Enriques surface with finite automorphism group (type VII) studied by Kondo in [9]. In this case S contains exactly 20 smooth rational curves and has root type A1 C A9 (see [ibid., Main Theorem]). Furthermore S5 is the full automorphism group of S.

8 Motivation and background The above study has been motivated by research on involutions of Enriques surfaces. The so-called Horikawa model is an important tool to understand the behaviour of involutions on Enriques surfaces (below we consider one of two Horikawa expressions in [8]). Consider a double quadric 2W1

X ! P 1 P 1 . P 3 / with branch divisor B of bi-degree .4; 4/ with only ADE singularities. By ramification formula X is a K3 surface. Assume that B is invariant under the small involution e

P 1 P 1 3 .x; y/ ! .x; y/ 2 P 1 P 1 : 5 ! For G D G3 , ƒ G .1=2/ has no primitive embeddings into H2 .Enriques surf.; Z /. The author made Conjecture 1.1 overlooking this fact. See §9.

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The involution e has two lifts to X . Let " be the anti-symplectic one. Then Fix."/ D ; (unless B passes through Fix.e/) and results in an Enriques surface S WD X=": S has an involution induced by the covering involution of X ! P 1 P 1 . The fixed point locus of is given as Fix. / D .B t .8 points//=" ;

(1)

because the involution we started from has 4 fixed points on P 1 P 1 , so we get 8 isolated fixed points on X . Now there are various cases to consider. The first extremal case is the generic one: Example 8.1. The curve B is smooth. In this case pg .B/ D 9, so the Euler number is top .B/ D 16. Therefore, from (1), we have 16 C 8 D 4: 2 Observe that the above (generic) case is parametrized by 10 moduli. top .Fix. // D

The other extreme is Example 8.2 (Barth–Peters Enriques surface). The curve B consists of a quadrangle and a smooth .2; 2/-curve running through its four vertices:

D4

D4

D4

D4

On the minimal resolution of the singularities of the double cover branched along 0 B we obtain the branch locus B D q81 P 1 C .an elliptic curve/ so 16 C 8 D 12: 2 This surface was first studied by Horikawa [8], later by Barth–Peters [3] and Mukai– Namikawa [11]. Barth–Peters [3] found out that the involution acts trivially on H2 .S; Z/. Observe that the number of moduli is 2 in this case. top .Fix. // D

403

Lecture notes on K3 and Enriques surfaces

The above examples are two extremal cases. In general, if g is an involution on an Enriques surface then 4  top .Fix.g//  12 and the condition “M-semi-symplectic” is exactly central in all possible involutions: the action of g is M-semi-symplectic , top .Fix.g// D 4 with 6 moduli. While studying the geometry of Enriques surfaces we found out that the Barth– Peters Enriques surface is characterized by the condition  R  E8  ker H2 .Enriques surf.; Z! / ! C; ˛ 7! ˛ ! : Therefore, the interplay between the geometry of Enriques surfaces and the root systems should be interesting. In the table below we collect the facts that (will) appear in various papers. Enriques surface S Gi (i D 1; 2; 4) has an M-semi-symplectic action on S

Root system of S A1 C A 9 ;

S has a cohomologically trivial involution (Example 8.2) S has an involution acting trivially on H 2 .S; Q)

A5 C A5

E8 E8 ;

E7 C A1 ;

D8

(see [13] and [9, Theorem 1.7]) S = quotient of f 1 .xt C yz/ C 2 .yt C xz/

E7

C 3 .zt C xy/g2 C 4 xyzt D 0 with four rational double points of type D4 by .x; y; z; t/ 7! .1=x; 1=y; 1=z; 1=t / S D H= , where H ´ Hessian quartic of a cubic surface P P also given by the equations 5iD1 xi D 5iD1 i =xi D 0

E6

in P 4 (see [6,  1]) and  .x1 ; : : : ; x5 / D .1=x1 ; : : : ; 1=x5 /

(E6 C A4 if all i ’s are equal)

Quotients of Jacobian Kummer surfaces by an involution ", i.e. S D Km.J.C //=" with " WD "G , where G ´ Göpel subgroup of J.C /2 " WD " , where is an even theta characteristic (see [14] for "G )

D6 C A 1 A7

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S. Mukai

To give a more precise explanation of the last row in the above table let us recall the following theorem: Theorem 8.3 (Ohashi [17]). Let C be a smooth projective curve of genus 2 such that X WD Km.J.C // is Picard general, i.e. .Km.J.C /// D 17, where Yx stands for the minimal desingularization of an algebraic surface Y . Then, the number of fixed point free involutions of X (up to conjugacy in Aut.X /) equals 31 D 25  1 D 15 C 10 C 6: Moreover, the number of the involutions of the "G -type is 15, whereas the number of the involutions of the " -type is 10. Remark 8.4. The types of fixed point free involutions in the above theorem correspond to index-2 sublattices of E7 . By removing an appropriate vertex of the extended Dynkin diagram Ez7 one obtains an index-2 root sublattice of the lattice in question. In this way one obtain two index-2 sublattices D6 C A1 and A7 . One can show that the following correspondence holds: D6 C A1 A7

! "G ; ! " :

Finally, the lattice E7 contains an index-2 sublattice L that is not a root lattice. The sublattice of L generated by roots is E6 . The involutions corresponding to the sublattice L are so-called Hutchinson–Weber ([6]) involutions, which is a special case of the second last raw of the above table.

9 New conjecture (by S. Mukai and H. Ohashi) Some progress is made on Conjecture 1.1 after the lectures. On one hand, we could construct M-semi-symplectic actions of the two groups .Z=2/3 and Z=2 Z=4 on Enriques surfaces. On the other hand, we could exclude such actions of the groups of order 16 and two groups of order 8; the cyclic group Z=8 and the quaternion group Q8 . Thus the Conjecture 1.1 does not hold true and the list of maximal groups Gi ’s in the conjecture should be modified. The following is our working hypothesis at present. Conjecture 9.1. For a finite group G the following conditions are equivalent: [A] G has an M-semi-symplectic action on an Enriques surface, and [Bnew ] G is a subgroup of G1 , G2 , G4 , Z=2 A4 or G ' Z=2 Z=4. (In particular G is a proper subgroups of the symmetric group S6 .)

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References [1] D. Allcock, The period lattice for Enriques surfaces. Math. Ann. 317 (2000), 483–488. [2] W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces. Ergeb. Math. Grenzgeb. (3) 4, Springer-Verlag, Berlin 1984. [3] W. Barth and C. Peters, Automorphisms of Enriques surfaces. Invent. Math. 73 (1983), 383–411. [4] A. Beauville, On the Brauer group of Enriques surfaces. Math. Res. Lett. 16 (2009), 1001– 1008. [5] A. Degtyarev, I. Itenberg, and V. Kharlamov, Real Enriques surfaces. Lecture Notes in Math. 1746, Springer-Verlag, Berlin 2000. [6] I. Dolgachev and J. Keum, Birational automorphisms of quartic Hessian surfaces. Trans. Amer. Math. Soc. 354 (2002), 3031–3057. [7] P. Griffiths and J. Harris, Principles of algebraic geometry. John Wiley & Sons, Inc., New York 1978. [8] E. Horikawa, On the periods of Enriques surfaces. I; II. Math. Ann. 234 (1978), 73–88; ibid. 235 (1978), 217–256. [9] S. Kondo, Enriques surfaces with finite automorphism groups. Japan. J. Math. (N.S.) 12 (1986), 191–282. [10] S. Kondo, Niemeier lattices, Mathieu groups, and finite groups of symplectic automorphisms of K3 surfaces. With an appendix by Shigeru Mukai. Duke Math. J. 92 (1998), 593–603. [11] S. Mukai, and Y. Namikawa, Automorphisms of Enriques surfaces which act trivially on the cohomology groups. Invent. Math. 77 (1984), 383–397. [12] S. Mukai, Finite groups of automorphisms of K3 surfaces and the Mathieu group. Invent. Math. 94 (1988), 183–221. [13] S. Mukai, Numerically trivial involutions of Kummer type of an Enriques surface. Kyoto Math. J. 50 (2010), 889–902. [14] S. Mukai, Kummer’s quartics and numerically reflective involutions of Enriques surfaces. J. Math. Soc. Japan. 64 (2012), 231–246. [15] Y. Namikawa, Periods of Enriques surfaces. Math. Ann. 270 (1985), 201–222. [16] V. V. Nikulin, On the description of the groups of automorphisms of Enriques surfaces. Dokl. Akad. Nauk SSSR 277 (1984), 1324–1327; English transl. Soviet Math. Dokl. 30 (1984), no. 1, 282–285. [17] H. Ohashi, Enriques surfaces covered by Jacobian Kummer surfaces. Nagoya Math. J. 195 (2009), 165–186. Shigeru Mukai, Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan E-mail: [email protected]

IMPANGA lecture notes on log canonical thresholds Notes by Tomasz Szemberg Mircea Musta¸ta˘

Introduction Let H  C n be a complex hypersurface defined by the polynomial f 2 CŒx1 ; : : : ; xn . The problem of understanding the singularities of H at a given point is classical. The topological study goes back to Milnor’s book [Mil]. In these notes, however, we will focus on an algebraic invariant, the log canonical threshold. The two best-known invariants of the singularity of f (or H ) at a point P 2 H are the multiplicity ordP .f / and the Milnor number P .f / (in the case when H has an isolated singularity at P ). They are both easy to define: ordP .f / is the smallest P ˛ j˛j with @@xf˛ .P / ¤ 0, where ˛ D .˛1 ; : : : ; ˛n / 2 Zn0 and j˛j D niD1 ˛i . If H is nonsingular in a punctured neighborhood of P , then P .f / D dimC OC n ;P =.@f =@x1 ; : : : ; @f =@xn /: Note that both these invariants are integers. They both detect whether P 2 H is a singular point: this is the case if and only ordP .f /  2, and (assuming that H is nonsingular in a punctured neighborhood of P ) if and only if P .f / > 0. In general, the more singular H is at P , the larger the multiplicity and the Milnor number are. In order to get a feeling for the behavior of these invariants, note that if f D x1a1 C  Cxnan , we have n Y ord0 .f / D min ai ; 0 .f / D .ai  1/: 1in

iD1

The Milnor number and other related information (such as the cohomology of the Milnor fiber, the monodromy action on this cohomology etc) play a fundamental role in the topological approach to singularities. However, this aspect will not feature much in these notes. The multiplicity, on the other hand, is a very rough invariant. Nevertheless, it can be very useful: maybe its most spectacular application is in resolution of singularities (see [Kol2]), where it motivates and guides the resolution process. The log canonical threshold lctP .f / of f at P is an invariant that, as we will explain in §1, can be thought of as a refinement of the reciprocal of the multiplicity. In order to compare its behavior with that of the multiplicity ˚andP of the Milnor number,  a1 n an 1 we note that if f D x1 C    C xn , then lct0 .f / D min 1; iD1 ai . Several features of the log canonical threshold can be seen on this example: in general, it is a rational number, it is bounded above by 1 (in the case of hypersurfaces), and it has roughly the same size as 1= ordP .f / (see §1 for the precise statement). If H

408

M. Musta¸ta˘

is nonsingular at P , then lctP .f / D 1. However, we may have lctP .f / D 1 even when P 2 H is a singular point: consider, for example, f D x 3 C y 3 C z 3 2 CŒx; y; z. The log canonical threshold first appeared implicitly in the paper of Atiyah [Ati], in connection with complex powers. In this paper Atiyah proved the following conjecture of Gelfand. Given f as above, one can easily see that for every s 2 C with Re.s/ > 0 one has aRdistribution on C n that takes a C-valued smooth function with compact support ' to C n jf .z/j2s '.z/dzd z. N I. M. Gelfand conjectured that this can be extended to C as a meromorphic map with values in distributions, and Atiyah proved1 that this is the case using resolution of singularities2 . His proof also shows, with current terminology, that the largest pole is bounded above by  lct.f /, where lct.f / D minP 2H lctP .f /. The first properties of the log canonical threshold (known at the time as the complex singularity exponent) have been proved by Varchenko in connection with his work on asymptotic expansions of integrals (similar to the integral we have seen above), and mixed Hodge structures on the vanishing cohomology, see [Var1], [Var2], and [Var3]. In this context, the log canonical threshold appears as one of the numbers in the spectrum of the singularity, a set of invariants due to Steenbrink [Ste]. It was Shokurov who introduced the log canonical threshold in the context of birational geometry in [Sho]. In this setting, one thinks of lctP .f / as an invariant of the pair .C n ; H /, giving the largest  > 0 such that the pair .C n ; H / is log canonical in some neighborhood of P (which explains the name). We mention that the notion of log canonical pairs is of central importance in the Minimal Model Program, since it gives the largest class of varieties for which one can hope to apply the program. In fact, in the context of birational geometry one does not require that the ambient variety is nonsingular, but only that it has mild singularities, and it is in this more general setting that one can define the log canonical threshold. Shokurov made a surprising conjecture, which in the setting of ambient nonsingular varieties asserts that the set of all log canonical thresholds lctP .f /, for f 2 CŒx1 ; : : : ; xn  with n fixed, satisfies ACC, that is, it contains no strictly increasing infinite sequences. The expectation was that a positive answer to this conjecture (in the general setting of possibly singular varieties) would be related to the so-called Termination of Flips conjecture in the Minimal Model Program, and Birkar showed such a relation in [Bir]. For more on this topic, see §3 below. Meanwhile, it turned out that the log canonical threshold came up in many other contexts having to do with singularities. The following is an incomplete list of such occurrences, but which can hopefully give the reader a feeling for the ubiquity of this invariant. • In the case of a polynomial f 2 ZŒx1 ; : : : ; xn  and of a prime p, the log canonical threshold of f is related to the rate of growth of the number of solutions of f in Z=p m . 1 At the same time, an independent proof of the same result, based on the same method, was given in [BG]. 2 In fact, Atiyah’s paper and Gelfand’s conjecture were in the context of polynomials with real coefficients. We have stated this in the complex case, since it then relates to what we will discuss in §1. For a treatment of both the real and the complex case, see [Igu].

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This is related to a p-adic analogue of the complex powers discussed above, see [Igu]. • Yet another integration theory (motivic integration) allows one to relate the log canonical thresholds to the rate of growth of the dimensions of the jet schemes of X , see [Mus2]. • The Bernstein polynomial of f is an invariant of the singularities of f that comes out of the theory of D-modules. The negative of the log canonical threshold is the largest root of the Bernstein polynomial, see [Kol3]. • Tian’s ˛-invariant is an asymptotic version of the log canonical threshold that provides a criterion for the existence of Kähler–Einstein metrics (see, for example [Tian], [DK] and [CS]). • The log canonical threshold appears implicitly or explicitly in many applications of vanishing theorems, due to its relation to multiplier ideals (see [Laz, Chapter 9]). An important example is the work of Angehrn and Siu [AS] on the global generation of adjoint line bundles. • Lower bounds for the log canonical threshold also come up in proving a strong version on non-rationality for certain Fano varieties of index one (for example, for hypersurfaces of degree n in Pn ). This is a point of view due to Corti [Cor] on the classical approach to the non-rationality of a quartic threefold of Iskovskikh and Manin [IM]. See for example [dFEM4] for an application of this point of view. The present notes are based on a mini-course I gave at the IMPANGA Summer school, in July 2010. The goal of the lectures was to introduce the log canonical threshold, and present some open problems and recent results related to it. I have tried to preserve, as much as possible, the informal character of the lectures, so very few complete proofs are included. In the first section we discuss the definition and some basic properties of the log canonical threshold, as well as some examples. The second section is devoted to an analogous invariant that comes up in commutative algebra in positive characteristic, the F -pure threshold. While defined in an entirely different way, using the Frobenius morphism, it turns out that this invariant is related in a subtle way to the log canonical threshold via reduction mod p. In Section 3 we discuss a recent joint result with T. de Fernex and L. Ein [dFEM2], proving Shokurov’s ACC conjecture in the case of ambient smooth varieties. We do not present the details of the proof, but rather describe following [dFM] a key ingredient of the proof, the construction of certain “limit power series” associated to a sequence of polynomials. The last section discusses following [JM] an asymptotic version of the log canonical threshold in the context of graded sequences of ideals, and a basic open question concerning this asymptotic invariant. The content of the first three sections follows roughly the three Impanga lectures, while the topic in the fourth section is a subsequent addition, that did not make it into the lectures because of time constraints. Acknowledgment. I am indebted to the organizers of the IMPANGA Summer school for the invitation to give this series of lectures and for the encouragement to publish the lecture notes. Special thanks are due to Tomasz Szemberg for the detailed notes he

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took during the lectures. During the preparation of this paper I was partially supported by NSF grant DMS-0758454 and by a Packard Fellowship.

1 Definition and basic properties In this section, we will work in the following setting. Let X be a nonsingular, irreducible, complex algebraic variety and a  OX a nonzero (coherent) ideal sheaf (often assumed to be principal). Since we are only interested in local aspects, we may and will assume that X D Spec R. Let P 2 V .a/ be a fixed closed point and mP the corresponding ideal. We refer to a regular system of parameters of OX;P as local coordinates at P . By a divisor over X we understand a prime divisor E on some model Y over X , that is, a nonsingular variety Y having a projective, birational morphism Y ! X . Every such divisor determines a valuation of the function field C.Y / D C.X / that is denoted by ordE . Explicitly, if f 2 R defines the divisor D on X , then ordE .f / is the coefficient of E in   .D/. We also put ordE .a/ D minfordE .f / j f 2 ag. The image of E on X is the center cX .E/ of E on X . We identify two divisors over X if they correspond to the same valuation. The multiplicity (or order) of a at P is the largest r 2 Z0 such that a  mPr . Of course, we have ordP .a/ D minf 2a ordP .f /. It is an easy exercise, using the Taylor expansion, to show that if x1 ; : : : ; xn are local coordinates at P , then ordP .f / is the ˛ smallest j˛j such that @@xf˛ .P / is nonzero. We can rephrase the definition of the order, as follows. If BlP .X / ! X is the blow-up of X at P , and F is the exceptional divisor, then ordP .a/ D ordF .a/. When defining the log canonical threshold we consider instead all possible divisors over X , not just F . On the other hand, we need to normalize somehow the values ordE .a/, as otherwise these are unbounded. This is done in terms of log discrepancies. Consider a projective birational morphism  W Y ! X of smooth, irreducible, n-dimensional varieties. We have the induced sheaf morphism   X ! Y which induces in turn the nonzero morphism n n   X ! nY D   X ˝ OY .KY =X /;

for some effective divisor KY =X , the relative canonical divisor, also known as the discrepancy of . Let us show that Supp.KY =X / is the inverse image of a closed subset Z of X of codimension  2, such that  is an isomorphism over X X Z (hence the support of KY =X is the exceptional locus of ). Indeed, it follows from definition that Y X Supp.KY =X / is the set of those y 2 Y such that  is étale at y (in which case, y is clearly isolated in  1 ..y//). On the other hand, since  is birational and X is normal, we have  .OY / D OX , and Zariski’s Main Theorem implies that all

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fibers of  are connected. In particular, if y 62 Supp.KY =X /, then  1 ..y// D fyg. This implies that Supp.KY =X / is the inverse image of a subset Z (which is closed in X since  is proper). Using the fact that  is a homeomorphism over X X Z and  .OY / D OX , we deduce that  is an isomorphism over X X Z. Since X is normal and Y is proper over X , it follows that  1 is defined in codimension one, which easily implies codim.Z; X /  2. Given a divisor E over X lying on the model Y over X , the log discrepancy of E is Logdisc.E/ WD 1 C ordE .KY =X /. It is easy to see that the definition is independent on the particular model Y we have chosen. The Arnold multiplicity of the nonzero ideal a at P 2 V .a/ is defined as ArnP .a/ D sup E

ordE .a/ ; Logdisc.E/

(1)

where the supremum is over the divisors E over X such that P 2 cX .E/. Note that we may consider the Arnold multiplicity as a more subtle version of the usual multiplicity. The log canonical threshold is the reciprocal of the Arnold multiplicity: lctP .a/ D 1=ArnP .a/. It is clear that ordE .a/ > 0 if and only if cX .E/ is contained in V .a/. By taking any divisor E with center P , we see that ArnP .a/ is positive, hence lctP .a/ is finite. We make the convention that lctP .a/ D 1 if P 62 V .a/. We will see in Property 1.18 below that since a is assumed nonzero, we have lctP .a/ > 0. Intuitively, the worse a singularity is, the higher the multiplicities ordE .a/ are, and therefore the higher ArnP .a/ is, and consequently the smaller lctP .a/ is. We will illustrate this by some examples in §1.2 below. 1.1 Analytic interpretation and computation via resolution of singularities. What makes the above invariant computable is the fact that it can be described in terms of a log resolution of singularities. Recall that a projective, birational morphism  W W ! X , with W nonsingular, is a log resolution of a if the inverse image a  OW is the ideal of a Cartier divisor D such that D C KY =X is a divisor with simple normal crossings. This means that at every point Q 2 W there are local coordinates y1 ; : : : ; yn such that D C KY =X is defined by .y1˛1 : : : yn˛n /, for some ˛1 ; : : : ; ˛n 2 Z0 . It is a consequence of Hironaka’s theorem on resolution of singularities that log resolutions exist in characteristic zero. Furthermore, since X is nonsingular, whenever it is convenient we may assume that  is an isomorphism over the complement of V .a/. The following theorem, that can be viewed as a finiteness result, is fundamental for working with log canonical thresholds. Theorem 1.1. Let f W W ! X be a log resolution of a, and consider a divisor with P simple normal crossings N iD1 Ei on W such that if a  OW D OW .D/, then we may write N N X X ai Di and KW=X D ki Ei : DD iD1

iD1

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In this case, we have lctP .a/ D

ki C 1 : ai iWP 2.Ei / min

(2)

One can give a direct algebraic proof of the above theorem: since every divisor over X appears on some log resolution of a, the assertion in the theorem is equivalent with the fact that the expression in (2) does not depend on the choice of resolution. For the proof of this statement, see [Laz, Theorem 9.2.18]. We prefer to give a different argument, involving an analytic description for the log canonical threshold. The advantage of this result is that it provides some more intuition for the log canonical threshold, making also the connection with the way it first appeared in the context of complex powers mentioned in the Introduction. Theorem 1.2. If a D .f1 ; : : : ; fr / is a nonzero ideal on the smooth, irreducible, complex affine algebraic variety X D Spec R, for every point P 2 X we have n o 1 is integrable around P : lctP .a/ D sup s > 0 j Pr  2 s iD1 jfi j Sketch of proof of Theorems 1.2 and 1.1. The assertions in both theorems follow if we show that given a log resolution  W W ! X of a as in Theorem 1.1, we have ki C 1 for all i with P 2 .Ei /: ai iD1 jfi (3) Let us choose local coordinates z1 ; : : : ; zn at P . Of course, for integrability questions we consider the corresponding structure of complex manifold on X . In particular, we say that a positive real function h is integrable around RP if for some open subset (in the classical topology) U  X containing P , we have U h dzd zN < 1 (it is easy to see that this is independent of the choice of coordinates). The key point is that the change of variable formula implies Z Z 1 1 N (4) Pr s dzd zN D Pr    .dz/  .d z/: 2 2 s 1 .U / jf j jf ı j U  i i iD1 iD1 Pr

1

s j2

is integrable around P iff s <

This is due to the fact that there is an open subset V  X such that  is an isomorphism over V , and U X V  U and  1 .U / X  1 .V /   1 .U / are proper closed analytic subsets, thus have measure zero. S It is easy to see that given a finite open cover  1 .U / D j Vj , the finiteness of the right-hand side of (4) is equivalent to the finiteness of the integrals of the same function on each of the Vj . Suppose that on Vj we have coordinates y1 ; : : : ; yn with the following properties: KVj =X is defined by .y1k1 : : : ynkn / and a  OVj is generated by y1a1 : : : ynan . Since  is a log resolution, we see that we may choose a cover as above, such that on each Vj we can find such a system of coordinates.

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Thus on Vj we can write fi ı  D ui y1a1    ynan ; for some regular functions u1 ; : : : ; ur on Vj , with no common zero. We also see that N   .dz/  .d zN / D wy12k1 : : : yn2kn dyd y; for some invertible regular function w on Vj . We conclude that Z Vj

Pr

1

2 iD1 jfi ı j

s   .dz/  .d zN / D

Z Vj

Pr

w

2 iD1 jui j

s

n Y

jyi j2ki 2sai dyd y: N

iD1

(5) Since  is proper,  1 .K/ is compact for every compact subset K of X . One can show Sj is compact, that by a suitableP choice of U and of the Vj , we may assume that each V r 2 S and both w and iD1 jui j extend to invertible functions on Vj . In particular, the right-hand side of (5) is finite of and only if Z

n Y Vj iD1

jyi j2ki 2sai dyd yN < 1:

(6)

R On the other hand, it is well known that U 0 jzj˛ dzd zN < 1 for some neighborhood of the origin U 0  C if and only if ˛ > 2. This implies via Fubini’s theorem that (6) holds if and only if 2ki  2sai > 2 for all i . Since we are allowed to replace U by a small neighborhood of P , the ki and ai that we see in the above conditions when we vary the Vj correspond precisely to those divisors Ei whose image contains P . We thus get the formula (3). Remark 1.3. One consequence of Theorem 1.1 is that lctP .a/ is a rational number. Note that the definition of the log canonical threshold makes sense also in positive characteristic, but the rationality of the invariant in not known in that context. There is also a global version of the log canonical threshold and of Arnold multiplicity: lct.a/ D min lctP .a/ and Arn.a/ D max ArnP .a/: P 2X

P 2X

With the notation in Theorem 1.1, we see that lct.a/ D min i

ki C 1 : ai

By definition, lct.a/ is infinite if and only if a D OX . Note also that we have lctP .a/ D maxU 3P lct.U; ajU /, where U varies over the open neighborhoods of P .

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1.2 Examples of log canonical threshold computations. In this subsection we collect some easy examples of log canonical thresholds. For details and further examples, we refer to [Laz, Chapter 9]. Example 1.4. Suppose that a D .f / is the ideal defining a nonsingular hypersurface. In this case, the identity map on X gives a log resolution of a, hence by Theorem 1.1 we have lctP .f / D 1 for every P 2 V .f /. Example 1.5. More generally, suppose that a is the ideal defining a nonsingular subscheme Z of pure codimension r. The blow-up W ! X of X along Z gives a log resolution of a, with KW=X D .r  1/E, where E is the exceptional divisor (check this!). It follows from Theorem 1.1 that lctP .a/ D r for every P 2 Z. In particular, if mP is the ideal defining P , we see that lctP .mP / D dim.X /. P Example 1.6. If f 2 O.X / is such that the divisor of f is riD1 ai Di , then by taking E D Di in the definition of the log canonical threshold, we conclude that if P 2 V .f /, then 1 lctP .f /  min  1: iWP 2Di ai Example 1.7. Suppose that f 2 CŒx; y has a node at P . In this case the blow-up W of A2 at P gives a log resolution of f in some neighborhood of P , and the inverse image of V .f / is D C E, where D is the proper transform, and E is the exceptional divisor. Since KW=A2 D E, it follows from Theorem 1.1 that lctP .f / D 1. Example 1.8. Let f 2 CŒx1 ; : : : ; xn  be a homogeneous polynomial of degree d , having an isolated singularity at the origin. If  W W ! An is the blow-up of the origin, and E is the exceptional divisor, then KW=An D .n  1/E and f  OW D O.D  dE/, where D is the proper transform of V .f /. Note that we have an isomorphism E ' Pn1 such that D \ E is isomorphic to the projective hypersurface defined by f , hence it is nonsingular. Therefore D C E is a divisor with simple normal crossings, and we see that  is ˚ a log  resolution of .f /. It follows from Theorem 1.1 that lct.f / D lct0 .f / D min 1; dn . Example 1.9. Suppose that a  CŒx1 ; : : : ; xn  is a proper nonzero ideal generated by monomials. For u 2 Zn0 , we write x u D x1u1    xnun . Given u D .u1 ; : : : ; un / and P v D .v1 ; : : : ; vn / in Rn , we put hu; vi D niD1 ui vi . The Newton polyhedron of a is   P .a/ D convex hull fu 2 Zn0 j x u 2 ag : Howald showed in [How] that lct.a/ D lct0 .a/ D maxf 2 R0 j .1; : : : ; 1/ 2   P .a/g: This follows rather easily using some basic facts about toric varieties (for these facts, see [Ful]). Indeed, if we consider the standard toric structure on An , the fact that

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a is generated by monomials says precisely that the .C  /n -action on An induces an action on the closed subscheme defined by a. By blowing up An along a, and then taking a toric resolution of singularities, we see that we can find a projective, birational morphism of toric varieties  W W ! X that gives a log resolution of a (indeed, in this case both KW=X and the divisor corresponding to a  OW are toric, hence have simple normal crossings, since W is nonsingular). Theorem 1.1 implies that in the definition of the log canonical threshold it is enough to consider torus invariant divisors on toric varieties Y having projective, birational, toric morphisms to X . Every such divisor E corresponds to a primitive nonzero integer vector v D .v1 ; : : : ; vn / 2 Zn0 such that ordE .a/ D minfhu; vi j u 2 P .a/g

Logdisc.E/ D v1 C    C vn : P Therefore lct.a/ is equal to the largest  such that niD1 vi    minu2P .a/ hu; vi for every v 2 Zn0 primitive and nonzero (equivalently, for every v 2 Qn0 ). It is then easy to see that this is equivalent to .1; : : : ; 1/ 2   P .a/. For example, suppose that a D .x1a1 ; : : : ;xnan /. It follows from definition that ˚ P P .a/ D .u1 ; : : : ; un / 2 Rn0 j niD1 uaii  1 . Howald’s formula gives in this case P lct 0 .a/ D niD1 a1i . P Example 1.10. Let a D .f1 ; : : : ; fr /, and consider f D riD1 i fi , where 1 ; : : : ; r are general complex numbers. Consider a log resolution  W W ! X of a that is an isomorphism over X X V .a/, and write a  OW D OW .D/. In this case f  OW D OW .D  F /, for some divisor F , and it is an easy consequence of Bertini’s theorem that F is nonsingular and F C D has simple normal crossings. If we write DD

N X iD1

ai Ei

and

and

KW=X D

N X

ki Ei ;

iD1

then ordEi .F / D 0 if ai > 0, and we have ai 2 f0; 1g for all i . Since  is an isomorphism over the complement of V .a/, it follows that ki D 0 if ai D 0. We then conclude from Theorem 1.1 that lctP .f / D minflctP .a/; 1g. Example 1.11. Let f D x1a1 C    C xnan , and consider a D .x1a1 ; : : : ; xnan /. Given any nonzero 1P ; : : : ; n , there is an isomorphism of An (leaving the origin fixed) a that ˚takes f to  niD1 i xi i . It follows from Examples 1.9 and 1.10 that lct0 .f / D Pn 1 min 1; iD1 ai . 1.3 Basic properties. We give a brief overview of the main properties of the log canonical threshold. For some applications of the log canonical threshold in birational geometry we refer to the survey [EM]. Property 1.12. If a  b are nonzero ideals on X , then lctP .a/  lctP .b/ for every P 2 X. Indeed, the hypothesis implies that ordE .a/  ordE .b/ for every divisor E over X.

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Property 1.13. We have lctP .ar / D lctPr.a/ for every r  1. Indeed, for every divisor E over X we have ordE .ar / D r  ordE .a/. Property 1.14. For every ideal a on X , we have lctP .a/  ordPn .a/ , where n D dim.X / (note that by convention, both sides are infinite if P 62 V .a/). The assertion follows from the fact that if r D ordP .a/ (which we may assume to be positive), then a  mPr , where mP is the ideal defining P . Using Example 1.5 and Properties 1.12 and 1.13, we conclude lctP .mP / n D : lctP .a/  lctP .mPr / D r r Property 1.15. If aN is the integral closure of a, then lct.a/ D lct.a/ N (see [Laz, §11.1] for definition and basic properties of integral closure). The key point is that for every N divisor E over X , we have ordE .a/ D ordE .a/. Property 1.16. If a and b are ideals on X , then Arn.a  b/  Arn.a/ C Arn.b/:

(7)

Indeed, for every divisor E over X we have ordE .a  b/ ordE .a/ ordE .b/ D C  Arn.a/ C Arn.b/: Logdisc.E/ Logdisc.E/ Logdisc.E/ By taking the maximum over all E, we get (7). Property 1.17. If H  X is a nonsingular hypersurface such that a  OH is nonzero, then lctP .a  OH /  lctP .a/ for every P 2 H . Note that this is compatible with the expectation that the singularities of a are at least as good as those of a  OH . This is one of the more subtle properties of log canonical thresholds, that is known as Inversion of Adjunction. It can be proved using either vanishing theorems (see [Laz, Theorem 9.5.1]), or the description of the log canonical threshold in terms of jets schemes (see [Mus2, Proposition 4.5]). More generally, if Y ,! X is a nonsingular closed subvariety such that a  OY is nonzero, then lctP .a  OY /  lctP .a/ for every P 2 Y . This follows by a repeated application of the codimension one case, by realizing Y in some neighborhood of P as H1 \    \ Hr , where r D codimX .Y / (note that in this case each H1 \    \ Hi is nonsingular at the points in Y ). Property 1.18. For every point P 2 X , we have lctP .a/  ordP1 .a/ . This is proved by induction on dim.X / using Property 1.17. Indeed, if dim.X / D 1 and t is a local coordinate at P , then around P we have a D .t r /, where r D ordP .a/, while lctP .a/ D 1=r. For the induction step, note that if x1 ; : : : ; xn are local coordinates at P , and if H is defined by 1 x1 C    C n xn , with 1 ; : : : ; n 2 C general, then H is nonsingular at P , and ordP .a/ D ordP .a  OH /, while lctP .a/  lctP .a  OH /.

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Property 1.19. If X and Y are nonsingular varieties, and a and b are nonzero ideals on X and Y , respectively, then lct .P;Q/ .p 1 .a/ C q 1 .b// D lctP .a/ C lct Q .b/ for every P 2 X and Q 2 Y , where p W X  Y ! X and q W X  Y ! Y are the canonical projections. This can be proved either as a consequence of the Summation Formula for multiplier ideals (see [Laz, Theorem 9.5.26]) or using the description of the log canonical threshold in terms of jet schemes (see [Mus2, Proposition 4.4]). Property 1.20. If a and b are ideals on X , then lctP .a C b/  lctP .a/ C lctP .b/ for every P 2 X. Indeed, we may apply Property 1.17 (in its general form) to the subvariety X ,! X  X, embedded diagonally. Indeed, using also Property 1.19 we get lctP .a C b/  lct.P;P / .p 1 .a/ C q 1 .b// D lctP .a/ C lctP .b/: Property 1.21. If mP is the ideal defining a point P 2 X , and a C mPN D b C mPN , then n j lctP .a/  lctP .b/j  ; N where n D dim.X /. Indeed, using Properties 1.12, 1.20 and 1.13, we obtain lctP .b/  lctP .b C mPN / D lctP .a C mPN /  lctP .a/ C lctP .mPN / D lctP .a/ C

n : N

By symmetry, we also get lctP .a/  lctP .b/ C Nn . In particular, if fN 2 CŒx1 ; : : : ; xn  is the truncation of f up to degree  N , then j lct0 .f /  lct 0 .fN /j  N nC1 . Property 1.22. Suppose that a is an ideal supported at a point on the smooth ndimensional complex variety X . In this case we have the following inequality relating the Hilbert-Samuel multiplicity e.a/ of a to the log canonical threshold: e.a/ 

nn : lct.a/n

(8)

This is proved in [dFEM3] by first proving a similar inequality for length: `.OX =a/ 

nn : nŠ lct.a/n

(9)

This in turn follows by considering a Gröbner deformation of a to a monomial ideal, for which the inequality follows from the combinatorial description of both `.OX =a/ and lct.a/.

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Suppose, for example, that a D .x1a1 ; : : : ; xnan /  CŒx1 ; : : : ; xn . It is easy to see, using the definition, that e.a/ D a1    an , while Example 1.9 implies that lct.a/ D P n 1 iD1 ai . Therefore the inequality (8) becomes Pn

1 iD1 ai

n



1 ; .a1    an /1=n

that is, the inequality between the arithmetic mean and the geometric mean. Property 1.23. Suppose that U is an affine variety, and a  O.U /Œx1 ; : : : ; xn  is an ideal contained in .x1 ; : : : ; xn /. For every t 2 U , we consider a t  C.t /Œx1 ; : : : ; xn  ' CŒx1 ; : : : ; xn . There is a disjoint decomposition of U into finitely many locally closed subsets Z1 ; : : : ; Zd , and ˛1 ; : : : ; ˛d such that for every t 2 Zi we have lct 0 .a t / D ˛i . Indeed, if  W Y ! U  An is a log resolution of a, then it follows from Generic Smoothness that there is an open subset U 0  U such that for every t 2 U 0 , if Y t is the fiber of Y over t , the induced morphism  t W Y t ! An gives a log resolution of a t in a neighborhood of 0. In particular, lct0 .a t / is independent of t 2 U 0 . After repeating this argument for an affine cover of U X U 0 , we obtain the desired cover. Property 1.24. A deeper property is the semicontinuity of the log canonical threshold. This says that in the context described in Property 1.23, for every t 2 U , there is an open neighborhood W of t such that lct0 .a t 0 /  lct 0 .a t / for every t 0 2 W . This was first proved by [Var1]. For other proofs, see [Laz, Corollary 9.5.39], [DK, Theorem 3.1] and [Mus2, Theorem 4.9]. Property 1.25. Suppose now that we are in the context of Property 1.23, but a D .f / is a principal ideal, such that for every t 2 U , the polynomial f t has an isolated singularity at 0. If U is connected and the Milnor number .f t / is constant for t 2 U , then also the log canonical threshold lct0 .f t / is constant. The only proof for this fact is due to Varchenko [Var3]. It relies on the fact that the log canonical threshold is one of the numbers in the spectrum of the singularity. One shows that all the spectral numbers satisfy a semicontinuity property analogous to Property 1.24. Since the sum of the spectral numbers is the Milnor number, and this is constant, these spectral numbers, and in particular the log canonical threshold, are constant. 1.4 The connection with multiplier ideals. A natural setting for studying the log canonical threshold is provided by multiplier ideals. In what follows we only give the definition and explain the connection with the log canonical threshold. For a thorough introduction to the theory of multiplier ideals, we refer to [Laz, Chapter 9]. As above, we consider a nonsingular, irreducible, affine complex algebraic variety X D Spec R. Let a D .f1 ; : : : ; fr / be a nonzero ideal on X . For every  2 R0 , the multiplier ideal J.a / consists of all h 2 R such that for every divisor E over X , we have ordE .h/ >   ordE .a/  Logdisc.E/: (10)

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In fact, in analogy with Theorem 1.1, one can show that it is enough to consider only those divisors E lying on a log resolution of a. One also has the following analytic description of multiplier ideals: h 2 J.a /

iff

Pr

jhj2

iD1

jfi j2

 is locally integrable:

Again, one can prove both these statements at the same time, arguing as in the proof we have sketched for Theorems 1.1 and 1.2. Since we only need to check conditions given by finitely many divisors, it is easy to show that the definition commutes with localization at a nonzero element in R, hence we get in this way coherent ideals on X. We have am  J.am / for every m 2 Z0 . It is clear from definition that if  < , then J.a /  J.a /. Furthermore, since it is enough to check the condition (10) for only finitely many divisors E, it follows that given any , there is " > 0 such that J.a / D J.at / for every t with   t   C ". 0 A positive  is a jumping number of a if J.a / ¤ J.a / for every 0 < . Note that this is the case if and only if there is h 2 J.a / and a divisor E over X such that ordE .h/ C Logdisc.E/ D   ordE .f /: In particular, it follows that all jumping numbers are rational. Furthermore, since we may consider only the divisors lying on a log resolution of a, the denominators of the jumping numbers are bounded, hence the set of jumping numbers is a discrete set of rational numbers. / By definition, J.a / D OX if and only if  < Logdisc.E for all divisors E, that ordE .a/ is,  < lct.a/. Therefore the smallest jumping number is the log canonical threshold lct.a/. The properties of the log canonical threshold discussed in the previous subsection have strengthening at the level of multiplier ideals. We refer to [Laz, Chapter 9] for this circle of ideas. If a D .f / is a principal ideal, then it is easy to see that for every   1 we have J.f  /  .f / (consider the condition in the definition when E runs over the irreducible components of V .f /). Furthermore, it follows from definition that f h 2 J.f  / if and only if h 2 J.f 1 /, hence J.f  / D f  J.f 1 / for every   1. In particular, this implies that   1 is a jumping number if and only if   1 is a jumping number. A deeper fact, known as Skoda’s theorem, says that for every ideal a, we have J.a / D a  J.a1 /

(11)

for every   n D dim.X /. The proof of this fact uses vanishing theorems, see [Laz, Chapter 9.6.C]. The name is due to the fact that (11) easily implies the theorem of Briançon–Skoda [BS]. Indeed, since every multiplier ideal is integrally closed (this is an immediate consequence of the definition (10)), the integral closure of an is contained in a: an  J.an / D J.an /  a:

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It is interesting to note that while the proof in [BS] relies on some analytic results obtained by Skoda via L2 methods, and the proof in [Laz] makes use of vanishing theorems, another proof of the Briançon–Skoda theorem was obtained by Hochster and Huneke in [HH] via characteristic p methods. We now turn to a different instance of such a connection between these three circles of ideas.

2 Connections with positive characteristic invariants In this section we describe an invariant defined in positive characteristic using the Frobenius morphism, the F -pure threshold. As we will see, this invariant satisfies properties similar to those of the log canonical threshold, and it is related with this one in a subtle way via reduction mod p. The F -pure threshold has been introduced by Takagi and Watanabe [TW] when the ambient variety is fairly general. In what follows we will focus on the case of ambient nonsingular varieties, in which case we can use a more direct asymptotic definition, following [MTW]. Let k be a perfect3 field of positive characteristic p. We consider a regular, finitely generated algebra R over k, and let X D Spec R. We denote by F W R ! R the Frobenius morphism on R that takes u to up . Note that since k is perfect (or, more generally, when k is F -finite), the morphism F is finite. Since R is nonsingular, F is also flat. Indeed, it is enough to show that the induced morphism on the completion OX;Q is flat for every Q 2 X ; since this local ring is isomorphic to k.Q/ŒŒx1 ; : : : ; xr , where k.Q/ is the residue field of Q, the Frobenius morphism is easily seen to be flat. Therefore R is projective as an R-module via F . Let a  R be a nonzero ideal, and P 2 V .a/ a closed4 point defined by the maximal ideal mP  R. Before defining the F -pure threshold, let us consider the following description of ordP .a/ (which also works in characteristic zero). For every integer r  1, let ˛.r/ WD largest i such that ai 6 mPr :

1

The condition ai 6 mPr is satisfied precisely when i  ordP .a/ < r, hence l m r ˛.r/ D  1: ordP .a/ D ordP1 .a/ . Therefore we have limr!1 ˛.r/ r We get the F -pure threshold by a similar procedure, replacing the usual powers of mP by Frobenius powers. Recall that for every ideal I and every e  1  e  e I Œp  D hp jh 2 I : A more natural condition in this context is the weaker condition that k is F -finite, that is, Œk W k p  < 1. The restriction to closed points does not play any role. We make it in order for some statements to parallel those in §1. 3 4

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If I is generated by h1 ; : : : ; hr , then

 e  e I Œp  D hpi j1  i  r :

For an integer e  1, let e

.e/ WD largest i such that ai ª mPŒp  : Note that since a  mP , each .e/ is finite. Whenever a is not understood from the e context, we write a .e/ instead of .e/. By definition, there exists h 2 a.e/ X mPŒp  . eC1

Since the Frobenius morphism on R is flat, we get hp 2 ap.e/ X mPŒp  , hence .e C 1/  p  .e/. It follows that supe1 .e/ D lime!1 .e/ , and this limit is pe pe the F -pure threshold of a at P , denoted by fptP .a/. We make the convention that fptP .a/ D 1 if P does not lie in V .a/.

2.1 Examples of computations of F -pure thresholds. We now give some easy examples of F -pure thresholds. The reader can compare the resulting values with the corresponding ones for log canonical thresholds in characteristic zero. Example 2.1. If dim.X / D n, then fptP .mP / D n. In fact, for every e  1 we have .e/ D .p e  1/n. Indeed, it is easy to check that if x1 ; : : : ; xn are local coordinates at e e e e P , then .x1 : : : xn /p 1 62 mPŒp  , but mP.p 1/nC1  mPŒp  . More generally, one can show that if a defines a nonsingular subvariety of codimension r at P , then .e/ D r.p e  1/ for every e  1, hence fptP .a/ D r. Example 2.2. It is a consequence of [HY, Theorem 6.10] that if a  kŒx1 ; : : : ; xn  is an ideal generated by monomials, then the F -pure threshold is given by the same formula as the log canonical threshold (see Example 1.9 above for the notation): fpt0 .a/ D maxf 2 R0 j .1; : : : ; 1/ 2   P .a/g: Example 2.3. Let f D x 2 C y 3 2 kŒx; y, where p D char.k/ > 3, and let P be the origin. In order to compute .1/, we need to find out the largest r  p  1 with the property that there are nonnegative i and j ˘D r such that 2i  p  1 and ˘ i C  j with p1 3j  p  1. We conclude that .1/ D p1 C , hence 2 3 ´ 5 .p  1/ if p 1 .mod 3/; .1/ D 65p7 if p 2 .mod 3/: 6 One can perform similar, but slightly more involved computations in order to get .e/ for every e  2, and one concludes (see [MTW, Example 4.3]) ´ 5 if p 1 .mod 3/; fpt0 .f / D 65 1  6p if p 2 .mod 3/: 6 Recall that in characteristic zero we have lct0 .x 2 C y 3 / D ple 1.11).

1 2

C

1 3

D

5 6

(see Exam-

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Example 2.4. Let f 2 kŒx; y; z be a homogeneous polynomial of degree 3, having an isolated singularity at the origin P . Therefore f defines an elliptic curve C in Pk2 . One can show that fpt0 .f /  1, with equality if and only if .1/ D p  1 (see [MTW, Example 4.6]). On the other hand, .1/ D p  1 if and only if f p1 62 .x p ; y p ; z p /, which is the case if and only if the coefficient of .xyz/p1 in f p1 is nonzero. This is equivalent to C being an ordinary elliptic curve. We refer to [Hart, §IV.4] for this notion, as well as for other equivalent characterizations. We only mention that C is ordinary if and only if the endomorphism of H 1 .C; OC / induced by the Frobenius morphism is bijective. A recent result due to Bhatt [Bha] says that if C is not ordinary (that is, C is supersingular), then fpt0 .f / D 1  p1 . 2.2 Basic properties of the F -pure threshold. Part of the interest in the F -pure threshold comes from the fact that it has similar properties with the log canonical threshold in characteristic zero. The reader should compare the following properties to those we discussed in §1.3 for the log canonical threshold. An interesting point is that some of the more subtle properties of the log canonical threshold (such as, for example, Inversion of Adjunction) are straightforward in the present context. Property 2.5. If a  b, then fptP .a/  fptP .b/ for every P 2 X . This is an immediate e e consequence of the fact that if ar 6 mPŒp  , then br 6 mPŒp  , hence b .e/  a .e/. Property 2.6. We have fptP .ar / D fptPr .a/ . Indeed, it follows easily from definition that r  ar .e/  a .e/  r.ar .e/ C 1/  1: Dividing by rp e , and letting e go to infinity, gives the assertion. Property 2.7. If dim.X / D n, then fptP .a/  ordPn .a/ . The proof is entirely similar to that of Property 1.14, using Example 2.1, and the properties we proved so far. Property 2.8. The analogue of Inversion of Adjunction holds in this case: if Y  X is a nonsingular closed subvariety such that a  OY is nonzero, then fptP .a/  fptP .a  OY / e for every P 2 Y . This follows from the fact that ai  mPŒp  implies .aOY /i  e .mP OY /Œp  , hence a .e/  aOY .e/ for every e  1. Property 2.9. For every P 2 X we have fptP .a/  ordP1 .a/ . This follows as in the case of Property 1.18, using Property 2.8 and the fact that when dim.X / D 1, we have fptP .a/ D ordP1 .a/ . Property 2.10. If a and b are nonzero ideals on X , then fptP .a C b/  fptP .a/ C fptP .b/ e

e

e

for every P . Indeed, note that if ar  mPŒp  and bs  mPŒp  , then .aCb/rCs  mPŒp  . Therefore aCb .e/  a .e/ C b .e/ C 1: Dividing by p e and taking the limit gives the assertion.

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Property 2.11. If a C mPN D b C mPN , then j fptP .a/  fptP .b/j 

n ; N

where n D dim.X /. The argument follows the one for Property 1.21, using the properties we proved so far. 2.3 Comparison via reduction mod p. As the above discussion makes clear, there are striking analogies between the log canonical threshold in characteristic zero and the F -pure threshold in positive characteristic. Furthermore, as Example 2.3 illustrates, there are subtle connections between the log canonical threshold of an ideal and the F -pure thresholds of its reductions mod p. For simplicity, we will restrict ourselves to the simplest possible setting, as follows. Let a  ZŒx1 ; : : : ; xn  be an ideal contained in .x1 ; : : : ; xn /. On one hand, we consider a  CŒx1 ; : : : ; xn , and with a slight abuse of notation we write lct 0 .a/ for the log canonical threshold of this ideal at the origin. On the other hand, for every prime p we consider the reduction ap D a  Fp Œx1 ; : : : ; xn  of a mod p. We correspondingly consider the F -pure threshold at the origin fpt0 .ap /, and the main question is what is the relation between lct0 .a/ and fpt0 .ap / when p varies. Example 2.3 illustrates very well what is known and what is expected in this direction. The main results in this direction are due to Hara and Yoshida [HY]. Theorem 2.12. With the above notation, for p 0 we have lct0 .a/  fpt0 .ap /. Theorem 2.13. With the above notation, we have limp!1 fpt0 .ap / D lct0 .a/. As we will explain in the next subsection, in fact the results of Hara and Yoshida concern the relation between the multiplier ideals in characteristic zero and the socalled test ideals in positive characteristic. The above results are consequences of the more general Theorems 2.18 and 2.19 below. It is worth mentioning that while the proof of Theorem 2.12 above is elementary, that of Theorem 2.13 relies on previous work (due independently to Hara [Ha] and Mehta and Srinivas [MeS]) using the action of the Frobenius morphism on the de Rham complex and techniques of Deligne–Illusie [DI]. The main open question in this direction is the following (see [MTW, Conjecture 3.6]). Conjecture 2.14. With the above notation, there is an infinite set S of primes such that lct0 .a/ D fpt0 .ap / for every p 2 S. P For example, it was shown in [MTW, Example 4.2] that if f D riD1 ci x ˛i 2 kŒx1 ; : : : ; xn  is such that the ˛i D .˛i;1 ; : : : ; ˛i;n / 2 Zn0 are affinely independent5 5

This means that if

Pr

iD1

i ˛i D 0, with i 2 Q such that

Pr

iD1

i D 0, then all i D 0.

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and ci 2 Z, then there is N such that lct0 .f / D fpt0 .fp / whenever p 1 (mod N ). For example, this applies for the diagonal hypersurface f D x1a1 C  Cxnan , when one can take N D a1 : : : an . We note that the condition p 1 (mod N ) can be rephrased by saying that p splits completely in the cyclotomic field generated by the N th roots of 1. A particularly interesting case is that of a cone over an elliptic curve. Suppose that f 2 ZŒx; y; z is a homogeneous polynomial of degree 3 such that the corresponding 2 projective curve Y ,! PZ has the property that YQ D Y Spec Z Spec Q is nonsingular. We denote by Yp the corresponding curve in PF2p , and we assume that p 0, so that Yp is nonsingular. Recall that by Example 1.8, we have lct 0 .f / D 1, while Example 2.4 shows that fpt 0 .fp / D 1 if and only if Yp is ordinary. The behavior with respect to p depends on whether YQ has complex multiplication. If this is the case, then Yp is ordinary if and only if p splits in the imaginary quadratic CM field. On the other hand, if YQ does not have complex multiplication, then it is known that the set of primes p such that Yp is ordinary has density one [Ser], but its complement is infinite [Elk]. This shows that unlike the case of Example 2.3, the set of primes p such that lct0 .f / D fpt 0 .fp / can be quite complicated. On the other hand, it is known that in the case of an elliptic curve, there is a number field K such that whenever a prime p splits completely in K, we have Yp ordinary (see [Sil, Exercise V.5.11]). It light of these two examples, one can speculate that there is always a number field L such that if p splits completely in L, then lct 0 .a/ D fpt0 .ap / (note that by Chebotarev’s theorem, this would imply the existence of a set of primes of positive density that satisfies Conjecture 2.14). We now describe another conjecture that this time has nothing to do with singulariN ties. If X  PQ is a projective variety, then we may choose homogeneous polynomials f1 ; : : : ; fr 2 ZŒx0 ; : : : ; xN  whose images in QŒx0 ; : : : ; xN  generate the ideal of X . For a prime p, we get a projective variety Xp  PFNp defined by the ideal generated by the images of f1 ; : : : ; fr in Fp Œx0 ; : : : ; xn . Given another choice of such f1 ; : : : ; fr , the varieties Xp are the same for p 0. Note that if X is smooth and geometrically connected6 , then for every p 0, the variety Xp is again smooth and geometrically connected. Similar considerations can be made when starting with a variety defined over a number field. Conjecture 2.15. If X is a smooth, geometrically connected, n-dimensional projective variety over Q, then there are infinitely many primes p such that the endomorphism induced by the Frobenius on H n .Xp ; OXp / is bijective. More generally, a similar assertion holds if X is defined over an arbitrary number field. We mention that this conjecture is open even in the case when X is a curve of genus  3. As the following result from [MuS] shows, this conjecture implies the expected relation between log canonical thresholds and F -pure thresholds. Theorem 2.16. If Conjecture 2.15 is true, then so is Conjecture 2.14. 6

S is connected Recall that this means that X Spec Q Spec Q

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2.4 Test ideals and F -jumping numbers. As we have seen in §2.2, the behavior of the F -pure threshold is similar to that of the log canonical threshold. There is however one property of the log canonical threshold that is more subtle in the case of the F -pure threshold, namely its rationality. In order to prove this for F -pure thresholds, one has to involve also the “higher jumping numbers”. In this section we give a brief introduction to test ideals. In the same way that the F -pure threshold is an analogue of the log canonical threshold in positive characteristic, the test ideals give an analogue of the multiplier ideals in the same context. They have been defined by Hara and Yoshida [HY] for rather general ambient varieties, and it was shown that they behave in the same way as the multiplier ideals do in characteristic zero. Their definition involved a generalization of the theory of tight closure of [HH] to the case where instead of dealing with just one ring, one deals with a pair .R; a /, where a is an ideal in R, and  2 R0 . In the case of an ambient nonsingular variety, it was shown in [BMS2] that one can give a more elementary definition. This is the approach that we are going to take. We will just sketch the proofs, and refer to [BMS2] for details. For a survey of test ideals in the general setting, see [ST]. Given any ideal b  R and e  1, we claim that there is a unique smallest ideal J e e such that b  J Œp  . Indeed, if .Ji /i is a family of ideals such that b  JiŒp  , then T Œpe  e T (the equality follows from the fact that R is a projective b  i JiŒp  D i Ji e module via the Frobenius morphism). We denote the ideal J as above by bŒ1=p  . Given a nonzero ideal a in R and  2 R0 , we consider for every e  1 the ideal e e Ie WD .adp e /Œ1=p  . It is easy to see using the minimality in the definition of the ideals Œ1=p e  b that we have Ie  IeC1 for every e  1. Since R is Noetherian, these ideals stabilize for e 0 to the test ideal .a /. It is not hard to check that this definition commutes with inverting a nonzero element in R, hence we get in this way coherent sheaves on X. In many respects, the test ideals satisfy the same formal properties that the multiplier ideals do in characteristic zero. It is clear from definition that if   , then  .a /   .a /. While it requires a little argument, it is elementary to see that given any , there is " > 0 such that .at / D .a / for every t with   t   C ". By analogy with the case of multiplier ideals, one says that  is an F -jumping number of a if  .a / ¤  .at / for every t < . It is easy to see from definition that in the case of a principal ideal we have  .f  / D f   .f 1 / for every   1. Furthermore, we also have an analogue of Skoda’s theorem: if a is generated by m elements, then  .a / D a   .a1 / for every   m. It is worth pointing out that the proof in this case (see [BMS2, Proposition 2.25]) is much more elementary than in the case of multiplier ideals. Note that if P is a closed point on X defined by the ideal mP , then fptP .a/ is the smallest  such that .a / 6 mP . Indeed, by definition the latter condition is  e Œ1=p e  equivalent with the existence of an e  1 such that adp e is not contained in Œp e  dp e e 6 mP . We can further rewrite this as mP , which in turn is equivalent to a

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a .e/  dp e e. Since fptP .a/ D supe0 a .e/ pe

a .e 0 / 0 , pe

it follows that if fptP .a/ > , then

> , hence a .e/  dp e e. Conversely, if  .a / 6 mP and there is e such that if we take " > 0 such that .a / D .aC" /, then the above discussion implies that there is e such that a .e/  d. C "/p e e, hence fptP .a/ 

a .e/ d. C "/p e e    C " > : pe pe

The global F -pure threshold fpt.a/ is the smallest F -jumping number, that is, the smallest  such that .a / ¤ R. It is clear from the above discussion that fpt.a/ D minP 2X fptP .a/ and fptP .a/ D maxU fpt.ajU /, where U varies over the affine open neighborhoods of P . The following result from [BMS2] gives the analogue for the rationality and the discreteness of the jumping numbers of the multiplier ideals of a given ideal. For extensions to various other settings, see [BMS1], [KLZ] and [BSTZ]. Theorem 2.17. If a is a nonzero ideal in R, then the set of F -jumping numbers of a is a discrete set of rational numbers. Sketch of proof. The new phenomenon in positive characteristic is that for every , we have (12) .a=p / D .a /Œ1=p : This follows from the fact that for e 0 we have    e Œ1=p eC1  e Œ1=p e  Œ1=p D adp e D  .a /Œ1=p :  .a=p / D adp e It is an immediate consequence of (12) that if  is an F -jumping number of a, then also p is an F -jumping number. The second ingredient in the proof of the theorem is given by a bound on the degrees of the generators of .a / in terms of the degrees of the generators of a, in the case when R D kŒx1 ; : : : ; xn . One shows that in general, if b  kŒx1 ; : : : ; xn  is an e ideal generated in degree  d , then bŒ1=p  is generated in degree  d=p e . This is e a consequence of the following description of bŒ1=p  . Consider the basis of R over e e e Rp D kŒx1p ; : : : ; xnp  given by the monomials w1 ; : : : ; wnpe of degree  p e  1 in each variable. If b is generated by h1 ; : : : ; hm , and if we write e

hi D

np X

e

upi;j wj ;

j D1 e

then bŒ1=p  D .ui;j j i  m; j  np e /. This follows from definition and the fact that e hi 2 J Œp  if and only if ui;j 2 J for all j . Suppose now that a is an ideal in kŒx1 ; : : : ; xn  generated in degree  d . Since  e Œ1=p e   .a / D adp e for all e 0, we deduce that  .a / is generated in degree  d .

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This implies that there are only finitely many F -jumping numbers of a in Œ0; . Indeed, otherwise we would get an infinite decreasing sequence of linear subspaces of the vector space of polynomials in x1 ; : : : ; xn of degree  d . We thus obtain the discreteness of the F -jumping numbers in the case of a polynomial ring. The rationality follows easily. If a is principal and  is an F -jumping number, then so are p and   1 (assuming  > 1). It follows that for every , the fractional part of p e  is an F -jumping number for every e  1. Since we have only finitely many such numbers in Œ0; 1, we conclude that  2 Q. The case of an arbitrary ideal is proved similarly, using the analogue of Skoda’s theorem. The case of an arbitrary regular ring R of finite type over k can be then reduced to that of a polynomial ring. An interesting feature of the analogy between test ideals and multiplier ideals is that some of the more subtle properties of multiplier ideals, whose proofs involve vanishing theorems (such as the Restriction Theorem, the Subadditivity Theorem and the Skoda Theorem) follow directly from definition in the case of test ideals. On the other hand, some properties of multiplier ideals that are simple consequences of the description in terms of resolution (for example, the fact that such ideals are integrally closed) can fail for test ideals. For this and related facts, see [MY]. The results that we mentioned relating the log canonical threshold and the F -pure threshold via reduction mod p have a stronger form relating the multiplier ideals and the test ideals. The following two results have been proved7 by Hara and Yoshida in [HY]. Note that they imply Theorems 2.12 and 2.13 above. We keep the notation in these two theorems. Using the description of the multiplier ideals in terms of a log resolution, one can show that all multiplier ideals of a  CŒx1 ; : : : ; xn  are obtained by base-extension from ideals in the ring ZŒ1=N Œx1 ; : : : ; xn  for some positive integer N . In particular, for every p > N we may define the reductions mod p of the multiplier ideals, that we denote by J.a /p . Theorem 2.18. If p 0, then .ap /  J.a /p for every . Theorem 2.19. For every  2 R0 , we have  .ap / D J.a /p for all p large enough (depending on ). The following is a stronger version of Conjecture 2.14. Conjecture 2.20. Given a, there is an infinite set of primes S such that  .ap / D J.a /p for every  2 R0 and every p 2 S. The result in [MuS] that we have already mentioned says that, in fact, Conjecture 2.15 implies Conjecture 2.20. On the other hand, it is shown in [Mus1] that a slightly more general version of Conjecture 2.20 (that deals with ideals in 7 Actually, the results in loc. cit. are in the context of local rings. However, using the arguments therein, one can get the global version of these results that we give.

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S 1 ; : : : ; xn ) implies Conjecture 2.15. Therefore the conjecture relating the mulQŒx tiplier ideals to the test ideals via reduction mod p is equivalent to the conjecture concerning the Frobenius action on the reductions to positive characteristic of a smooth projective variety.

3 Shokurov’s ACC conjecture In this section we turn to Shokurov’s ACC conjecture for log canonical thresholds from [Sho]. This has been proven in the case of ambient smooth varieties in [dFEM2], building on work from [dFM] and [Kol1]. Recall that a set satisfies the ascending chain condition (ACC, for short) if it contains no infinite strictly increasing sequences. Theorem 3.1. For every n, the set Tn of all log canonical thresholds lctP .a/, where a is a nonzero ideal on an n-dimensional nonsingular complex algebraic variety X and P 2 V .a/, satisfies ACC. Remark 3.2. As we have already mentioned in the introduction, Shokurov’s conjecture is formulated when the ambient variety is not necessarily smooth, but only has klt singularities, and in fact more generally, when one deals with a pair .X; D/ with klt singularities, where D is an effective Q-divisor on X , with a suitable condition on the coefficients. We refer to [Bir] for the precise statement8 . The interest in this conjecture (aside from its intrinsic appeal) comes from the connections with one of the remaining open problems in the Minimal Model Program. As an aside, let us mention that after the recent breakthrough in [BCHM], there are two fundamental remaining open problems in this program: • Termination of Flips (proved for certain sequences of flips in the case of varieties of general type in [BCHM]). • Abundance, that is KX nef implies KX semiample. Via work of Birkar [Bir], the ACC conjecture is related to Termination of Flips, as follows. Suppose that Termination of Flips is known in dimension n, and that Shokurov’s ACC conjecture (in its general form mentioned in Remark 3.2) is known in dimension n C 1, then Termination of Flips follows in dimension n C 1 for sequences of flips of pairs .X; D/ such that KX C D is numerically equivalent to an effective Q-divisor. While this is the most interesting case (this is when one expects at the end of the Minimal Model Program to get a minimal model), this extra condition on .X; D/ which does not appear in the inductive hypothesis, does not allow to deduce in general Termination of Flips from the ACC conjecture. 8 In a very recent breakthrough, a proof of the general version of the ACC conjecture was announced in [HMX]. That proof goes far beyond the scope of these notes, relying heavily on techniques from the Minimal Model Program.

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The general case of Shokurov’s conjecture in known in dimensions 2 and 3, by work of Shokurov [Sho] and Alexeev [Ale]. The methods used to prove Theorem 3.1 above, also allow to prove the same result under weaker assumptions on the singularities of the ambient variety: • X with quotient singularities, see [dFEM2]. • X with locally complete intersection singularities, see [dFEM2]. The key point is that Inversion of Adjunction works well in this setting.

1

• .X; P / with ”bounded singularities”, in the sense that one assumes that OX;P is isomorphic to some OY;P , where Y is defined in a fixed AN by equations of bounded degree, see [dFEM1]. Note that this bounds the embedding dimension of .X; P /, and this is a key obstruction towards proving the general case of Shokurov’s conjecture by these methods.

1

We do not give the proof of Theorem 3.1, but explain instead an idea that goes into the proof. We show how this is used in order to prove the following result from [dFM] and [Kol1]. Theorem 3.3. For every n  1, the set Tndiv of all log canonical thresholds lctP .f /, where P is a point on an n-dimensional nonsingular complex algebraic variety, and f 2 O.X/ vanishes at P , is a closed subset of R. There are two important points concerning the proofs of Theorem 3.3. First, it is convenient to work with log canonical thresholds of formal power series f 2 kŒŒx1 ; : : : ; xn , where k is an arbitrary field of characteristic zero. The basic properties of log canonical thresholds that we discussed extend to this setting, see [dFM]. The key point is that results of [Tem] provide existence of log resolutions in this setting. A second idea is that given a sequence of polynomials .fm /m1 in CŒx1 ; : : : ; xn  such that limm!1 lct 0 .fm / D ˛, there is F 2 KŒŒx1 ; : : : ; xn  such that lct.F / D ˛, for some algebraically closed field K containing C. Once this is done, an easy argument shows that there is a polynomial f 2 CŒx1 ; : : : ; xn  such that lct0 .f / D lct.F /. The construction of F can be achieved in two ways: using ultrafilters (as in [dFM]) or using an infinite sequence of generic points (as in [Kol1]). In what follows we discuss the former method. Let us begin by reviewing the definition of ultrafilters. Definition 3.4. A filter on N D Z>0 is a collection U of subsets of N such that 1) ; 62 U; 2) A; B 2 U H) A \ B 2 U; 3) A 2 U; B  A H) B 2 U. A filter is called an ultrafilter if it is maximal, in the sense that it is not properly contained in another filter. Equivalently,

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4) For every A  N, either A or its complement N n A is in U. An ultrafilter is called principal if there is a 2 N that is contained in all A 2 U (in which case, by maximality, we have U D fA  N j a 2 Ag). It is easy to see that an ultrafilter U is non-principal if and only if the complement of every finite proper subset of N is in U. One can show using the Kuratowski–Zorn Lemma that there are ultrafilters containing the filter fN X A j A  N finiteg, hence there are non-principal ultrafilters 9 . Let us fix such a non-principal ultrafilter U. Given a set A, its non-standard extension is 

A WD AN =

where the equivalence relation on AN is defined by .am / .bm / if fm 2 N j am D bm g 2 U (in this case, one also says that am D bm for almost all m). The class of a sequence .am / in A is denoted by Œam . There is an injective map A ,! A that takes a 2 A to Œa (the class of the constant sequence). The principle is that whatever algebraic structure A has, this extends to A. For example, if k is a field, then k is a field, with addition and multiplication defined by Œam  C Œbm  D Œam C bm 

and

Œam   Œbm  D Œam  bm :

Let us see, for example, that every nonzero element in k has an inverse (of course, the zero element in k is the image of the zero element in k): if Œam  ¤ 0, then the set 1 T D fm j am ¤ 0g lies in U. If we put bm D am for m 2 T , and bm 2 k arbitrary for m 62 T , then Œam   Œbm  D 1. u1 un n u 1 ; : : : ; un / 2 Z0 , we put x D x1    xn and juj D P Recall that for u D .u   i ui . We may identify . k/Œx1 ; : : : ; xn  with the set of those Œfm  2 .kŒx1 ; : : : ; xn / such that there is an integer P d with deg.fm /  d for all d . Indeed, given Œfm  2  .kŒx1 ; : : : ; xn /, with fm D u2Zn ;jujd au;m x u , the corresponding polynomial in 0 P f 2 .k/Œx1 ; : : : ; xn  is u2Zn ;jujd Œau;m x u . Therefore we write f D Œfm  (note 0 that this is compatible with our previous convention). However, a general element in  .kŒx1 ; : : : ; xn / is not a polynomial in .k/Œx1 ; : : : ; xn . If f D Œfm  2 .k/Œx1 ; : : : ; xn , and a D Œam  2 k, then f .a/ D Œfm .am /. In particular, we have f .a/ D 0 if and only if fm .am / D 0 for almost all m. It is then easy to see that if k is algebraically closed, then k is algebraically closed, as well. Suppose now that fm 2 CŒx1 ; : : : ; xn  are such that fm .0/ D 0 for all m, and limm!1 lct0 .fm / D ˛. We may consider Œfm  2 .CŒx1 ; : : : ; xn /. While this is not in generalPa polynomial, it determines a formal P power series F with coefficients in C: u if fm D u2Zn am;u x for all m, then F D u2Zn Œam;u x u 2 .C/ŒŒx1 ; : : : ; xn . 0

9

0

In fact, the existence of non-principal ultrafilters is equivalent to the Kuratowski–Zorn Lemma, hence to the axiom of choice.

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Claim. We have lct.F / D ˛. Given any polynomial or power series h, let us denote by hN the truncation of h of degree  N . It is enough to show that for every N , we have lct0 .FN / D lct0 ..fm /N /

(13)

for almost all m (hence this holds, in particular, for an infinite set of values of m). Indeed, it follows from an extension of Property 1.21 to power series that j lct.F /  lct0 .FN /j 

n N C1

and j lct 0 .fm /  lct 0 ..fm /N /j 

n : N C1

Since j lct0 .fm /  ˛j  N nC1 for all m 0, we deduce from (13) that j lct.F /  ˛j  3n , and this happens for all N , hence the claim. N C1 Note that FN is the polynomial in .C/Œx1 ; : : : ; xn  corresponding to the sequence ..fm /d /. After replacing each fm by .fm /d we may assume that deg.fm /  d for every m, so that F is a polynomial in .C/Œx1 ; : : : ; xn  of degree  d . If we parametrize polynomials in n variables, of degree  d and vanishing at the origin, nCd  by their coefficients, we find a polynomial ring R D Œy1 ; : : : ; yr  (with r D d  1), and a polynomial h 2 RŒx1 ; : : : ; xn  with h.0/ D 0, such that every polynomial in CŒx1 ; : : : ; xn  corresponds to h t for a unique closed point of Spec R. Furthermore, every polynomial in .C/Œx1 ; : : : ; xn  of degree  d and vanishing at the origin corresponds to a closed point of Spec.R ˝C C/. Using Property 1.23, we obtain a disjoint decomposition of Spec R in locally closed subsets Z1 ; : : : ; Zd , and ˛1 ; : : : ; ˛d such that for every closed point t 2 Zi , we have lct0 .h t / D ˛i . This gives a decomposition of N according to which Zi contains the point corresponding to fm . Since U is an ultrafilter, it follows that there is i such that fm 2 Zi for almost all m. The condition for a polynomial fm to belong to some Zj is given by finitely many polynomial expressions in the coefficients of fm being zero or nonzero. We thus conclude that since fm 2 Zi for almost all m, then F belongs to Zi Spec C Spec C, and by construction of the Zi , this implies that lct0 .F / D ˛i . This completes the proof of the claim. In the above discussion, we started with a sequence of polynomials .fm / and obtained a formal power series F . If we start, more generally, with a sequence of ideals .am / in CŒx1 ; : : : ; xn  vanishing at 0, one obtains an ideal A  .C/ŒŒx1 ; : : : ; xn  contained in the maximal ideal. A similar argument to the one given above can be used to show that if limm!1 lct 0 .am / D ˛, then lct.A/ D ˛. For details, we refer to [dFM]. We can now sketch the proof of Theorem 3.3. Note first that by Example 1.6, we have Tndiv  Œ0; 1. One can show using Property 1.21 that if X is an n-dimensional nonsingular variety and f 2 O.X / vanishes at some P 2 X , we may write lctP .f / as the limit of a sequence .lct 0 .hm //m1 , for suitable hm 2 CŒx1 ; : : : ; xn  vanishing at 0. Therefore in order to prove Theorem 3.3, it is enough to show that if fm 2 CŒx1 ; : : : ; xn  are polynomials vanishing at 0, with ˛ D limm!1 lct0 .fm /, then there is another such polynomial f with ˛ D lct0 .f /.

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The above construction gives a formal power series F 2 .C/ŒŒx1 ; : : : ; xn  with lct.F / D ˛. Let E be a divisor over Spec ..C/ŒŒx1 ; : : : ; xn / that computes lct.F /, and let be the generic point of the center of E. Note that the completion OX; is isomorphic to a formal power series ring k. /ŒŒx1 ; : : : ; xs , for some s  n. If k D k. / is an algebraic closure of k. /, then we may replace F 2 .C/ŒŒx1 ; : : : ; xn  by its image G in kŒŒx1 ; : : : ; xn , and lct.G/ D lct.F / D ˛. The advantage is that we now have a divisor Ez over Spec.kŒŒx1 ; : : : ; xn / that computes lct.G/, and whose center is equal to the closed point. In this case lct..G/ C m` / D lct.G/ for ` 0, where m is the ideal defining the closed point. Indeed, if ` > ordEz .G/, then ordEz ..G/ C m` / D ordEz .G/, which implies

b

lct..G/ C m` / 

z z Logdisc.E/ Logdisc.E/ D lct.G/; D ` ordEz .G/ ordEz ..G/ C m /

while the inequality lct.G/  lct..G/ C m` / is a consequence of Property 1.12. The ideal .G/ C m` is the image of an ideal b  kŒx1 ; : : : ; xn  vanishing at zero, hence ˛ D lct..G/ C m` / D lct0 .b/. Since ˛  1, it follows from Example 1.10 that if g is a linear combination of the generators of b with general coefficients in k, then lct 0 .g/ D ˛. Let d D deg.g/. As we have seen before, we have a disjoint decomposition Z1 t    t Zr of the space parametrizing complex polynomials in n variables of degree  d , such that points of each Zi have constant log canonical threshold. If g corresponds to a point in Zi Spec C Spec k, we see that a polynomial f 2 CŒx1 ; : : : ; xn  corresponding to a point in Zi has lct 0 .f / D ˛. This completes the (sketch of) proof of Theorem 3.3. For simplicity, in the above we have restricted the discussion to the case of principal ideals. Minor modifications of the argument allow to prove that the set Tn in Theorem 3.1 is closed in R. Furthermore, the same circle of ideals allow the proof of the following statement, conjectured by Kollár, concerning decreasing sequences of log canonical thresholds. Theorem 3.5. With the notation in Theorem 3.1, the limit of every strictly decreasing div . sequence of elements in Tndiv is in Tn1 The key point is to show (using the notation used for the proof of Theorem 3.3 above) that if E is a divisor computing the log canonical threshold of F 2 .C/ŒŒx1 ; : : : ; xn , then the center of E is not equal to the closed point. In this case, after localizing at the generic point of this center, we end up in a ring of power series in at most .n  1/-variables. The proof of Theorem 3.1 is more involved. In addition to the ideas used above, one has to use the following ingredient. Theorem 3.6. Let a be an ideal on a smooth complex variety X , and E a divisor over X that computes lctP .a/, for some P 2 X . If E has center equal to P on X , then for every ideal b on X such that a C mP` D b C mP` , where mP is the ideal of P and ` > ordE .a/, we have lctP .b/ D lctP .a/.

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A proof of this result was given in [Kol1] using the results in the Minimal Model Program from [BCHM]. A more elementary proof, only relying on the Connectedness Theorem of Shokurov and Kollár, was given in [dFEM2].

4 Asymptotic log canonical thresholds In this section we discuss following [JM] an asymptotic version of the log canonical threshold, in the context of graded sequences of ideals. In particular, we explain a question concerning the computation of asymptotic log canonical thresholds by quasimonomial valuations. For proofs and details we refer to [JM]. 4.1 Definition and basic properties. Let X be a smooth, connected, complex algebraic variety. A graded sequence of ideals a on X is a sequence .am /m1 of ideals that satisfies ap  aq  apCq for every p; q  1. All our graded sequences are assumed to be nonzero, that is, some ap is nonzero. A trivial example of such a sequence if given by am D I m , where I is a fixed nonzero ideal on X . The most interesting example is related to asymptotic base loci of line bundles. Suppose that X is projective and L is a line bundle on X such that h0 .X; Lm / ¤ 0 for some m  1. If we take ap to be the ideal defining the base locus of the complete linear series jLp j, then a is a graded sequence of ideals. For other examples of graded sequences we refer to [Laz, Chapter 11.1]. L We note that if the graded OX -algebra OX ˚ . m1 am / is finitely generated10 , then there is p  1 such that amp D apm for every m  1 (see [Bour, Chap. III, §1, Prop. 2]). In this case, we consider the graded sequence as essentially trivial. The interest in the study of graded sequences and of their asymptotic invariants arises precisely when this algebra is not finitely generated (or at least, when this finiteness is not known a priori). Since we are interested in the behavior of singularities, we may, as before, assume that X D Spec R is affine. We denote by ValX the space of real valuations of the fraction field of R that are nonnegative on R. For example, if E is a divisor over X , then all positive multiplies of ordE lie in ValX . Given a graded sequence of ideals a , one can extend “asymptotically” usual invariants of ideals, to obtain invariants for the sequence. More precisely, suppose that ˛./ is an invariant of ideals that satisfies the following two conditions: 1) If a  b, then ˛.a/  ˛.b/. 2) ˛.a  b/  ˛.a/ C ˛.b/. 10 For example, if am defines the base locus of Lm , as above, this condition holds if the section C-algebra ˚m0 .X; Lm / is finitely generated.

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Examples of such an invariants are given by ˛.a/ D v.a/ WD minfv.f / j f 2 ag, where v 2 ValX . Another example is given by ˛.a/ D Arn.a/ (the fact that Arn satisfies 1) and 2) above follows from Properties 1.12 and 1.16). Given a graded sequence of ideals a and an invariant ˛ as above, we see that ˛.apCq /  ˛.ap  aq /  ˛.ap / C ˛.aq /: It is easy to deduce from this (see [JM, Lemma 2.3]) that inf

m1

˛.am / ˛.am / D lim : m!1 m m

We denote this limit by ˛.a /. In particular, we have v.a / when v 2 ValX , and 1 (with the convention that this is infinite if Arn.a /. We define lct.a / D Arn.a / Arn.a / D 0). Of course, using the local Arnold multiplicity, one can define in the same way ArnP .a / and lctP .a /. Example 4.1. Suppose that X D AnC and a is a graded sequence of ideals on X, all of them generated by monomials. Using the notation introduced in Example 1.9, let Pm denote the Newton polyhedron of am (see Example 1.9 for definition). S 1 Since ap  aq  apCq , we have Pp C Pq  PpCq . Let P .a / be the closure of m m P .am /. n To every v 2 RP one associates a “monomial” valuation val of C.x ; : : : ; xn / v 1 0 such that for f D u cu x u 2 CŒx1 ; : : : ; xn  we have valv .f / WD minfhu; vi j cu ¤ 0g: Note that this is a (multiple of a) divisorial valuation precisely when v 2 Qn0 . Since valv .am / D minu2Pm hu; vi, we see that valv .a / D min hu; vi: u2P .a /

Furthermore, since lct.am / D maxf  0 j .1; : : : ; 1/ 2 Pm g, it is easy to see that lct.a / D maxf  0 j .1; : : : ; 1/ 2 P .a /g: Note that P .a / is a nonempty convex subset of Rn0 with the property that P .a/ C u  P .a / for every u 2 Rn0 . Conversely, given Q  Rn0 that satisfies these properties, we may define am D .x u j u 2 mQ/: One can check that a is a graded sequence of ideals such that Q D P .a /. In order to study asymptotic invariants, it is convenient to also consider the associated sequence of asymptotic multiplier ideals of a . Recall that these are defined as follows (for details, see [Laz, Chapter 11.1]). Let  2 R0 be fixed. For every

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=m =mp m; p  1 we have J.a=m m /  J.amp /. Indeed, if h 2 J.am /, then for every divisor E over X we have

ordE .h/ >

  ordE .am /  Logdisc.E/  ordE .amp /  Logdisc.E/; m mp

hence h 2 J.a=mp mp /. By the Noetherian property, it follows that we have an ideal,  denoted J.a /, that is equal to J.a=m m / if m is divisible enough. This is the asymptotic multiplier ideal of a of exponent . For every p  1, we put bp D J.ap /, and let b D .bm /m1 . The following properties are an immediate consequence of the definition: i) If p < q, then bq  bp . 1=m / D ii) We have ap  bp for every p (this follows from ap  J.ap /  J.apm p J.a / for suitable m).

A more subtle property is a consequence of the Subadditivity Theorem (see [Laz, Theorem 11.2.3]): bmp  bpm for all m; p  1. Using these properties one shows that for every valuation v 2 ValX , we have v.bm / v.bm / D lim ; m!1 m m m1

v.b / WD sup and similarly,

Arn.bm / Arn.bm / D lim : m!1 m m m1

Arn.b / WD sup

We also put lct.b / D 1= Arn.b /. The basic principle, that was first exploited in [ELMNP], is that the two sequences a and b have the same asymptotic invariants. More precisely, we have the following: Property 4.2. For every divisor E over X , and every p  1, ordE .a / 

Logdisc.E/ ordE .bp / ordE .ap / <  : p p p

(14)

The second inequality follows from ap  bp . For the first one, let m be divisible enough, so that bp D J.a1=m mp /. It follows from the definition of multiplier ideals that ordE .J.a1=m ordE .amp / Logdisc.E/ ordE .bp / mp //  D > pm p p p Logdisc.E/ :  ordE .a /  p By letting p go to infinity in (14), we conclude that ordE .a / D ordE .b /.

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Property 4.3. One also has lct.a / D lct.b /. For this, see [JM, Proposition 2.13]. As a consequence, one gets a formula describing the asymptotic log canonical threshold in terms of asymptotic orders of vanishing, just as for one ideal. Property 4.4. For every graded sequence of ideals a , we have lct.a / D inf E

Logdisc.E/ ; ordE .a /

(15)

where the infimum is over all divisors E over X . The inequality “” follows from / lct.am /  Logdisc.E by multiplying by m, and letting m go to infinity. For the reverse ordE .am /

inequality, given m  1, there is a divisor Fm over X such that lct.bm / D Using Property 4.2, we deduce m  lct.bm / D

Logdisc.Em / . ordEm .bm /

Logdisc.E/ Logdisc.Em / Logdisc.Em / Logdisc.Em /  D  inf : E ordEm .bm /=m ordEm .b / ordEm .a / ordE .a /

Letting m go to infinity, and using Property 4.3, we get the inequality “” in (15). 4.2 A question about asymptotic log canonical thresholds. As we will see in Example 4.6 below, the infimum in (15) is not, in general, a minimum. In order to have a chance to get a valuation that realizes that infimum, we need to enlarge the class of valuations we consider. A quasi-monomial valuation v of the function field of X is a valuation v 2 ValX that is monomial in a suitable system of coordinates on a model over X . More precisely, there is a projective, birational morphism  W Y ! X , with Y nonsingular, ˛ D .˛1 ; : : : ; ˛n / 2 Rn0 , and local coordinates y1 ; : : : ; yn at a point P 2 Y such that if P f 2 OY;P is written as f D ˇ 2Zn cˇ y ˇ in OY;P , then

1

0

v.f / D minfh˛; ˇi j cˇ ¤ 0g: If Ei  Y is the divisor defined at P by .yi /, we put Logdisc.v/ WD

n X

˛i  Logdisc.Ei /:

iD1

One can show that this definition is independent of the model Y we have chosen. We refer to [JM, §3] for this and for other basic facts about quasi-monomial valuations, and in particular, for their description as Abhyankar valuations. Note that if E is a divisor over X, and ˛ is a non-negative real number, then ˛  ordE is a quasi-monomial valuation, and Logdisc.˛  ordE / D ˛  Logdisc.E/. It is easy to see that if v is a quasi-monomial valuation and a is a graded sequence of ideals, we still have lct.a /  Logdisc.v/ . The following conjecture was made in [JM] v.a / (in a somewhat more general form). It says that the asymptotic log canonical threshold of a graded sequence of ideals can be computed by a quasi-monomial valuation.

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Conjecture 4.5. If a is a graded sequence of ideals on X , then there is a quasimonomial valuation v of the function field of X such that lct.a / D

Logdisc.v/ : v.a /

Note that the conjecture is trivially true if lct.a / D 1. Indeed, in this case any valuation v such that v.a / D 0 satisfies the required condition (for example, we can take v D ordE , where E has center at some point not contained in V .am /, where m is such that am is nonzero). Example 4.6. Suppose that a is a graded sequence of ideals in CŒx1 ; : : : ; xn , all of them generated by As in Example 4.1, we put Pm D P .am /, and let P .a / Smonomials. 1 be the closure of m m Pm . We put e D .1; : : : ; 1/ 2 Rn . It follows from Example 4.1 that if lct.a / < 1, then Arn.a /  e lies on the boundary of the convex set P .a /. In this case, there is a nonzero affine linear function h such that P .a /  fu j h.u/  0g and h.Arn.a /  e/ D 0 (see, for example, [Bro, Theorem 4.3]). If h.x1 ; : : : ; xn / D ˛1 x1 C    C ˛n xn C b, then it is easy to see that .˛1 ; : : : ; ˛n / 2 Rn0 X f0g, and the ˛/ “monomial” valuation w˛ satisfies lct.a / D Logdisc.w . w˛ .a / On the other hand, it is easy to construct examples of such sequences a for which / there is no divisor E over X such that lct.a / D Logdisc.E . Indeed, suppose that ordE .a /

Q D f.u1 ; u2 / 2 R20 j .u1 C 1/u2  1g: As in Example 4.1, we take am D .x a y b pj .a; b/ 2 mQ/, so that P .a / D Q. In particular, we get Arn.a / D D 1C2 5 . One can show that since all am are / generated by monomials, every E with lct.a / D Logdisc.E is a toric divisor (see [JM, ordE .a / Proposition 8.1]). In this case, if ordE .x/ D ˛ and ordE .y/ D ˇ, then ordE .a / D minfu1 a C u2 b j .u1 ; u2 / 2 R20 ; .u1 C 1/u2  1g: p One deduces ordE .a / D 2 ˛ˇ ˛, and a simple computation implies ˛=ˇ D 1 62 Q, a contradiction. The space ValX has a natural topology. This is the weakest topology that makes all maps ValX 3 v ! v.f / 2 R0 continuous, where f 2 R. One can extend the log discrepancy map from quasi-monomial valuations to get a lower-semicontinuous function Logdisc W ValX ! R0 , such that Logdisc.v/ > 0 if v is nontrivial11 . The rough idea is to approximate each nontrivial valuation by quasi-monomial valuations, and to take the supremum of the log discrepancies of these valuations. See [JM, §5] for the precise definition, which is a bit technical. The following is one of the main results in [JM]. 11

The trivial valuation is the one that takes value zero on every nonzero element of the fraction field of R.

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Theorem 4.7. If a is a graded sequence of ideals on X , then there is a valuation v 2 ValX such that Logdisc.v/ lct.a / D : (16) v.a / We expect that every valuation as in the above theorem has to be quasi-monomial (in particular, this would give a positive answer to Conjecture 4.5). Conjecture 4.8. If a is a graded sequence of ideals on X with lct.a / < 1, and if v 2 ValX is a nontrivial valuation such that (16) holds, then v is a quasi-monomial valuation. Theorem 4.9 ([JM]). Conjecture 4.8 holds when dim.X / D 2. In the above discussion we only considered the asymptotic invariant lct.a /, constructed from the log canonical threshold. One can consider also asymptotic versions constructed from the higher jumping numbers of multiplier ideals, as follows. If q is a fixed nonzero ideal on X , and if a is a proper ideal, then lct q .a/ WD minf 2 R0 j q 6 J.a /g: When a is fixed and we let q vary, we obtain in this way all the jumping numbers of a. If a is a graded sequence of ideals of X , one defines lctq .a / WD sup m  lct q .am / D lim m  lct q .am /: m

m!1

The results in this section work if we replace lct.a / by lct q .a /, and the conjectures also make sense in this more general setting. For technical reasons, as well as for possible applications in the analytic setting (see below), it is convenient to work in a more general setting, when X is an excellent scheme. It is shown in [JM] that the above results on asymptotic invariants also hold in this setting, and furthermore, in order to prove the above conjectures in the general setting, it is enough to prove them when X D AnC . One can interpret Conjecture 4.5 as predicting a finiteness statement for arbitrary graded sequences of ideals. One can consider it as an algebraic analogue of the Openness Conjecture of Demailly and Kollár [DK]. Let us briefly recall this conjecture. Suppose that ' is a psh (short for plurisubharmonic) function12 on an open subset U  C. The complex singularity exponent of ' at P is cP .'/ WD supfs > 0 j exp.2s'/ is locally integrable around P g (compare with the analytic definition of the log canonical threshold in the case when ' D 'a ). The Openness Conjecture asserts that the set of those s > 0 such that 12 We do not give the precise definition, but recall that to an ideal a of regular (or holomorphic) functions Pn on U  C, generated by f1 ; : : : ; fr , one associates a psh function 'a .z/ D 12 log. iD1 jfi .z/j2 /. More interesting examples are obtained by taking suitable limits of such functions.

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exp.2s'/ is integrable around P is open; in other words, that exp.2cP .'/'/ is not integrable around P . In the case when ' D 'a for an ideal a of regular (or holomorphic) functions, then this assertion can be proved using resolution of singularities, in the same way that we proved Theorem 1.2. There is no graded sequence of ideals associated to a psh function. However, one can associate to such a function a sequence of ideals b of holomorphic functions that behaves in a similar fashion with the sequence of asymptotic multiplier ideals of a graded sequence of ideals (see [DK] for a description of this construction). We hope that using this formalism, one can show that a positive answer to Conjecture 4.5 would imply the Openness Conjecture.

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J. Kollár, Which powers of holomorphic functions are integrable? Preprint, arXiv:0805.0756v1 [math.AG].

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J. Kollár, Lectures on resolution of singularities. Ann. of Math. Stud. 166, Princeton University Press, Princeton, NJ, 2007.

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J. Kollár, Singularities of pairs. In Algebraic geometry, Santa Cruz 1995, Proc. Symp. Pure Math. 62, Part 1, Amer. Math. Soc., Providence, RI, 1997, 221–286.

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R. Lazarsfeld, Positivity in algebraic geometry II. Ergeb. Math. Grenzgeb. 49, Springer-Verlag, Berlin 2004.

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V. B. Mehta and V. Srinivas, A characterization of rational singularities. Asian J. Math. 1 (1997), 249–271.

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J. Milnor, Singular points of complex hypersurfaces. Ann. of Math. Stud. 61, Princeton University Press, Princeton, NJ.; University of Tokyo Press, Tokyo 1968.

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M. Musta¸ta˘ , Ordinary varieties and the comparison between multiplier ideals and test ideals II. Proc. Amer. Math. Soc. 140 (2012), no. 3, 805–810.

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M. Musta¸ta˘ , Singularities of pairs via jet schemes. J. Amer. Math. Soc. 15 (2002), 599–615.

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M. Musta¸ta˘ and V. Srinivas, Ordinary varieties and the comparison between multiplier ideals and test ideals. Nagoya Math. J. 204 (2011), 125–157.

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M. Musta¸taˇ , S. Takagi and K.-i. Watanabe, F-thresholds and Bernstein-Sato polynomials. In European Congress of Mathematics (Stockholm 2004), Eur. Math. Soc., Zürich 2005, 341–364.

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M. Musta¸ta˘ and K.-I. Yoshida, Test ideals vs. multiplier ideals. Nagoya Math. J. 193 (2009), 111–128.

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K. Schwede and K. Tucker, A survey of test ideals. In Progress in Commutative Algebra 2, De Gruyter Proc. Math., de Gruyter, Berlin 2012, 39–100.

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V. V. Shokurov, Three-dimensional log perestroikas. With an appendix in English by Yujiro Kawamata. Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), 105–203; English transl. Russian Acad. Sci. Izv. Math. 40 (1993), 95–202.

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Mircea Musta¸ta˘ , Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, U.S.A. E-mail: [email protected]

On Schur function expansions of Thom polynomials Özer Öztürk and Piotr Pragacz

Why should we expect a city to cure us of our spiritual pains? Perhaps because we cannot help, loving our city like a family. But we still have to decide which part of the city we love and invent the reasons why. Orhan Pamuk, Istanbul: memories and the city

1 Introduction A prototype of the formulas considered in the present paper, is the following classical result. Let f W M ! N be a holomorphic, surjective map of compact Riemann surfaces. For x 2 M , we set ex WD number of branches of f at x: Then the ramification divisor of f is equal to X .ex  1/x: The Riemann–Hurwitz formula asserts that X   .ex  1/ D 2g.M /  2  deg.f / 2g.N /  2 :

(1)

x2M

(See, e.g., [14].) The right-hand side of Eq. (1) can be rewritten as f  c1 .N /  c1 .M /; and gives us the Thom polynomial of the singularity A1 of maps between curves. In general, according to the monograph [1], the global behavior of singularities of maps f W M ! N of complex analytic manifolds, is governed by their Thom polynomials. Knowing the Thom polynomial of a singularity class †, one can compute the cohomology class represented by the †-points of a map f . We shall recall the definition of a Thom polynomial in Section 3. The term “Thom polynomial” has nowadays rather wide meaning. In the present paper, however, it will mean a classical Thom polynomial of the singularity classes of maps (cf. [40]). We shall work here with complex manifolds1 .  Research supported by the MNiSW grant N N201 608040, and the Japanese JSPS Grant-in-Aid for Scientific Research (B) 22340007. 1 A manifold here is always smooth.

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An explicit2 computation of a Thom polynomial is usually a difficult task. At first, the computations of Thom polynomials were performed in the basis of monomials in the Chern classes. But around 2004, two papers: [5] and [31] appeared independently, with computations of some Thom polynomials in the basis of Schur functions. (The two papers concerned different singularity classes.) One should stress that even with a powerful theory of symmetric functions from [20] and [17], a passage from the monomial basis to the Schur basis is rather difficult: it is possible “in theory” but it is rather difficult in practice (of course, we speak here about “large” expressions). It is, by no means, reasonable to ask why to work with Schur function expansions? One of the aims of the present paper is (to try) to answer this question. Of course, an important role of Schur functions in geometry was known earlier, e.g., by the Schubert Calculus (see also [16], [28], [10] – to mention just a few references). The latter reference gives a wide geometric motivation of the importance and ubiquity of Schur functions in algebraic geometry. A basic property of Schur function expansions of Thom polynomials is the nonnegativity of the coefficients proved by Andrzej Weber and the second named author in [35] (see also [36]). These positive coefficients often have a pleasant algebraic structure, e.g., satisfy some recursions. This allows one to organize the computations of them in a pretty systematic way. Among these coefficients, we find numbers appearing in different contexts in enumerative geometry, e.g., complete quadrics (see [32]). More, as it follows from a recent paper [22], the positivity of the coefficients of Schur function expansions of classical Thom polynomials leads to upper bounds for the coefficients of Legendrian Thom polynomials expanded in an appropriate basis. Another feature comes from the fact that Thom polynomials are closely related with degeneracy loci of the cotangent map f  W T  NM ! T  M (by T  NM we denote the cotangent bundle of N pull backed by f to M ). Polynomials supported on such degeneracy loci were described using Schur functions in [28]; this helps to study the Schur function expansions of Thom polynomials of other singularity classes. In the present article, we survey basically only those papers, where the Schur function expansions of Thom polynomials play a significant role in the process of their computations or/and help in understanding their structure. In [5], the authors computed the Thom polynomial of the second order Thom– Boardman singularity classes †i;j W M m ! N miC1 via its Schur function expansion, and conjectured the positivity of Schur function expansion for all Thom–Boardman singularity classes. 2

Even the word “explicit” has different meanings for different authors working on Thom polynomials.

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In [31], the second author stated some formulas for Thom polynomials of singularities I2;2 ; A3 W M m ! N mCk (any k) and some partial result for Ai W M m ! N mCk (any i; k). These expressions had the form of Schur function expansions. The details were given in [32], [19] and [33]. In Sections 7 and 8, we discuss some essential computations from these papers. In [23], [24], the first author computed Schur function expansions for A4 W M m ! mCk N (k D 2; 3) and III2;3 W M m ! N mCk (any k). This paper is organized as follows. In Section 2, we recall the definition and properties of Schur functions, including: cancellation-, vanishing-, basis-, and factorization property. In Section 3, we recall the notion of a singularity class, and, following Thom [40], attach to a singularity class its Thom polynomial. In Section 4, we discuss the P -polynomials of singularity classes. From the structure of the P -ideal of †i , we deduce some result on a rectangle containment for partitions appearing in the Schur function expansion of a Thom polynomial of †  †i (Theorem 13). In Section 5, We discuss a way of computing of Thom polynomials of the closures of single R-L orbits in a space of jets of maps: .C ; 0/ ! .CCk ; 0/, called there “singularities” after [37]. This is a “method of restriction equations” that we learned from [37]. In Section 6, we collect formulas for the Chern and Euler classes of singularities, and show by an example, how one can compute them. In Section 7, we state some general properties of the Schur function expansions of Thom polynomials of singularities. Theorem 13 is reinterpreted for singularities. We discuss the Thom polynomial of III2;3 for any k. For any i; k, we give the 1-part of the Thom polynomial of Ai . We discuss also recent results of Féher and Rimányi [7] giving a bound on the lengths of partitions appearing in Schur function expansions, and certain basic recursion (on k). In Section 8, we recall Pascal staircases, and survey Schur function expansions of Thom polynomials of I2;2 and A3 from [32] and [19]. Their coefficients obey some (other) recursions on k. We provide details of two computations with extensive use of the algebra of Schur functions and multi-Schur functions. In Section 9, we discuss the Schur function expansions of the Thom polynomials of III3;3 . In Section 10, we discuss some properties of the Thom polynomials of I2;3 . In the appendices (Section 11 and 12), we give the Schur function expansions of the Thom polynomials of III3;3 and I2;3 for several k. This is basically a survey paper. Some new material is gathered in the last four sections. We lectured on this material at IMPANGA seminars in Warsaw and Cracow. Acknowledgments We gratefully thank Alain Lascoux. He taught the second named author the Schur functions in 1979, and discussed with him the Schur function expansions of Thom polynomials in 2004. This was the starting point of the project surveyed

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in the present paper. He also taught, in 2008, the first named author how to write clever algorithms for computations with Schur functions. We thank Maxim Kazarian for mailing us [12]. We are also grateful to Alexander Klyachko and Andrzej Weber for helpful discussions. Finally, we thank the referee whose comments led to the improvement of the exposition. A part of the present article was written during the stay of the second named author at RIMS in Kyoto, in March 2011. He thanks this institute, and especially Shigeru Mukai, for the warm welcome there.

2 Schur functions The main reference for this section, for the conventions and notation, is [17]. This book studies (among others) multi-Schur functions which are a useful generalization of Schur functions. We shall need them in this paper. But we start our discussion with Schur functions. For m 2 N, by an alphabet A of cardinality m we shall mean a finite set of indeterminates A D fa1 ; : : : ; am g. Sometimes, to point out the cardinality of an alphabet fa1 ; : : : ; am g we shall denote it by Am . We shall often identify an alphabet fa1 ; : : : ; am g with the sum a1 C    C am . Definition 1. Given two alphabets A, B, the complete functions Si .AB/ are defined by the generating series (with z an extra variable): X Y Y Si .AB/z i WD .1  bz/= .1  az/: (2) b2B

a2A

We see that Si .AB/ interpolates between Si .A/ - the complete symmetric function of degree i in A and Si .B/ - the elementary symmetric function of degree i in B times .1/i . For example, S3 .A  B/ is equal to S3 .A  B/ D S3 .A/  S2 .A/ƒ1 .B/ C S1 .A/ƒ2 .B/  ƒ3 .B/; where ƒi .B/ denotes the ith elementary symmetric function in B. A weakly increasing sequence .i1 ; i2 ; : : : ; is / of nonnegative integers is called a partition. The number it divides into parts, jI j D i1 C i2 C    C is , is called the weight of I . The nonzero ip are called the parts of I . The number of nonzero parts is called the length of I . Given two partitions I D .i1 ; i2 ; : : : ; is / and J D .j1 ; j2 ; : : : ; j t /, we shall say that I is contained in J , and write I  J , if for any p D 0; 1; 2; : : :, we have isp  j tp . Following [17], we give Definition 2. Given a partition I D .i1 ; i2 ; : : : ; is / 2 Ns , and alphabets A and B, the Schur function SI .A  B/ is ˇ ˇ (3) SI .A  B/ WD ˇSiq Cqp .AB/ˇ1p;qs :

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In other words, we put on the diagonal from to bottom: Si1 ; Si2 ; : : : ; Sis , and then, in each column, the indices of the successive Sj ’s should increase by one from bottom to top. For example, if I D .1; 3; 3; 4; 5/, then ˇ ˇS1 .AB/ ˇ ˇ 1 ˇ ˇ  0 SI .A B/ D ˇ ˇ 0 ˇ ˇ 0

S4 .AB/ S3 .AB/ S2 .AB/ S1 .AB/ 1

S5 .AB/ S4 .AB/ S3 .AB/ S2 .AB/ S1 .AB/

S7 .AB/ S6 .AB/ S5 .AB/ S4 .AB/ S3 .AB/

ˇ S9 .AB/ˇˇ S8 .AB/ˇˇ S7 .AB/ˇˇ : S6 .AB/ˇˇ S5 .AB/ˇ

These functions are often called supersymmetric Schur functions or Schur functions in difference of alphabets. See [39], [3], [29], [34], [20] and [17] for their study. We have the following cancellation property: for alphabets A, B, C, SI ..A C C/  .B C C// D SI .A  B/:

(4)

We shall use the simplified notation i1 i2 : : : is or i1 ; i2 ; : : : ; is for a partition .i1 ; i2 ; : : : ; is / (the latter one if is  10). Also, we shall write .i s / for the partition .i; : : : ; i/ (s times). A partition I has a graphical representation due to Ferrers, called its diagram: it is a diagram of left packed square boxes with i1 ; i2 ; : : : ; is the number of boxes in the successive rows. For example, the diagram of the partition .2; 5; 6; 8/ is:

Given two partitions I and J , if we put their diagrams in such a position that they share the lowest row and the leftmost column, then “ I  J ” iff the set of boxes of the diagram of I is contained in the set of boxes of the diagram of J . We record the following property: SI .A  B/ D .1/jI j SJ .B  A/ D SJ .B  A /;

(5)

where J is the conjugate partition of I (i.e. the consecutive rows of the diagram of J are the transposed columns of the diagram of I ), and A denotes the alphabet fa1 ; a2 ; : : : g. Fix two positive integers m and n. Let I be a partition. Suppose that the diagram of I and the following .m; n/-hook:

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n

-

6 m ?

share the lowest row and the leftmost column. If the diagram of I is contained in this hook, then we say that the partition I is contained in the .m; n/-hook. We record the following vanishing property. Given alphabets A and B of cardinalities m and n, if the diagram of a partition I is not contained in the .m; n/-hook, then SI .A  B/ D 0: (6) For instance, I D .2; 5; 6; 8/ is not contained in the .2; 4/-hook 

4 *

6 2 ?

Therefore S2568 .A2  B4 / D 0. This vanishing property is an immediate consequence of the factorization property (see Eq. (9)). Moreover, we have the following result. Theorem 3. If A and B are alphabets of cardinalities m and n, then the Schur polynomials SI .A  B/, where I runs over partitions contained in the .m; n/-hook, are Z-linearly independent. (I.e., they form a basis of the abelian group of supersymmetric Schur functions in A and B.) For a proof, see, e.g., [34, Proposition 2.3]. Note 4. We shall often identify partitions with their diagrams, as is customary. It is handy to adopt the following Convention 5. Instead of introducing in the argument of a symmetric function, formal variables which will be specialized, we write r for a variable which will be specialized to r (r can be 2x1 , x1 C x2 ,…). For example,   S2 .x1 C x2 / D x12 C x1 x2 C x22 but S2 x1 Cx2 D .x1 C x2 /2 D x12 C 2x1 x2 C x22 : This convention stems from [18] where the reader can find instructive examples of its use.

On Schur function expansions of Thom polynomials

Definition 6. Given two alphabets A; B, we set Y R.A; B/ WD .a  b/;

449

(7)

a2A; b2B

the resultant of A; B. Thus R.A; B/ is the resultant of the polynomials R.x; A/ D R.fxg; A/ and R.x; B/. We now record some properties of Schur functions that are used in our computations with Thom polynomials. The first one is the following linearity formula. We have (see [17]) Sj .E  Bn / D Sj .E  Bn1 /  bn Sj 1 .E  Bn1 /:

(8)

This equality is used quite often to estimate the sizes of partitions indexing Schur function expansions of Thom polynomials (see, e.g., [32], [19], [23], [24], [25]). It serves also to establish an extremely useful Transformation Lemma (see Lemma 7). The second one is the following factorization property [3]. For partitions I D .i1 ; : : : ; im / and J D .j1 ; : : : ; js /, we have S.j1 ;:::;js ;i1 Cn;:::;im Cn/ .Am  Bn / D SI .Am / R.Am ; Bn / SJ .Bn /:

(9)

For example, with m D 4, n D 2, I D .2; 3/, J D .1; 3/, we have

D

B4 R

A2

S1367 .A2  B4 / D S23 .A2 /R.A2 ; B4 /S13 .B4 /: This factorization property is useful to simplify the h-parts (cf. the end of Section 4) of Thom polynomials (see Section 8, and [32], [19], [23], [24]). Cf. also [26]. We shall also need multi-Schur functions. Given s, two sets fA1 ; A2 ; : : : ; As g, 1 fB ; B2 ; : : : ; Bs g of alphabets, and a partition I D .i1 ; : : : ; is /, following [17] we define the multi-Schur function ˇ ˇ (10) SI .A1  B1 ; : : : ; As  Bs / D ˇSiq Cqp .Aq Bq /ˇ1p;qs :

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In case where the alphabets are repeated, we indicate by a semicolon the corresponding bloc separation. For example, Si;i Ii .A  CI B  D/ D Si;i;i .A  C; A  C; B  D/: We record the following Transformation Lemma (see [17, Lemma 1.4.1]). Lemma 7. Let D 0 ; D 1 ; : : : ; D s1 be a family of alphabets such that card.D i /  i for 0  i  s  1. Then the multi-Schur function SI .A1  B1 ; : : : ; As  Bs / is equal to the determinant ˇ ˇ ˇSi Cqp .Aq Bq D sp /ˇ : q 1p;qs In other words, one does not change the value of a multi-Schur function by replacing in row p the difference A  B by A  B  D sp . We leave it to the reader to prove this result.

3 Thom polynomials of singularity classes of maps n Fix m; n; p 2 N. Consider the space J p .Cm 0 ; C0 / of p-jets of analytic functions from Cm to Cn which map 0 to 0. Consider the natural right-left action of the group p n Aut pm  Autpn on J p .Cm 0 ; C0 /, where Aut n denotes the group of p-jets of automorn phisms of .C ; 0/. By a singularity class we shall mean a closed algebraic right-left n m invariant subset of J p .Cm and N n , a 0 ; C0 /. Given complex analytic manifolds M m n p p singularity class †  J .C0 ; C0 / defines the subset †.M; N /  J .M; N /, where J p .M; N / is the space of p-jets from M to N . n Theorem 8. Let †  J p .Cm 0 ; C0 / be a singularity class. There exists a universal polynomial T † over Z in m C n variables c1 ; : : : ; cm , c10 ; : : : ; cn0 which depends only on †, m and n such that for any complex analytic manifolds M m , N n and for almost any map3 f W M ! N , the class of

†.f / WD fp1 .†.M; N // is equal to T † .c1 .M /; : : : ; cm .M /; f  c1 .N /; : : : ; f  cn .N //; where fp W M ! J p .M; N / is the p-jet extension of f . This is a theorem due to Thom, see [40]. If a singularity class † is stable (e.g. closed under the contact equivalence, see, e.g., [7]), then T † depends on ci .TM  T NM /. 3 The Riemann–Hurwitz formula quoted in Introduction holds for any surjective f . In the theory of Thom polynomials we restrict ourselves only to almost all maps, i.e., the maps from some open subset in the space of all maps.

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Let f W M ! N be a map of complex analytic manifolds. In the present paper, we shall work with the cotangent map f  W T  NM ! T  M;

(11)

rather than with the tangent one. Given a partition I , we define SI .T  M  T  NM / to be the effect of the following specialization of SI .AB/: the indeterminates of A are set equal to the Chern roots of T  M , and the indeterminates of B to the Chern roots of T  NM . Given a singularity class †, the Poincaré dual of †.f /, for almost any map f W M ! N , will be written in the form X ˛I SI .T  M  T  NM / (12) I

with integer coefficients ˛I . Accordingly, we shall write T†D

X

˛I SI ;

(13)

I

where SI is identified with SI .AB/ for the universal Chern roots A and B. For example, consider the singularity class † D †i . So, m  i  n, and looking at the .m  i/th degeneracy locus of the cotangent map (11), we have T

†i

D S.nmCi/i ;

the Giambelli–Thom–Porteous formula (see [27]). A basic result on Schur function expansions of Thom polynomials of singularity classes is Theorem 9 ([35]). Let † be a nontrivial stable singularity class. Then for any partition I , the coefficient ˛I in the Schur function expansion of the Thom polynomial X T†D ˛I SI ; is nonnegative and

P I

˛I > 0.

This result was conjectured in [5]. Thus, it is not obvious. But its proof is almost obvious. The original proof in [35] used the classification space of singularities and the Fulton–Lazarsfeld theorem [9]. We now give an outline of another proof (of nonnegativity only), communicated to the second author by Klyachko (Ankara, 2006) and, independently, by Kazarian [12].

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Sketch of proof of Theorem 9. First, using some Veronese map, we “materialize” all singularity classes in sufficiently large Grassmannians. We fix a singularity class † and take the Schur function expansion of T † . We take sufficiently large Grassmannian containing † and such that specializing T † in the Chern classes of the tautological (quotient) bundle Q, we do not lose any Schur summand. We identify by the Giambelli formula (see [8], p. 146 and [10], p. 18, p. 27), a Schur polynomial of Q with the corresponding Schubert cycle. To test a coefficient in the Schur function expansion of T † , we intersect Œ† with the corresponding dual Schubert cycle (see [8], p. 150). Using the Bertini–Kleiman theorem [13], we put the cycles in a general position, so that we can reduce to settheoretic intersection, which is nonnegative. Note 10. If ˛I ¤ 0, then we shall say that I belongs to the indexing set of the Schur function expansion of T † , or that the partition I appears in the Schur function expansion of T † , or just I appears in T † . It appears that this positivity result can be used to find upper bounds for the coefficients of expansions of Legendrian Thom polynomials in a suitable basis, see [22]. z For the Lagrangian Thom polynomials, this is the basis of the so called Q-functions, see [21]. We record now a variant valid for not necessary stable singularity classes. Theorem 11 ([36]). Let † be a nontrivial singularity class. Then for any partitions I , J , the coefficient ˛I;J in the Schur function expansion of the Thom polynomial X ˛I;J SI .T  M /SJ .T NM / T†D is nonnegative, and

P I;J

˛I;J > 0.

(It is important that we use the cotangent bundle to the source M and the tangent bundle to the target N .) The latter result implies the former, see [36]. This last paper contains also some variations on positivity of generalized Thom polynomials, and emphasizes the role of cone classes for globally generated and ample vector bundles, following Fulton and Lazarsfeld.

4 P -ideals of singularity classes More generally, it is natural to consider the P -ideal of a singularity class †, denoted by P † . This is the subset in the polynomial ring ZŒc1 ; : : : ; cm ; c10 : : : ; cn0 , consisting of all polynomials P which satisfy the following universality property. For any complex analytic manifolds M m , N n and almost any map f W M ! N , P .c1 .M /; : : : ; cm .M /; f  c1 .N /; : : : ; f  cn .N //

On Schur function expansions of Thom polynomials

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is supported on †.f /. (This means – see [28], [10] – that the class of a cycle on M in H.M; Z/ is in the image of H.†.f /; Z/ ! H.M; Z/.) Note 12. These ideals were first studied (1988) in [28] for the classes † D †i . They were rediscovered (2004) in [6] in the context of group actions. For † D †i , P † is simply the ideal of polynomials which – after specialization to the Chern classes of M and N – support cycles in the locus D, where   dim Ker.f W TM ! T NM /  i: for almost any map f W M ! N . (This means that the class of a cycle on M in H.M; Z/ is in the image of H.D; Z/ ! H.M; Z/.) Note that in terms of the cotangent map, D is the locus where   rank f  W T  NM ! T  M  m  i; for almost any map f W M ! N . Of course, the component of minimal degree of P † is generated over Z by T † . Usefulness of P -ideals come from the following observation. Suppose that †  †0 , 0 where †0 is another singularity class. Then T † belongs to P † . Thus if one knows the 0 algebraic structure of P † , one can use it to compute T † . In this way, the degeneracy loci of the cotangent map (11) appear to be useful objects to study Thom polynomials. i Set P i WD P † . By [28] and [29], one knows the algebraic structure of P i , i.e., a certain finite set of its algebraic generators (cf. [28, Proposition 6.1]), and its Z-basis (cf. [28, Proposition 6.2]). The arguments combine geometry of Grassmann bundles with algebra of Schur functions. Before proceeding further, let us state the following result which is rather useful to compute the Schur function expansions of Thom polynomials. Its setting is the same as that of Theorem 8. Theorem 13 ([28], [32]). Suppose that a stable singularity class † is contained in †i . Then all summands in the Schur function expansion of T † are indexed by partitions containing .n  m C i /i . Thus the partitions not containing this rectangle cannot appear in the Schur function expansion of T † . This result seems to be quite obvious. However, its proof is not obvious. Let A and B be two alphabets such that Y X Y X .1 C a/ and cj0 D .1 C b/: ci D a2A

b2B

We have Proposition 14. No nonzero ZŒc1 ; : : : ; cm -linear combination of the Schur functions SI .A  B/’s, where all I ’s do not contain .n  m C i /i , belongs to P i .

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The idea of the proof is to interpret P i as a “generalized resultant”, and use some specialization trick. For details, we refer the reader to the proof of “Claim” on p. 164 in [29]. Thus, in particular, no nonzero Z-linear combination of the SI .A  B/’s, where all I ’s do not contain .n  m C i /i , belongs to P i . Also, we have Proposition 15. Any SI .A  B/, where I contains .n  m C i /i belongs to P i . The idea of the proof is to use a desingularization of D in the product of two Grassmann bundles, and apply appropriate pushforward formulas. For details, see [28, Proposition 3.2]. We are now ready to justify the theorem. Since † is contained in †i , the Thom polynomial T † belongs to P i . By the stability assumption, the Thom polynomial T † is a (unique) Z-linear combination of the SI .A  B/’s. Propositions 15 and 14 imply that only Schur functions indexed by partitions containing the rectangle .n  m C i /i appear in this sum. In the computations of Thom polynomials, it is convenient to “split” them into pieces supported on the consecutive degeneracy loci of the cotangent map (11). Let T be the Thom polynomial of a singularity class. Following [33], by the h-part of T we mean the sum of all Schur functions appearing in T (multiplied by their coefficients) such that the corresponding partitions satisfy the following condition: I contains the rectangle partition .n  m C h/h , but it does not contain the larger diagram .n  m C h C 1/hC1 . The polynomial T is a sum of its h-parts, h D 1; 2; : : :.

5 Single R-L orbits In the present paper, we shall mostly study Thom polynomials of singularities. Let k  0 be a fixed integer and  2 N. Two stable germs 1 ; 2 W .C ; 0/ ! .CCk ; 0/ are said to be right-left equivalent if there exist germs of biholomorphisms ' of .C ; 0/ and of .CCk ; 0/ such that ı 1 ı ' 1 D 2 : A suspension of a germ is its trivial unfolding: .x; v/ 7! ..x/; v/. Consider the equivalence relation (on stable germs .C ; 0/ ! .CCk ; 0/) generated by right-left equivalence and suspension. A singularity  is an equivalence class of this relation.4 According to Mather’s classification ([4] or [1]), singularities are in one-to-one correspondence with finite dimensional (local) C-algebras. We shall use the following notation of Mather: – Ai will stand for the stable germs with local algebra CŒŒx=.x iC1 /;

4

i  0I

This terminology stems from [37]; a singularity corresponds to a single R-L orbit.

On Schur function expansions of Thom polynomials

455

– Ia;b (of Thom–Boardman type †2;0 ) for stable germs with local algebra CŒŒx; y=.xy; x a C y b /;

b  a  2I

– IIIa;b (of Thom–Boardman type †2;0 ) for stable germs with local algebra CŒŒx; y=.xy; x a ; y b /;

b  a  2 (here k  1).

With a singularity , there is associated Thom polynomial T  in the formal variables c1 ; c2 ; : : : which after the substitution of ci to ci .f  T N  TM / D Œc.f  T N /=c.TM /i ;

(14)

for a general map f W M ! N between complex analytic manifolds, evaluates the Poincaré dual of Œ.f /, where .f / is the cycle carried by the closure of the set fx 2 M W the singularity of f at x is g:

(15)

By codim./, we mean the codimension of .f / in X . Codimensions of above singularities are as follows (cf. [4, Chapter 8]): – Ai associated with maps .C ; 0/ ! .CCk ; 0/), where i  0 and k  0 has codimension .k C 1/i. – Ia;b associated with maps .C ; 0/ ! .CCk ; 0/), where b  a  2 and k  0 has codimension .k C 1/.a C b  1/ C 1. – IIIa;b associated with maps .C ; 0/ ! .CCk ; 0/), where b  a  2 and k  1 has codimension .k C 1/.a C b  2/ C 2. We shall now follow the approach in [37]. Let  W .Cn ; 0/ ! .CnCk ; 0/ be a prototype of a singularity . It is possible to choose a maximal compact subgroup G of the right-left symmetry group Aut  D f.'; / 2 Autn  Aut nCk W

ı  ı ' 1 D g;

(16)

such that images of its projections to the factors Aut n and Aut nCk are linear 5 . That is, projecting on the source Cn and the target CnCk , we obtain representations 1 ./ and 2 ./. Let E0 and E denote the vector bundles associated with the universal principal G -bundles EG ! BG that correspond to 1 ./ and 2 ./, respectively. The total Chern class, c./ 2 H  .BG ; Z/, and the Euler class, e./ 2 H 2 codim./ .BG ; Z/, of  are defined by c./ WD

c.E / c.E0 /

and e./ WD e.E0 /:

(17)

We end this section by recalling the method of restriction equations due to Rimányi et al. 5

By Autn we mean here the space of automorphisms of .Cn ; 0/.

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Theorem 16 ([37]). Let  be a singularity. Suppose that the number of singularities of codimension less than or equal to codim./ is finite. Moreover, assume that the Euler classes of all singularities of codimension smaller than codim./ are not zero-divisors. Then we have 1. if  ¤  and codim./  codim./, then T  .c.// D 0; 2. T  .c.// D e./. This system of equations (taken for all such ’s) determines the Thom polynomial T  in a unique way. Solving of these equations is rather difficult. This method is well suited for computer experiments, though the bounds of such computations are quite sharp.

6 Computing the Chern and Euler classes The Chern and Euler classes recalled in the present section were given in: [37], [32], [23], [24] and [25]. Let  W .C ; 0/ ! .CCk ; 0/ be a singularity in the sense of Section 5. For  D Ai , a suitable maximal compact subgroup can be chosen as GAi D U.1/  U.k/. The Chern class is c.Ai / D

k 1 C .i C 1/x Y .1 C yj /; 1Cx

(18)

j D1

where x and y1 ,…, yk are the Chern roots of the universal bundles on BU.1/ and BU.k/. The Euler class is e.Ai / D i Š x

i

k Y

.yj  ix/ : : : .yj  2x/.yj  x/:

(19)

j D1

In case of  D I2;2 , we consider the extension of U.1/  U.1/ by Z=2Z. Denoting this group by H , a maximal compact subgroup is G D H  U.k/ for all k  0. But to make computations easier, we use the subgroup U.1/  U.1/  U.k/ as G (cf. [37], p. 502)). We have c.I2;2 / D

k .1 C 2x1 /.1 C 2x2 / Y .1 C yj /: .1 C x1 /.1 C x2 /

(20)

j D1

Here x1 ; x2 and y1 ; : : : ; yk are the Chern roots of the universal bundles on two copies of BU.1/ and on BU.k/. The Euler class is e.I2;2 / D x1 x2 .2x1  x2 /.2x2  x1 /

k Y

.yj  x1 /.yj  x2 /.yj  x1  x2 /: (21)

j D1

457

On Schur function expansions of Thom polynomials

Next, we consider  D III2;2 . This time we use the maximal compact group G D U.2/  U.k 1/ for k  1. We have k1 .1C2x1 /.1C2x2 /.1Cx1 Cx2 / Y .1 C yj /; c.III2;2 / D .1Cx1 /.1Cx2 /

(22)

j D1

where x1 ; x2 and y1 ; : : : ; yk1 denote the Chern roots of the universal bundles on BU.2/ and BU.k 1/. The Euler class is e.III2;2 / D .x1 x2 /2 .x1 2x2 /.x2 2x1 /

k1 Y

k1 Y

j D1

j D1

.x1 yj /

.x2 yj /:

(23)

For the singularity III2;3 , we can use the action of the U.1/  U.1/  U.k  1/. We have k1 .1C2x1 /.1C3x2 /.1Cx1 Cx2 / Y .1 C yj /: (24) c.III2;3 / D .1Cx1 /.1Cx2 / j D1

This time x1 ; x2 and y1 ; : : : ; yk are the Chern roots of the universal bundles on two copies of BU.1/ and on BU.k  1/. The Euler class is e.III2;3 / D 4x12 x23 .x1  x2 /.x1  3x2 /.x2  2x1 / 

k1 Y

.x1  yj /.x2  yj /.2x2  yj /:

(25)

j D1

For the singularity III3;3 , the maximal compact group is U.2/  U.k  1/. The Chern class is c.III3;3 / D

k1 .1 C 3x1 /.1 C 3x2 /.1 C x1 C x2 / Y .1 C yj /; .1 C x1 /.1 C x2 /

(26)

j D1

where x1 ; x2 and y1 ; : : : ; yk1 are the Chern roots of the universal bundles BU.2/ and BU.k  1/. The Euler class is e.III3;3 / D 4x13 x23 .3x1  x2 /.3x1  2x2 /.3x2  x1 /.3x2  2x1 / 

k1 Y

.x1  yj /.2x1  yj /.x2  yj /.2x2  yj /:

(27)

j D1

We display now the Chern or/and Euler classes of some other singularities (we omit to interpret the variables xi and yj ). We have c.Ia;b / D

.1 C .1 C

aCb x /.1 gcd.a;b/ 1 a x /.1 gcd.a;b/ 1

C C

ab Y x / k1 gcd.a;b/ 2 .1 b x / gcd.a;b/ 2 j D1

C yj /I

(28)

458

e.Ia;b / D

Ö. Öztürk and P. Pragacz a b1 k  Y gcd.a; b/ aŠbŠab1 b a1 x aCb Y  Y gcd.a; b/ .i / .i / I x  y x  y j j b a gcd.a; b/aCb j D1 iD1 iD1 (29) k1 Y .1 C ax1 /.1 C bx2 /.1 C x1 C x2 / .1 C yj /I (30) c.IIIa;b / D .1 C x1 /.1 C x2 / j D1

e.IIIa;b / D .a  1/Š.b  1/Š 

b1 Y

a1 Y

iD1

iD1

.ax1  ix2 /

k1 Y  a1 Y

b1 Y

j D1

iD1

.yj  ix1 /

iD1

.bx2  ix1 / 

(31)

.yj  ix2 / :

A general strategy for computing the Chern and Euler classes of singularities was described in [37]. We show now, following [24], how to compute the Euler class of III2;3 . Assume that k D 1 and consider the germ g.x; y/ D .x 2 ; y 3 ; xy/. A prototype of III2;3 can be written as the unfolding 8 X ui hi ; gC iD1

where hi form a basis of the space

mx;y

m3x;y ;   @x ; @g C C3  I.g/ @y ˚ @g

and where I.g/ is the subspace generated by the component functions of g. We shall work with the basis consisting of the following germs: h1 .x; y/ D .x; 0; 0/;

h5 .x; y/ D .0; y; 0/;

h2 .x; y/ D .y; 0; 0/;

h6 .x; y/ D .0; y 2 ; 0/;

h3 .x; y/ D .y 2 ; 0; 0/; h4 .x; y/ D .0; x; 0/;

h7 .x; y/ D .0; 0; x/; h8 .x; y/ D .0; 0; y/:

Let hi denote the representation of the action of the group U.1/  U.1/ on the space generated by hi . Then, denoting the one-dimensional representations of the first and the second copies of U.1/ by  and , we have h5 D 2 ;

h1 D ; h2 D 2 ˝ 1 ;

h6 D ;

2

;

h7 D ;

˝ ;

h8 D :

h3 D  ˝  1

h4 D 

2 3

On Schur function expansions of Thom polynomials

Therefore for k D 1, using the representation

L

459

hi , we can write the Euler class as

e.III2;3 / D 4x12 x23 .x1  x2 /.x1  3x2 /.x2  2x1 /;

(32)

where x1 and x2 denote the Chern roots of the universal bundles on the two copies of BU.1/. For k D 2, in addition to hi above, we need to consider the representations of the action of the group U.k  1/ D U.1/ on the spaces generated by .x; y/ 7! .0; 0; 0; x/, .x; y/ 7! .0; 0; 0; y/ and .x; y/ 7! .0; 0; 0; y 2 /. These can be written as ˝ 1 , ˝ 1 and ˝ 2 , where denotes the one-dimensional representation of this copy of U.1/. Hence, in this case, the Euler class can be written as e.III2;3 / D 4x12 x23 .x1  x2 /.x1  3x2 /.x2  2x1 /.x1  y1 /.x2  y1 /.2x2  y1 /; (33) where xi are as above and y1 denotes the Chern root of the universal bundle on BU.1/. For k  1, we need to consider U.k 1/ instead of U.1/, giving rise to y1 ; : : : ; yk1 Q (and respectively to the product jk1 D1 .x1  yj /.x2  yj /.2x2  yj /) instead of y1 (and respectively of .x1  y1 /.x2  y1 /.2x2  y1 /). We shall need the following alphabets: Definition 17. We set D D x1 C x 2 C x 1 C x 2 ; E D 2x1 C 2x2 ; F D 2x1 C 3x2 C x1 C x2 ; G D 3x1 C 3x2 C x1 C x2 ; H D 2x1 C 4x2 C x1 C x2 :

Notation 18. In the rest of the paper we shall use the shifted parameter r WD k C 1:

(34)

When we need to emphasize the dependence on r we shall write .r/ for the singularity  W .C ; 0/ ! .CCr1 ; 0/, and denote the Thom polynomial of .r/ by Tr , or Tr for short. (In this notation, the result of Thom, TrA1 D Sr , has a transparent form.) We now specify, with the help of these alphabets, some equations characterizing Thom polynomials Tr imposed by different singularities. Note 19. The variables below will be specialized to the Chern roots of the cotangent bundles.

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First, we give the vanishing equations coming from the Chern classes of singularities. Let Bj denote an alphabet of cardinality j . We have the following equations:   Ai .r/ W Tr x Br1  .i C 1/x D 0 for i D 0; 1; 2; : : : I (35) I2;2 .r/ W I2;3 .r/ W

Tr



Tr .X2  E  Br1 / D 0I

(36)

 2x C 3x  5x  6x  Br1 D 0I

(37)

III2;2 .r/ W

Tr .X2  D  Br2 / D 0I

(38)

III2;3 .r/ W

Tr .X2  F  Br2 / D 0I

(39)

III2;4 .r/ W

Tr .X2  H  Br2 / D 0:

(40)

Using the Chern classes displayed above, one can write down other vanishing equations. We give now some normalizing equations coming from the Euler classes of singularities. We have     Ai .r/ W Tr x Br1  .i C1/x D R x C 2x C 3x C    C ix ; Br1 C .i C1/x I (41) I2;2 .r/ W

Tr .X2 EBr1 / D x1 x2 .x1 2x2 /.x2 2x1 / R.X2 C x1 Cx2 ; Br1 /I (42)   I2;3 .r/ W Tr 2x C 3x  5x  6x  Br1   D 2xR 2x C 3x ; 5x C 6x C Br1 (43) r1 Y .4x  bj /.6x  bj /I  j D1

III2;2 .r/ W

Tr .X2  D  Br2 / D R.X2 ; D C Br2 /I

III2;3 .r/ W Tr .X2  F  Br2 / D 2x2 .x1 x2 /R.X2 ; F CBr2 /

r2 Y

(44)

.2x2 bj /I (45)

j D1

III3;3 .r/ W Tr .X2  G  Br2 / D x1 x2 .3x1  2x2 /.3x2  2x1 /  R.X2 ; G C Br2 /

r2 Y

.2x1  bj /.2x2  bj /:

j D1

(46)

On Schur function expansions of Thom polynomials

461

Using the Euler classes displayed above, one can write down other normalizing equations.

7 Thom polynomials of singularities In this section, we shall study, for singularities , Schur function expansions Thom polynomials T  written in the form (13) (cf. also (12)): X ˛I SI : TD I

It is interesting to find bounds on partitions appearing in Schur function expansions of Thom polynomials of singularities. One such follows immediately from Theorem 13. Proposition 20. Suppose that a singularity  is of Thom–Boardman type †i;::: . Then all summands in the Schur function expansion of Tr are indexed by partitions containing the rectangle partition .r C i  1/i . For example, consider the singularity III2;3 .r/. As its Thom–Boardman type is †2;0 , all partitions in the Schur function expansion of T III2;3 .r/ contain the partition .r C 1; r C 1/. This Thom polynomial is characterized by the equations (35), i D 0; 1; 2; 3, (38) and (45). Its Schur function expansion is given by the following expression: Theorem 21 ([24], [7]). We have III2;3

Tr

D

rC1 X

2i SrC1i;rC1;rCi :

(47)

iD1

7.1 On Morin singularities Ai .r/. One of the most important problems in global singularity theory is to write down the explicit Schur function expansion of the Thom polynomials for Morin singularities Ai .r/. We now describe, following [33], the 1-part A of Tr i for any i and r. Let A be an alphabet of cardinality m. Consider the function F .A; /, defined for any difference of alphabets G  H by X F .A; G  H/ WD SI .A/Snim ;:::;ni1 ;nCjI j .G  H/; (48) I

where the sum is over partitions I D .i1 ; i2 ; : : : ; im / such that im  n. A basic link of this function to resultants is given by the following result. Lemma 22 ([33], Lemma 8). For a variable x and an alphabet B of cardinality n, we have F .A; x  B/ D R.x C Ax; B/: (49)

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Ö. Öztürk and P. Pragacz

Next, we define the following function Fr.i / ./: Fr.i/ .G  H/ D

X

SJ



2 C 3 C  C i



Srji 1 ;:::;rj1 ;rCjJ j .G  H/; (50)

J

where the sum is over partitions J  .r i1 /, and for i D 1 we understand Fr.1/ ./ D Sr ./. The following result gives the key algebraic property of Fr.i/ . Proposition 23. We have

  Fr.i/ .x  Br / D R x C 2x C 3x C    C ix ; Br :

(51)

Proof. The assertion follows from Lemma 22 with m D i  1, n D r, and A D 2 C 3 C  C i : With the help of Proposition 23, the following result on Thom polynomials was established: Ai

Theorem 24 ([33], pp. 173–174). For any i , r, the 1-part of Tr

is equal to Fr.i/ .

We shall now use a couple of functions Fr.i/ to rephrase some results from [40], [38], folklore, [11] and [37], respectively: Fr.1/ D Sr D TrA1 I X 2j Srj;rCj D TrA2 I Fr.2/ D j r

F1.3/

D S111 C 5S12 C 6S3 D T1A3 I

F1.4/ C 10S22 D S1111 C 9S112 C 26S13 C 24S4 C 10S22 D T1A4 I F2.3/ C 5S33 D S222 C 5S123 C 6S114 C 19S24 C 30S15 C 36S6 C 5S33 D T2A3 ([33], pp. 174–176). The reader can find in [33] more examples of the functions Fr.i/ . In the next section, we shall discuss the Schur function expansions of TrA3 for all r. Definition 25. For a positive integer p We denote by ˆp the linear endomorphism on the Z-module spanned by Schur functions indexed by partitions of length  p that sends a Schur function Sj1 ;:::;jp to Sj1 C1;:::;jp C1 . Example 26. For any i; r  1, we have .i/ /; Fr.i/ D Fr.i/ C ˆi .Fr1

(52)

where the first summand gathers the Schur functions indexed by partitions of length < i. In [2], the author discusses another approach to Thom polynomials of Morin singularities.

On Schur function expansions of Thom polynomials

463

7.2 A basic recursion. In the forthcoming section, we shall discuss some recursions for Thom polynomials. The following result was recently obtained in [7, Proposition 7.15, Theorem 7.14]. Let Q denote the local algebra of the singularity . Theorem 27. Let  be a stable singularity. Then the length of any partition, appearing in the Schur function expansion of Tr , is  dim.Q /1. Moreover, by erasing one column of length dim.Q /1 from all the diagrams of partitions appearing in Tr , we  get all the diagrams of partitions appearing in Tr1 (we disregard the partitions whose diagrams have no such a column). In other words, for p D dim.Q /  1, the following equation holds:  /; Tr D TSr C ˆp .Tr1

(53)

where the first summand gathers the Schur functions indexed by partitions of length < p. This result was earlier established for the singularities I2;2 .r/, A3 .r/, A4 .r/, III2;3 .r/ and III3;3 .r/ from the restriction equations which they obey, with help of Eq. (8) (see [32], [19], [23], [24] and [25]). This recurrence relation is quite easy to observe, especially by computing examples with the help of computer. It is, however, not sufficient to compute Thom polynomials. As the matter of fact, Schur function expansions of Thom polynomials often contain many terms, where the first column is shorter than the maximal possible. So these “initial terms”, denoted by TSr in (53), cannot be obtained by the operation of adding a maximal possible column. Another interesting question is to find upper bounds of the coefficients in Schur function expansions of Thom polynomials. This will be a subject of some future study.

8 Pascal staircases and two recursions We invoke first some results from [32] and [19]. We start with useful algebraic identity associated with Pascal staircases (cf. [19]). Then we discuss the Schur function computations of the Thom polynomials of I2;2 .r/ and A3 .r/. 8.1 Pascal staircases. The material of this subsection stems from [19]. Consider an infinite matrix P D Œps;t  with rows and columns numbered by s; t D 1; 2; : : : . We suppose that p1;t D p2;t D 0 for t  2, p3;t D p4;t D 0 for t  3, p5;t D p6;t D 0 for t  4 etc. The first column is an arbitrary sequence v D .v1 ; v2 ; : : :/. In the case when this sequence is the sequence of coefficients of the Taylor expansion of a function f .z/, we write Pf for the corresponding matrix P . To define the remaining ps;t ’s, we use the recursive formula psC1;t D ps;t1 C ps;t :

(54)

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Ö. Öztürk and P. Pragacz

We visualize this definition by a

b 

a

)

b aCb

We thus get the following Pascal staircase P D Œpi;j i;j D1;2;::: : v1 v2 v3 v4 v5 v6 v7 :: :

0 0 v2 v3 Cv2 v4 Cv3 Cv2 v5 Cv4 Cv3 Cv2 v6 Cv5 Cv4 Cv3 Cv2 :: :

0 0 0 0

v3 Cv2 v4 C2v3 C2v2 v5 C2v4 C3v3 C3v2 :: :

0 0 0 0 0 0 v4 C2v3 C2v2 :: :

0 0 0 0 0 0 0 :: :

::: ::: ::: ::: ::: ::: :::

Given an integer d  0, and an alphabet A, we define the function W .d / D W .d; A/ by X pd C1i;j C1 Si .A/ Sj;d ij .X2 /: (55) W .d; A/ D i;j

The function W .d; A/ is linear in the elements of the first column of P . Hence it is sufficient to restrict to the case v D .1; y; y 2 ; : : :/, i.e., to take P D P1=.1zy/ to determine it. Lemma 28. If P D P1=.1zy/ and A D x1 C x2 , then W .0/ D 1 and for d  1   (56) W d; x1 C x2 D .y  1/y d 1 Sd .X2 /: For the proof, see [19]. Let B be another alphabet. Taking now A D x1 C x2 C B instead of x1 C x2 , and using X   pd C1ik;j C1 Si  x1 C x2 Sj;d ij k .X2 /Sk .B/ W .d; A/ D i;j;k

D

X

  W d  k; x1 C x2 Sk .B/

k

D .1  y 1 /

X

y d k Sd k .X2 /Sk .B/ D y t .1  y 1 /Sd .X2  y 1 B/;

k

we get the following corollary. Corollary 29. If P D P1=.1zy/ and B is an arbitrary alphabet, then (apart from initial values) we have   W d; x1 C x2 C B D .y  1/y d 1 Sd .X2  y 1 B/: (57)

On Schur function expansions of Thom polynomials

465

8.2 Recursions for I2 ;2 .r/. The material of this subsection stems from [32]. I I The codimension of I2;2 .r/, r  1, is 3r C 1. Set Tr WD Tr 2;2 and TSr D TSr 2;2 . We have T1 D TS1 D S22 . A partition appearing in the Schur function expansion of Tr contains the partition .r C 1; r C 1/ and has at most three parts. In particular, if Si1 ;i2 appears in the Schur function expansion of Tr , then i1 D r C 1 C p and i2 D 2r  p, where 0  2p  r  1. Invoke the map ˆ3 from Definition 25. We have, for r  2, the following recursive equation: S Sr2 / C    C ˆr1 Tr D TSr C ˆ3 .TSr1 / C ˆ23 .T 3 .T1 /:

(58)

So we are left with computation of TSr . Consider the matrix whose .i; j /th entry is the partition .i C j; 1 C 2i  j / with the convention that .i C j; 1 C 2i  j / is the empty partition for 2j > i C 1: 2 3 22 ; ; ; ; ::: 6 34 ; ; ; ; : : :7 6 7 6 46 55 ; ; ; : : :7 6 7 6 58 67 ; ; ; : : :7 6 7 : 66; 10 79 88 ; ; : : :7 6 7 67; 12 8; 11 9; 10 ; ; : : :7 6 7 68; 14 9; 13 10; 12 11; 11 ; : : :7 4 5 :: :: :: :: :: : : : : : Note that the rth row of the above matrix consists of partitions appearing in TSr . It turns out that the coefficients of their Schur functions are given by the corresponding entries of the Pascal staircase P D ŒPi;j iD1;:::Ij D1;::: , associated with the sequence f2i  1giD1;2;::: : 2

1 0 0 0 6 3 0 0 0 6 6 7 3 0 0 6 6 15 10 0 0 6 P D 6 31 25 10 0 6 6 63 56 35 0 6 6127 119 91 35 4 :: :: :: :: : : : : Namely, we have TSr D

X 2j rC1

0 0 0 0 0 0 0 :: :

0 0 0 0 0 0 0 :: :

::: ::: ::: ::: ::: ::: :::

Pr;j SrCj;2rC1j :

3 7 7 7 7 7 7 7: 7 7 7 7 5

(59)

466

Ö. Öztürk and P. Pragacz

Example 30. We have the following values of TS1 ; : : : ; TS7 : TS1 D S22 ; TS2 D 3S34 ; TS3 D 7S46 C 3S55 ; TS4 D 15S58 C 10S67 ; TS5 D 31S6;10 C 25S79 C 10S88 ; TS6 D 63S7;12 C 56S8;11 C 35S9;10 ; TS7 D 127S8;14 C 119S9;13 C 91S10;12 C 35S11;11 : In this case, the algebra of Schur functions combined with one of the equations characterizing the Thom polynomial, yields quickly an expression for TSr . Of course, TSr is uniquely determined by its value on X2 . The following result gives this value. Proposition 31. For any r  1, we have TSr .X2 / D .x1 x2 /rC1 Sr1 .D/:

(60)

We show the induction step. Suppose that the assertion is true for TSi , where i < r. Let I D .j; r C 1 C p; r C 1 C q/ be a partition appearing in the Schur function expansion of Tr . By the factorization property (9), we get SI .X2  D  Br2 / D R  Sj .D  Br2 /  Sp;q .X2 /; where R D R.X2 ; D C Br2 /. Therefore, using Eq. (58), we obtain Tr .X2  D  Br2 / D R 

X r1 j D0

 TSrj .X2 / Sj .D  Br2 / : .x1 x2 /rj C1

(61)

By the induction assumption, for positive j  r  1, we have TSrj .X2 / D .x1 x2 /rj C1 Sr1j .D/: We use now the fact that among the equations characterizing Tr is (38) (because the codimension of III2;2 .r/ is smaller than codim.I2;2 .r//). Substituting this to (61), we obtain r1 X TSr .X2 / Sj .D  Br2 /Sr1j .D/ C D 0: (62) .x1 x2 /rC1 j D1

But we also have, by a formula for addition of alphabets, r1 X j D1

Sj .D  Br2 /Sr1j .D/ C Sr1 .D/ D Sr1 .Br2 / D 0:

(63)

On Schur function expansions of Thom polynomials

467

Combining (62) and (63), gives TSr .X2 / D .x1 x2 /rC1 Sr1 .D/; that is, the induction assertion. The Schur function expansion of Si .D/ was described in [28], [15] andAppendixA3 in [30], in the context of the Segre classes of the second symmetric power of a rank 2 vector bundle. Indeed, D is the alphabet of the Chern roots of the second symmetric power of a rank 2 bundle with the Chern roots x1 , x2 . The recursions encoded by the Pascal diagram (59) express the recursions for the coefficients of the Segre classes of the second symmetric power of a rank 2 vector bundle (loc.cit.). 8.3 Recursions for A3 .r/. The material of this subsection stems from [19]. We set   X Fr WD Sj1 ;j2 2 C 3 Srj2 ;rj1 ;rCj1 Cj2 :

(64)

j1 j2 r

This function is the 1-part of TrA3 (see Section 7). In [37], the author gave Thom polynomials for A3 .1/ and A3 .2/. Their Schur function expansions are T1A3 D S111 C 5S12 C 6S3 D F1

(65)

and T2A3 D S222 C 5S123 C 6S114 C 19S24 C 30S15 C 36S6 C 5S33 D F2 C 5S33 : (66) Note that the 2-part of T2A3 is 5S33 . We now pass to the case of general r. Since A3 .r/ has codimension 3r, a partition appearing in the 2-part of TrA3 has weight 3r and its diagram contains the partition .r C 1; r C 1/. Moreover, it can have at most three rows. Consider the matrix whose .i; j /th entry is the partition .1 C i C j; 2 C 2i  j / with the convention that .1 C i C j; 2 C 2i  j / is the empty partition for 2j > i C 1: 2 3 33 ; ; ; ; ::: 6 45 ; ; ; ; : : :7 6 7 6 57 7 66 ; ; ; : : : 6 7 6 69 78 ; ; ; : : :7 6 7 67; 11 8; 10 99 ; ; : : :7 6 7 68; 13 9; 12 10; 11 ; ; : : :7 6 7 69; 15 10; 14 11; 13 12; 12 ; : : :7 4 5 :: :: :: :: :: : : : : : xr whose Schur summands are indexed We now want to define a symmetric function H by partitions from the .r 1/st row of the above matrix. Their coefficients will be given

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by the corresponding entries of the following Pascal staircase. Consider the following Taylor expansion: 5  6z .1  z/.1  2z/.1  3z/ D 5 C 24z C 89z 2 C 300z 3 C 965z 4 C 3024z 5 C 9329z 6 C    :

f .z/ D

The Pascal staircase associated with f is the following infinite matrix: 2 3 5 0 0 0 0 ::: 6 24 0 0 0 0 ::: 7 6 7 6 89 24 0 0 0 ::: 7 6 7 6 300 113 0 0 0 ::: 7 6 7 P D 6 965 413 113 : 0 0 ::: 7 6 7 6 3024 1378 526 7 0 0 : : : 6 7 6 9329 4402 1904 526 0 : : : 7 4 5 :: :: :: :: :: : : : : : For r  2, we set

xr WD H

X

Pr1;j SrCj;2rj :

(67)

2j r

Sr , r D 2; : : : ; 7: Example 32. We have the following values of H x2 D 5S33 ; H x3 D 24S45 ; H x4 D 24S66 C89S57 ; H x5 D 113S78 C300S69 H x6 D 113S99 C413S8;10 C965S7;11 ; H x7 D 526S10;11 C1378S9;12 C3024S8;13 : H We define by induction on r xr C ˆ3 .H xr1 / C ˆ23 .H xr2 / C    C ˆr2 x Hr D H 3 .H2 /: With this definition of Hr , we state the following result. Theorem 33 ([19]). We have

TrA3 D Fr C Hr :

In other words, the function Hr is the 2-part of TrA3 , and its h-parts are zero for h  3. Note also that we recover the recurrence (52): Fr D Fxr C ˆ3 .Fr1 /:

On Schur function expansions of Thom polynomials

469

We show now, following [19], the essential computations in the proof of Theorem 33. As explained in [19], it is crucial to show the vanishing (38) of TrA3 at the Chern class c.III2;2 .r//. I.e., it suffices to show the equality .Fr C Hr /.x1 C x2  D  Br2 / D 0:

(68)

Due to the factorization property (9), each Schur function occurring in the expansion of Hr is such that Sc;rC1Ca;rC1Cb .X2 D Br2 / D R.X2 ; D CBr2 /  Sc .D Br2 /  Sa;b .X2 /: We set Vr .X2 I Br2 / WD

Hr .X2  D  Br2 / ; R.X2 ; D C Br2 /

(69)

so that Vr .X2 I Br2 / D

r2 X

X

eri;j Si .D  Br2 / Sj;rij 2 .X2 /: (70)

iD0 fj 0W iC2j r2g

We have the following recursive relation which follows from the observation that the coefficient of br2 in Vr .X2 I Br2 / is equal to Vr1 .X2 I Br3 /. Lemma 34. For r  2, we have Vr .X2 I Br2 / D

r2 X

Vri .X2 I 0/ Si .Br2 /:

(71)

iD0

Thus it is sufficient to compute Vr .X2 I 0/. Proposition 35. For r  2, we have   Vr .X2 I 0/ D 3r2 3Sr2 .X2 /  2S1;r3 .X2 / :

(72)

(In particular, V2 .X2 I 0/ D 5 and V3 .X2 I 0/ D 9S1 .X2 /.) We now apply Corollary 29 from Subsection 8.1 with B D 2x1 C 2x2 . Expanding    Sd X2  y 1 2x1 C 2x2 D Sd .X2 / 

2x1 C 2x2 x1 x2 Sd 1 .X2 / C 4 2 Sd 2 .X2 /; y y

we get, for d  3, W .d; D/ D y d 2 .y  1/.y  2/Sd .X2 //  2y d 3 .y  1/.y  2/S1;d 1 .X2 / (73)

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and initial conditions W .0/ D 1; W .1/ D .y  3/S1 .X2 /; W .2/ D .y  1/.y  2/S2 .X2 /  2.y  3/S11 .X2 /: We come back to Proposition 35, and we take the Pascal staircase (59). Then for d D r  2, the function W .d; D/ is the function Vr .X2 I 0/. We thus have to specialize y into 1; 2; 3 successively. Apart from initial values, only y D 3 contributes, and we get, for d  3, W .d; D/ D 3d C1 Sn .X2 /  2  3d S1;d 1 .X2 /: This proves Proposition 35, checking the cases r D 2; 3; 4 directly. We now pass to the specialization Fr .X2  D  Br2 /. It is rather straightforward to prove the following lemma (cf. [19]). Lemma 36. The resultant R.X2 ; D C Br2 / divides Fr .X2  D  Br2 /. We set Ur .X2 I Br2 / WD

Fr .X2 D Br2 / : R.X2 ; D C Br2 /

(74)

Observe that each variable b 2 Br2 appears at most with degree 3 in Fr .X2 D Br2 /, and hence at most with degree 1 in Ur .X2 I Br2 /. We have the following precise recursive relation which follows from the observation that the coef3 ficient of br2 in Fr .X2 D Br2 / is equal to Fr1 .X2 D Br3 /. Lemma 37. For r  2, we have Ur .X2 I Br2 / D

r2 X

Uri .X2 I 0/ Si .Br2 /:

(75)

iD0

Let be the endomorphism of the C-vector space of functions of x1 ; x2 , defined by

  x1 f .x1 ; x2 /  x2 f .x2 ; x1 /

f .x1 ; x2 / D : x1  x2

For any i; j 2 N, we have

.x1j x2i / D Si;j .X2 /:

(76)

The proof of the following proposition will make use of multi-Schur functions (see the end of Section 2). Proposition 38. For r  2, we have   Fr .X2  D/ D 3r2 R.X2 ; D/.x1 x2 /r2 3Sr2 .X2 /  2S1;r3 .X2 / :

(77)

471

On Schur function expansions of Thom polynomials

Proof. The identity is true for r D 2. To prove the assertion for r  3, we compute in two different ways the action of on the multi-Schur function Sr;rIr .X2 C 2x1 C 3x1  DI x1  D/:

(78)

Firstly, expanding (78), we have  

Sr;rIr .X2 C 2x1 C 3x1  DI x1  D/  X    D

Sj1 ;j2 2x1 C 3x1 Srj2 ;rj1 Ir .X2  DI x1  D/ j1 j2 r

D

 X

Sj1 ;j2

j1 j2 r

X

D

Sj1 ;j2





2 C 3

2 C 3







Srj2 ;rj1 IrCj1 Cj2 .X2  DI x1  D/

Srj2 ;rj1 ;rCj1 Cj2 .X2  D/

j1 j2 r

D Fr .X2  D/: Secondly, using Lemma 7, we subtract x1 from the arguments in the first two rows of the determinant (78) without changing its value. We get the determinant ˇ ˇ     ˇ Sr x2 C 2x1 C 3x1  D SrC1 x2 C 2x1 C 3x1  D SrC2 .D/ ˇˇ ˇ ˇ ˇ   ˇ: ˇ Sr1 x2 C 2x1 C 3x1  D  S C 2x C 3x  D S .D/ x r 2 1 1 rC1 ˇ ˇ ˇ ˇ ˇSr2 X2 C 2x1 C 3x1  D  Sr1 X2 C 2x1 C 3x1  D  Sr .x1  D/ˇ Since the elements in the first two rows of the third column are zero, this determinant is equal to   Sr;r x2 C 2x1 C 3x1  D  Sr .x1  D/: Since x2 C 2x1 C 3x1  D D x2 C 3x1  2x2  x1 C x2 and the following two factorizations hold:   Sr;r x2 C 3x1  2x2  x1 C x2 D 3r2 .x2  2x1 /.x1 x2 /r1 .3x1  2x2 / and

Sr .x1  D/ D x1r2 x2 .x1  2x2 /;

we infer that   Sr;rIr X2 C 2x1 C 3x1  DI x1  D D 3r2 R.X2 ; D/.x1 x2 /r2 x1r3 .3x1  2x2 /:

(79)

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By (76), the result of applying to (79) is   3r2 R.X2 ; D/.x1 x2 /r2 3Sr2 .X2 /  2S1;r3 .X2 / : Comparison of both computations of applied to (78) yields the proposition. In terms of Ur , we rewrite Proposition 38 into Corollary 39. For r  2, we have

  Ur .X2 I 0/ D 3r2 3Sr2 .X2 /  2S1;r3 .X2 / :

(80)

These are the essential computations with Schur functions leading to the proof of Theorem 33.

9 Towards the Thom polynomial of III3;3 .r/ The singularity III3;3 .r/ has codimension 4r C 2. So, the partitions that we need to consider have weight 4r C2. Moreover, all diagrams contain the partition .r C1; r C1/, have at most 4 rows and the length of the second row is at most r. Let D r denote the set of all such diagrams. By D r;2 ; D r;3 and D r;4 we shall denote the subsets of D r , that consist of diagrams with 2,3 and 4 rows, respectively. III Set Tr WD Tr 3;3 . Then, the part of Tr corresponding to the partitions in D r;4 is given by ˆ4 .Tr1 /. The Thom polynomial Tr must satisfy the following system of equations: (35) for i D 0; 1; 2; 3; 4, (36), (37), (38), (39), (40) together with the normalizing equation (46). For a partition I 2 D r , we have   SI  Br1 / DSI .x  Br1  2x   DSI x Br1  3x   DSI x Br1  4x D 0: Hence Eqs. (35) for i D 0; 1; 2; 3; 4 are satisfied automatically by any linear combination of Schur functions indexed by partitions in D r . Moreover, Eq. (36) implies Eq. (38) by the substitution br1 D x1 C x2 . Hence we can replace the former set of equations by a smaller set of equations consisting of Eqs. (36), (37), (39), (40) and (46). Note that in these equations, the alphabets we need to consider, are suitable for the factorization property (9) associated with a pair of alphabets of cardinalities r C 1 and 2. In [26], we give an algorithm based on ACE (cf. [41]) which solves the latter system of equations. Using this algorithm, we get the (unique) Tr for r D 2; : : : ; 8, expanded in the Schur function basis. In the next example, we give Tr for r D 2; 3, and in Section 11, we give T4 ; : : : ; T8 .

On Schur function expansions of Thom polynomials

473

Example 40. We have T2 D 4S37 C 16S46 C 28S55 C 20S145 C 6S136 C 7S235 C 3S244 C 2S1135 C 3S1234 C 6S1144 C S2233 I T3 D 8S4;10 C 40S59 C 88S68 C 120S77 C 12S149 C 52S158 C 100S167 C 14S248 C 50S257 C 20S266 C 15S347 C 10S356 C ˆ4 .T2 /: In [25], the author proposes a conjecture about the recursion for the coefficients in the Schur function expansion of Tr . This recursion is checked for 2  r  8, with the help of an algorithm in [26].

10 On the Thom polynomial of I2 ;3 .r/ I

Set Tr WD Tr 2;3 . The singularity I2;3 .r/ has codimension 4r C 1. So, the partitions that we need to consider have weight 4r C 1. Moreover, all diagrams contain the partition .r C 1; r C 1/ and have at most 4 rows. Then, the part of Tr corresponding to the partitions with 4 rows is given by ˆ4 .Tr1 /. The Thom polynomial Tr must satisfy the following system of equations: (35) for i D 0; 1; 2; 3, (36), (38), (39) together with the normalizing equation (43). An algorithm analogous to the one in [26], allows us to get the (unique) solutions Tr of this system of equations for r D 1; : : : ; 7, expanded in Schur function basis. In the next example, we give Tr for r D 1; 2; 3, and in Section 12, we give T4 ; : : : ; T7 . Example 41. We have T1 D 2S122 C 4S23 I T2 D 32S36 C 24S45 C 24S135 C 12S144 C 12S234 C 3S333 C ˆ4 .T1 /I T3 D 208S49 C 208S58 C 112S67 C 168S148 C 152S157 C 56S166 C 100S247 C 76S256 C 50S346 C 24S355 C 18S445 C ˆ4 .T2 /:

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Ö. Öztürk and P. Pragacz III3;3

11 Appendix 1: Tr III3;3

Let Tr D Tr

, r D 4; : : : ; 8

. We have

T4 D 16S5;13 C 96S6;12 C 256S7;11 C 416S8;10 C 496S9;9 C 24S1;5;12 C 128S1;6;11 C 304S1;7;10 C 448S189 C 28S2;5;11 C 128S2;6;10 C 264S279 C 100S288 C 30S3;5;10 C 112S369 C 70S378 C 31S459 C 25S468 C 10S477 C ˆ4 .T3 /I T5 D 32S6;16 C 224S7;15 C 704S8;14 C 1344S9;13 C 1824S10;12 C 2016S11;11 C 48S1;6;15 C 304S1;7;14 C 864S1;8;13 C 1504S1;9;12 C 1904S1;10;11 C 56S2;6;14 C 312S2;7;13 C 784S2;8;12 C 1232S2;9;11 C 448S2;10;10 C 60S3;6;13 C 284S3;7;12 C 616S3;8;11 C 364S3;9;10 C 62S4;6;12 C 238S4;7;11 C 182S4;8;10 C 70S499 C 63S5;6;11 C 56S5;7;10 C 35S589 C ˆ4 .T4 /I T6 D 64S7;19 C 512S8;18 C 1856S9;17 C 4096S10;16 C 6336S11;15 C 7680S12;14 C 8128S13;13 C 96S1;7;18 C 704S1;8;17 C 2336S1;9;16 C 4736S1;10;15 C 6816S1;11;14 C 7872S1;12;13 C 112S2;7;17 C 736S2;8;16 C 2192S2;9;15 C 4032S2;10;14 C 5392S2;11;12 C 1904S2;12;12 C 120S3;7;16 C 688S3;8;15 C 1800S3;9;14 C 2976S3;10;13 C 1680S3;11;12 C 124S4;7;15 C 600S4;8;14 C 1348S4;9;13 C 980S4;10;12 C 364S4;11;11 C 126S5;7;14 C 492S5;8;13 C 420S5;9;12 C 252S5;10;11 C 127S6;7;13 C 119S6;8;12 C 91S6;9;11 C 35S6;10;10 C ˆ4 .T5 /I T7 D 128S8;22 C 1152S9;21 C 4736S10;20 C 11904S11;19 C 20864S12;18 C 28032S13;17 C 31616S14;16 C 32640S15;15 C 192S1;8;21 C 1600S1;9;20 C 6080S1;10;19 C 14144S1;11;18 C 23104S1;12;17 C 29376S1;13;16 C 32064S1;14;15 C 224S2;8;20 C 1696S2;9;19 C 5856S2;10;18 C 12448S2;11;17 C 18848S2;12;16 C 22752S2;13;15 C 7872S2;14;14 C 240S3;8;19 C 1616S3;9;18 C 4976S3;10;17 C 9552S3;11;16 C 13392S3;12;15 C 7296S3;13;14 C 248S4;8;18 C 1448S4;9;17 C 3896S4;10;16 C 6696S4;11;15 C 4656S4;12;14 C 1680S4;13;13 C 252S5;8;17 C 1236S5;9;16 C 2844S5;10;15 C 2328S5;11;14 C 1344S5;12;13

On Schur function expansions of Thom polynomials

475

C 254S6;8;16 C 1002S6;9;15 C 912S6;10;14 C 672S6;11;13 C 252S6;12;12 C 255S7;8;15 C 246S7;9;14 C 210S7;10;13 C 126S7;11;12 C ˆ4 .T6 /I T8 D 256S9;25 C 2560S10;24 C 11776S11;23 C 33280S12;22 C 65536S13;21 C 97792S14;20 C 119296S15;19 C 128512S16;18 C 130816S17;17 C 384S1;9;24 C 3584S1;10;23 C 15360S1;11;22 C 40448S1;12;21 C 74496S1;13;20 C 104960S1;14;19 C 122880S1;15;18 C 129536S1;16;17 C 448S2;9;23 C 3840S2;10;22 C 15104S2;11;21 C 36608S2;12;20 C 62592S2;13;19 C 83200S2;14;18 C 93952S2;15;17 C 32064S2;16;16 C 480S3;9;22 C 3712S3;10;21 C 13184S3;11;20 C 29056S3;12;19 C 45888S3;13;18 C 57728S3;14;17 C 30624S3;15;16 C 496S4;9;20 C 3392S4;10;20 C 10688S4;11;19 C 21184S4;12;18 C 30880S4;13;17 C 20688S4;14;16 C 7296S4;15;15 C 504S5;9;20 C 2976S5;10;19 C 8160S5;11;18 C 14432S5;12;17 C 11352S5;13;16 C 6336S5;14;15 C 508S6;9;19 C 2512S6;10;18 C 5872S6;11;17 C 5172S6;12;16 C 3672S6;13;15 C 1344S6;14;14 C 510S7;9;18 C 2024S7;10;17 C 1914S7;11;16 C 1584S7;12;15 C 924S7;13;14 C 511S8;9;17 C 501S8;10;16 C 456S8;11;15 C 336S8;12;14 C 126S8;13;13 C ˆ4 .T7 /: I2 ;3

12 Appendix 2: Tr I2;3

Let Tr D Tr

, r D 4; : : : ; 7

. We have T4 D 1280S5;12 C 1024S7;10 C 1408S6;11 C 480S89 C 1056S1;5;11 C 1120S1;6;10 C 736S179 C 240S188 C 656S2;5;10 C 656S269 C 368S278 C 360S359 C 328S368 C 124S377 C 180S458 C 134S467 C 75S557 C 36S566 C ˆ4 .T3 /I

T5 D 7744S6;15 C 8832S7;14 C 7168S8;13 C 4544S9;12 C 1984S10;11 C 6432S1;6;14 C 7232S1;7;13 C 5632S1;8;12 C 3232S1;9;11 C 992S1;10;10 C 4048S2;6;13 C 4448S2;7;12 C 3264S2;8;11 C 1616S2;9;10 C 2280S3;6;12 C 2416S3;7;11 C 1632S3;8;10 C 560S3;9;9 C 1204S4;6;11 C 1208S4;7;10 C 692S489 C 602S5;6;10 C 542S579 C 206S588 C 270S669 C 201S678 C ˆ4 .T4 /I

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Ö. Öztürk and P. Pragacz

T6 D 46592S7;18 C 53888S8;17 C 45824S9;16 C 32640S10;15 C 19200S11;14 C 8064S12;13 C 38784S1;7;17 C 44608S1;8;16 C 37248S1;9;15 C 25408S1;10;14 C 13568S1;11;13 C 4032S1;12;12 C 24512S2;7;16 C 27936S2;8;15 C 22720S2;9;14 C 14624S2;10;13 C 6784S2;11;12 C 13920S3;7;15 C 15632S3;8;14 C 12256S3;9;13 C 7312S3;10;12 C 2384S3;11;11 C 7472S4;7;14 C 8200S4;8;13 C 6128S4;9;12 C 3152S4;10;11 C 3864S5;7;13 C 4100S5;8;12 C 2812S5;9;11 C 980S5;10;10 C 1932S6;7;12 C 1924S6;8;11 C 1108S6;9;10 C 903S7;7;11 C 813S7;8;10 C 309S7;9;9 C ˆ4 .T5 /I T7 D 279808S8;21 C 325376S9;20 C 282368S10;19 C 212224S11;18 C 140544S12;17 C 79104S13;16 C 32512S14;15 C 233088S1;8;20 C 270464S1;9;19 C 232832S1;10;18 C 171392S1;11;17 C 108672S1;12;16 C 55680S1;13;15 C 16256S1;14;14 C 147520S2;8;19 C 170560S2;9;18 C 145088S2;10;17 C 103872S2;11;16 C 62272S2;12;15 C 27840S2;13;14 C 84000S3;8;18 C 96544S3;9;17 C 80736S3;10;16 C 55776S3;11;15 C 31136S3;12;14 C 9856S3;13;13 C 45328S4;8;17 C 51600S4;9;16 C 42160S4;10;15 C 27888S4;11;14 C 13536S4;12;13 C 23688S5;8;16 C 26568S5;9;15 C 21080S5;10;14 C 12928S5;11;13 C 4304S5;12;12 C 12100S6;8;15 C 13284S6;9;14 C 10032S6;10;13 C 5232S6;11;12 C 6050S7;8;14 C 6388S7;9;13 C 4400S7;10;12 C 1540S7;11;11 C 2898S8;8;13 C 2886S8;9;12 C 1662S8;10;11 C ˆ4 .T6 /:

References [1] V. Arnold, V. Vasilev, V. Goryunov, and O. Lyashko, V. I. Arnold, V. V. Goryunov, O. V. Lyashko, and V. A. Vasilliev, Singularity theory I. Reprint of Dynamical systems VI, Encyclopaedia Math. Sci. 6, Springer-Verlag, Berlin 1998. [2] G. Bérczi, Moduli of map germs, Thom polynomials and the Green-Griffiths conjecture. In Contributions to algebraic geometry, EMS Ser. Congr. Rep., EMS Publ. House, Zurich 2012, 141–167. [3] A. Berele and A. Regev, Hook Young diagrams with applications to combinatorics and to representation theory of Lie superalgebras. Adv. in Math. 64 (1987), 118–175. [4] A. Du Plessis and C. T. C. Wall, The geometry of topological stability. London Math. Soc. Monogr. (N.S.) 9, Oxford University Press, New York 1995.

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[5] L. Fehér and B. Komuves, On second order Thom-Boardman singularities. Fund. Math. 191 (2006), 249–264. [6] L. Fehér and R. Rimányi, Calculation of Thom polynomials and other cohomological obstructions for group actions. In Real and complex singularities (San Carlos 2002), T. Gaffney and M. Ruas, eds., Contemp. Math. 354, Amer. Math. Soc., Providence, RI, 2004, 69–93. [7] L. Fehér and R. Rimányi, Thom series of contact singularities. Ann. of Math., to appear; preprint, arXiv:0809.2925v3 [math.AG]. [8] W. Fulton, Young tableaux. London Math. Soc. Stud. Texts 35, Cambridge University Press, Cambridge 1997. [9] W. Fulton and R. Lazarsfeld, Positive polynomials for ample vector bundles. Ann. of Math. 118 (1983), 35–60. [10] W. Fulton and P. Pragacz, Schubert varieties and degeneracy loci. Lecture Notes in Math. 1689, Springer-Verlag, Berlin 1998. [11] T. Gaffney, The Thom polynomial of †1111 . In Singularities, Proc. Symposia Pure Math. 40 (1), Amer. Math. Soc., Providence, RI, 1983, 399–408. [12] M. É. Kazarian, On the positivity of Schur expansions of Thom polynomials (after M. Mikosz, P. Pragacz and A. Weber). Private communication (03.04.2009). [13] S. Kleiman, The transversality of a general translate. Compositio Math. 38 (1974), 287–297. [14] S. Kleiman, The enumerative theory of singularities. In Real and complex singularities (Oslo 1976), P. Holm, ed., Sijthoff and Noordhoff, Alphen aan den Rijn 1977, 297–396. [15] D. Laksov, A. Lascoux, and A. Thorup, On Giambelli’s theorem for complete correlations. Acta Math. 162 (1989), 143–199. [16] A. Lascoux, Fonctions de Schur et grassmanniennes. I. C. R. Acad. Sci. Paris Sér. A-B 281 (1975), 813–815. [17] A. Lascoux, Symmetric functions and combinatorial operators on polynomials. CBMS Regional Conf. Ser. in Math. 99, Amer. Math. Soc., Providence, RI, 2003. [18] A. Lascoux, Addition of ˙1: application to arithmetic. Sém. Lothar. Combin. B52a (2004), 9 pp. [19] A. Lascoux and P. Pragacz, Thom polynomials and Schur functions: the singularities A3 ./. Publ. Res. Inst. Math. Sci. 46 (2010), 183-200. [20] I. G. Macdonald, Symmetric functions and Hall-Littlewood polynomials. Second Edition, Oxford Math. Monographs, Oxford University Press, New York 1995. [21] M. Mikosz, P. Pragacz, and A. Weber, Positivity of Thom polynomials II; the Lagrange singularities. Fund. Math. 202 (2009), 65–79. [22] M. Mikosz, P. Pragacz, and A. Weber, Positivity of Legendrian Thom polynomials. J. Differential Geom. 89 (2011), no. 1, 111–132. [23] Ö. Öztürk, Thom polynomials and Schur functions: the singularities A4 ./. Serdica Math. J. 33 (2007), 301–320. [24] Ö. Öztürk, Thom polynomials and Schur functions: the singularities III2;3 . Ann. Polon. Math. 99 (2010), 295–304.

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[25] Ö. Öztürk, Selected topics on Thom polynomials and vector bundles. Ph.D. Thesis, IMPAN, Warsaw 2010. [26] Ö. Öztürk, Addendum: ACE algorithms for Thom polynomials of III3;3 .r/. Available at http://www.impan.pl/ pragacz/download/algIII33.pdf [27] I. Porteous, Simple singularities of maps. In Proc. Liverpool Singularities I, Lecture Notes in Math. 192, Springer-Verlag, Berlin 1971, 286–307. [28] P. Pragacz, Enumerative geometry of degeneracy loci. Ann. Sc. Ec. Norm. Sup. 21 (1988), 413–454. [29] P. Pragacz, Algebro-geometric applications of Schur S - and Q-polynomials. In Topics in invariant theory, Séminaire d’Algèbre Dubreil-Malliavin 1989-1990, M.-P. Malliavin, ed., Lecture Notes in Math. 1478, Springer-Verlag, Berlin 1991, 130–191. [30] P. Pragacz, Symmetric polynomials and divided differences in formulas of intersection theory. In Parameter spaces, P. Pragacz, ed., Banach Center Publ. 36, Polish Academy of Sciences, Warsaw 1996, 125–177. [31] P. Pragacz, Thom polynomials and Schur functions I. Preprint, arXiv:math/0509234v4 [math.AG]. [32] P. Pragacz, Thom polynomials and Schur functions: the singularities I2;2 ./. Ann. Inst. Fourier 57 (2007), 1487–1508. [33] P. Pragacz, Thom polynomials and Schur functions: towards the singularities Ai ./. In Real and complex singularities (Sao Carlos 2006), M. J. Saja and J. Seade, eds., Contemp. Math. 459, Amer. Math. Soc., Providence, RI, 2008, 165–178. [34] P. Pragacz and A. Thorup, On a Jacobi-Trudi identity for supersymmetric polynomials. Adv. in Math. 95 (1992), 8–17. [35] P. Pragacz and A. Weber, Positivity of Schur function expansions of Thom polynomials. Fund. Math. 195 (2007), 85–95. [36] P. Pragacz and A. Weber, Thom polynomials of invariant cones, Schur functions and positivity. In Algebraic cycles, sheaves, shtukas, and moduli, P. Pragacz, ed., Trends Math., Birkhäuser, Basel 2007, 117–129. [37] R. Rimányi, Thom polynomials, symmetries and incidences of singularities. Invent. Math. 143 (2001), 499–521. [38] F. Ronga, Le calcul des classes duales aux singularitiés de Boardman d’ordre 2. Comment. Math. Helv. 47 (1972), 15–35. [39] J. Stembridge, A characterization of supersymmetric polynomials. J. Algebra 95 (1985), 439–487. [40] R. Thom, Les singularités des applications différentiables. Ann. Inst. Fourier 6 (1955–56), 43–87. [41] S. Veigneau, ACE, an algebraic combinatorics environment for the computer algebra system MAPLE. User’s reference manual, Version 3.0, 1998; available at http://igm.univ-mlv.fr/~veigneau/public.html.

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Özer Öztürk, Department of Mathematics of Mimar Sinan Fine Arts University, Çıra˘gan C., Çi˘gdem S., No. 1, 34349, Be¸sikta¸s, Istanbul, Turkey E-mail: [email protected] ´ Piotr Pragacz, Institute of Mathematics of Polish Academy of Sciences, Sniadeckich 8, 00-956 Warszawa, Poland E-mail: [email protected]

A note on the kernel of the norm map Marek Szyjewski

1 Introduction For a fixed integer p and a field K let g.K/ D K  =K p be the p-th powers class group. For p D 2 there is the well-known Gross–Fischer exact sequence:   p  N f1; ag ,! g.K/ ! g K p a ! g .K/ (1.1) (cf. [3], p. 203). The group g.K/ may be expressed as Galois cohomology group     g.K/ D H 1 K; p D H 1 G .Ks =K/ ; p .Ks /   ; p .K/ , provided which is the group Hom G .Ks =K/   K contains a primitive p-th 1 1 root of unity. The norm map H L;  ! H K;  p p is corestriction. In the case p  a , the above sequence may be included in the long exact sequence p D 2, L D K [.a/

   !H i1 .K; 2 / ! H i .K; 2 / ! H i .L; 2 / [.a/

! H i .K; 2 / ! H iC1 .K; 2 / !    (see e.g. [1], Corollary 4.6). A generalization of the sequence (1.1) for p D 2 and several square roots (a multiquadratic extension) appeared in Theorem 2.1 of [2]. We are interested in a direct generalization for other values of p, assuming that K contains all p-th roots of unity. We show that in general the sequence (1.1) need not be exact even for p D 3. We show that this sequence is exact for p prime if K is a finite or local field, except in the case that p is the characteristic of the residue field. Thus we produce counterexamples that show that the well-known zero sequence   res   cor   H 1 K; p ! H 1 L; p ! H 1 K; p need not be exact for p > 2.

2 Notation and basic facts Let p be fixed positive integer. In this section we do not need p to be a prime. With a field K we associate an abelian group g.K/ – the cokernel of the homomorphism K W x 7! x p . The usual notation is the following: K p D im .K / ; K  =K p D g.K/

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(the notation K p should not be confused with the one for the cartesian p-th power). The operation g is functorial: an embedding r W K ! L induces a homomorphism rM W g.K/ ! g.L/:       coim .r/ M D K  =r 1 Lp Š r K  Lp =Lp D im .r/ M : If L=K is a finite field extension, then there is a norm homomorphism N D NL=K which commutes with : N B L D K B N I thus N W L ! K  induces a homomorphism NM W g .L/ ! g.K/. For every finite extension L=K of degree p (the same p fixed in the beginning to define g), if r W K ! L is a K-embedding, then NM B rM D 0 where 0 is a trivial homomorphism g.K/ ! g.K/ (this follows from N Br D N jK  D K .) In other words: the sequence NM

rM

g.K/ ! g.L/ ! g.K/

(2.1)

is a zero-sequence, or a complex, for .L W K/ D p. A natural question is whether for a degree p extension the image of rM is the kernel of NM , that is, whether this sequence is exact. The answer is positive for: • p D 2 and all K of characteristic different from 2 (Gross–Fischer theorem); • finite K and either an arbitrary p dividing jKj  1 or a prime p different from char .K/; • local K and a prime p different from the characteristic of the residue field. Proposition 1. If K is a finite field and either p divides jKj1 or p is a prime different form char .K/, .L W K/ D p, then the sequence (2.1) is exact. Proof. A finite field K has a unique extension L of degree p. Let v be a generator of the cyclic group L . Its norm is a product of its conjugate 2

NL=K .v/ D v 1CjKjCjKj

CCjKjp1

D v .jKj

p

1/=.jKj1/

and has order jKj  1. Thus NL=K W L ! K  is surjective, and so is NM W g.L/ ! g.K/. p  The assumption that p divides jKj  1 yields that L D K p u , where u is a generator of K  : K  D hui. Moreover p .K/ D Ker .K / D hu.jKj1/=p i

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483

and g .K/ is a cyclic group of order p. Thus im .K / is a cyclic group of order jKj1 p is a cyclic group of order p. Since jKj  1 divides jLj  1, the same holds for L: jg .K/j D jg.L/j D p: A generator uK p of g.K/ is a p-th power in L, so rM W g.K/ ! g.L/ is trivial and N W g.L/ ! g.K/ is surjective; hence N W g.L/ ! g.K/ is bijective. In the case of a prime p not dividing jKj it is easy to see that gcd .p; jLj  1/ D p p p jKj jLj jKj jKj .p; gcd  1/ since D  .mod p/. Thus L contains K u (and    p K p u W K D gcd .p; jKj  1/), rM W g.K/ ! g.L/ is trivial and jg .K/j D jg.L/j D gcd .p; jKj  1/; hence N is bijective.

3 The first counterexample Let p D 3. Let moreover L D C .t/ be the field of rational functions in one variable  t, and K D C t 3 . K is also a field of rational functions in one variable t 3 (we find p p the standard notation K D C .X /, t D 3 X cumbersome). Choose " D 1C2 3 , a primitive root of 1. Proposition 2. If p D 3, L D C.t / and K D C.t 3 /, then the norm of h.t / D a cube, while h.t / is not the product of an element of K and a cube.

t1 "t1

is

Proof. L=K is cyclic and the automorphism  of L, defined by  .t / D "t;

 jC D idC ;

generates the Galois group G.L=K/. It is easy to express the norm NL=K in terms of decomposition of irreducibles in C Œt:    k NL=K a .t  b/k D a3 t 3  b 3 : Let ' W L ! Z3 (a cartesian product here) be a homomorphism ' .f .t// D .v t1 .f .t// ; v"t1 .f .t // ; v"2 t1 .f .t /// which assigns orders of zeros in 1, "2 , " to a rational function f .t /. Firstly note that   ' L3 D 3Z3 : Secondly

  ' K  D Z  .1; 1; 1/ :

The first observation enables a reduction mod 3:   'M W g.L/ ! Z33 ; 'M f L3 D '.f / .mod 3/

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where Z33 is again a cartesian power. The second observation yields that 'M .rM .g .K/// D lin ..1; 1; 1// is a line through .1; 1; 1/ in Z33 . Now the rational function h.t / D

t 1 t 1 D 2 L "t  1  .t  1/

has norm 1, NL=K .h .t// D 1, so the coset h.t /L3 is in the kernel of NM W g.L/ ! g.K/. On the other hand,   'M h.t /L3 D .1; 1; 0/ does not belong to the line 'M .rM .g .K/// D lin ..1; 1; 1//, hence h.t /L3 does not belong to rM .g .K//, i.e. is not a product of an element of K and a cube.

4 Local fields We shall prove that for a prime p and for local K containing a primitive p-th root of unity and L=K cyclic, the sequence (2.1) is exact except in the case when p is the characteristic of the residue field. Lemma 1. For a finite extension L=K of degree p the equality Ker.NM / D im .r/ M holds if and only if every ˛ in L such that NL=K .˛/ D 1 is of the form ˛ D xˇ p for some x 2 K  , ˇ 2 L . Proof. If Ker.NM / D im .r/ M and N .˛/ D 1, then ˛Lp 2 Ker.NM /, so ˛Lp D rM .x/  for suitable x 2 K ; therefore ˛Lp D xLp . Conversely, if N .˛/ D 1 implies that ˛Lp D rM .x/ and  2 L is such that NM . / D K p , then N . / D y p   N y 1  D 1;

for suitable y 2 K  ;

and the substitution ˛ D y 1  shows that y 1  D xˇ p ;

 D yxˇ p ;

Lp 2 im .r/ M :

Thus Ker.NM /  im.r/. M Theorem 1. If p is a prime, K is a local field with the residue field Kx of characteristic different from p, K contains  p  a primitive degree p root of unity, L=K is a cyclic extension and L D K p a , then the image of rM W g.K/ ! g .L/ is the kernel of NM W g .L/ ! g .K/.

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485

Note that for p D 2 (the case of Gross–Fischer theorem), every field K of characteristic different from 2 contains a primitive degree p root of 1 and every extension of degree p is cyclic. x D q, let OK be the ring of integers, and let x 7! xN be the residue Proof. Let jKj x By assumption K contains a p-th primitive root " of 1; the homomorphism OK ! K. residue "N 2 Kx is a primitive p-th root of 1, so p j q  1. Consider the following two cases: x is the residue field of the Case 1. L=K is unramified. If L=K is unramified and L x Kx is cyclic. If NL=K .˛/ D 1, then N x x .˛/ local field L, then L= L=K N D 1; thus there   x x exist t 2 K and b 2 L such that ˛N D t b p : If  2 K  has residue N D t, then the polynomial X p   1 ˛ 2 OK ŒX  x thus X p   1 ˛ has a root ˇ in L by Hensel’s Lemma; therefore has a root b in L, ˇ p   1 ˛ D 0;

˛ D ˇ p :

The lemma above yields that Ker.NM / D im .r/. M p  x D Kx and L D K p  , where Case 2. L=K is ramified. Since p is a prime, L  generates the maximal ideal of the ring OK . Let N .˛/ D 1. Then ˛N is a p-th root of 1: N .˛/ D 1;

˛N p D 1:

Let  2 K  be a p-th root of 1 such that N D ˛. N Obviously,    p N 1 ˛ D 1 N .˛/ D 1 and 1 ˛ D 1: The polynomial

X p  1 ˛ 2 OK ŒX 

x hence it has root ˇ in L (even in K); has root 1 in L, ˇ p  1 ˛ D 0;

˛ D ˇ p

and the lemma above yields that Ker.NM / D im .r/. M x In this case there is another counterexample. The other case is p D char.K/. p  p  Proposition 3. If p D 3, K D Q3 3 , Kx D F3 , L D K 6 3 , then the image of rM W g.K/ ! g .L/ is smaller than the kernel of NM W g .L/ ! g .K/.

486

M. Szyjewski

Proof. The subring OK =3OK of the factor ring OL =3OL Š F3 ŒX  =.X 6 /

  corresponds to F3 X 3 =.X 6 /. It is easy to see that 

a0 C a1 X C a2 X 2 C a3 X 3 C a4 X 4 C a5 X 5

3

D a0 C a1 X 3 ;

so any product x˛ 3 with x 2 OK and ˛ 2 OL reduces mod 3 to an element of F3 X 3 =.X 6 /. p

" D 31 is a primitive root of unity. If  is the generator of Galois group 2 G .L=K/ such that p  p 6 6  3 D " 3; p   p  1  6 3 1  " 6 3 D p p 1  6 3 1  6 3

then

has norm 1. Since p  p p 2  1 1p 1  " 6 3 6 6 6 C 3 1 C 3 C 3 D p p 6 2 1  3 1  6 3 p 4  p 5  p 6 6 6 6 D1C 3 C 3 C 3 10 p 6 7  p 8  p 9  3 1  p 6 6 6 C C 3 C 3 C 3 p ; 2 1  6 3 if

p 1" 6 3 p 6 1 3

is a product x˛ 3 with x 2 K, ˛ 2 L, then clearing denominators one may

 , ˛ 2 OL . Thus assume that x 2 OK 3

6

p 1" 6 3 p 6 1 3

should reduce mod 3 to an invertible

element of F3 ŒX =.X /, while actually it reduces to 1 C X 4 C X 5 .

5 Global fields Theorem 2. Let p be a prime, p > 2, and let K be a global field. If L=K is a cyclic Galois extension of degree p, then the factor group Ker.NM /=im .r/ M is infinite. Proof. Denote by R, S the ring of integers in K, L respectively. Let  be a generator of the Galois group G .L=K/. There exist infinitely many prime ideals q of R which split completely in S: qS D q   .q/   2 .q/ : : :  p1 .q/ : There exists c 2 q n q2 which is coprime with qS  q1 D  .q/   2 .q/ : : :  p1 .q/ :

A note on the kernel of the norm map

487

The choice of c yields that the q-adic valuation of c equals 1 and the q-adic valuation c of  .c/ and  2 .c/ is 0. The element h .q/ D .c/ mod Lp belongs to Ker.NM /. There   is no x 2 K and ˇ 2 L such that hD

c D xˇ p ;  .c/

because this would imply that p  c h ˇ xˇ p  .c/ D ; D  c D  .h/ x .ˇ/p  .ˇ/   .c/ p  ˇ c 2 .c/ h D ; D  .h/  .ˇ/ . .c//2 while the q-adic valuation of

c 2 .c/ . .c//2

is exactly 1, so it is not divisible by p.

Thus there is an infinite set of distinct elements c Lp 2 Ker.NM / hLp D  .c/ which are not in im .r/. M ta Remark 1. In the setup of Proposition 2 one may use h.t / D "ta for a 2 C  to see that Ker.NM /=im.r/ M has cardinality of the continuum. One may use an algebraically closed field of arbitrary transfinite cardinality to obtain the same cardinality of Ker.NM /=im.r/.

Acknowledgment. We thank the referee for pointing out to us the argument in the proof of Theorem 2.

References [1] J. Kr. Arason, Cohomologische Invarianten quadratischer Formen. J. Algebra 36 (1975), 448–491. [2] R. Elman, T.Y. Lam, and A. R. Wadsworth, Quadratic forms under multiquadratic extensions. Nederl. Akad. Wetensch. Indag. Math. 42 (1980), no. 2, 131–145. [3] T. Y. Lam The algebraic theory of quadratic forms. W. A. Benjamin, Reading, Mass., 1973. Marek Szyjewski, ul. Mieszka I 15/97, 40-877 Katowice, Poland E-mail: [email protected]

Seshadri and packing constants Halszka Tutaj-Gasi´nska

1 Introduction In this note we consider a certain connection between the (local) positivity of line bundles on algebraic varieties and the symplectic packing of balls into symplectic manifolds. Let X be a smooth complex projective variety with an ample line bundle L. Local positivity of the bundle may be measured by Seshadri constants (introduced by Demailly in [9] and then generalized by Xu, [25]). These constants measure, roughly speaking, how small can be the ratio between the degree of a curve and the sum of the curve’s multiplicities in given points. On the other hand, if X is a smooth projective variety, it is also a symplectic manifold, with the symplectic form given by c1 .L/. We may then consider a symplectic packing of X, i.e. a symplectic embedding of a disjoint union of standard balls into X . The amount of the volume of X which may be filled by the images of the symplectically embedded balls is measured by so called packing constants – introduced and investigated by Gromov, McDuff, Polterovich and Biran, see [11], [15], [3]. In the first part of this note we collect some results concerning both kinds of constants and then we show a connection between them. In the second part of the paper we consider the special case of toric varieties, and we show that there is a formula connecting Seshadri constants in fixed points of a toric variety with packing constants, where packing in this case is symplectic and equivariant.

2 Seshadri constants In this chapter we recall the definition and basic facts about Seshadri constants. Let X be a smooth complex projective variety of dimension n, with an ample line bundle L and let P1 ; : : : ; PN be N different points on X . Definition 1. The Seshadri constant of L in P1 ; : : : ; PN is defined as the number ² ³ LC ".X; L; P1 ; : : : ; PN / WD inf ; multP1 C C    C multPN C where the infimum is taken over the set of all curves on X passing through at least one Pi . Equivalently ".X; L; P1 ; : : : ; PN / WD sup f" j   L  ".E1 C    C EN / is numerically effectiveg;

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H. Tutaj-Gasi´nska

where  W Xz ! X is the blow up of X in P1 ; : : : ; PN , with exceptional divisors E1 ; : : : ; EN . If the points P1 ; : : : ; PN are very general on X (i.e. they are outside a countable sum of algebraic subsets of HilbN .X /) we will write ".X; L; N / instead of ".X; L; P1 ; : : : ; PN /. (For the fact that ".X; L; P1 ; : : : ; PN /  ".X; L; Q1 ; : : : ; QN / if P1 ; : : : ; PN are very general on X and Q1 ; : : : ; QN 2 X see [14], Example 5.1.11). Remark 2. From the Seshadri criterion of ampleness (see e.g. [14], Theorem 1.4.13) it follows that for an ample line bundle L on X we have r n n L : 0 < ".X; L; P1 ; : : : ; PN /  N Finding the exact value of Seshadri constants is in most cases a difficult problem. For X D P 2 with L D OP 2 .1/ the exact values of ".X; L; N / are known only if N  9 or N D k 2 ; k 2 N. Namely, for N D 1; : : : ; 9 we have ".X; L; N / D 6 1 1; 12 ; 12 ; 12 ; 25 ; 25 ; 38 ; 17 ; 3 respectively; for N D k 2 , ".X; L; N / D k1 . Also, for X D P 1  P 1 with the line bundle of type .1; 1/, we know the values of 8 1 ".X; L; N / only for N  8 or N D 2k 2 ; k 2 N: ".X; L; N / D 1; 1; 23 ; 23 ; 35 ; 47 ; 15 ;2 1 2 for N D 1; : : : 8 respectively and ".X; L; N / D k for N D 2k . q The famous conjecture of Nagata states that ".P 2 ; OP 2 .1/; N / D N1 (so it is maximal possible) for all integers N  10 (cf. [12], [16]). We do not know so far a single example of a Seshadri constant with an irrational value. The main obstacle in proving that the constantqis irrational is that at present we are able to compute the constant only when either find a curve C such that

n

Ln N

is rational or when we can

LC : multP1 C C    C multPN C q n Then ".X; L; P1 ; : : : ; PN / is rational (see [17]). If it is less than n LN , we say that the constant is submaximal, and C is called a submaximal curve. The problem is that we do not have (at the moment) many ways of proving the nonexistence of submaximal curves. This makes it difficult proving that the Seshadri constants are maximal. ".X; L; P1 ; : : : ; PN / D

3 Packing numbers Let us now look at the symplectic side of the problem. Recall that a symplectic manifold is a smooth real manifold X (of real dimension 2n) with a closed nondegenerate 1 ^n differential 2-form !, so the volume form on X is given by nŠ ! . A basic example 2n is R with the 2-form !0 WD dx1 ^ dy1 C    C dxn ^ dyn .

Seshadri and packing constants

491

As mentioned in the introduction, another example of a symplectic manifold is produced by a smooth complex projective variety X with an ample line bundle L. This variety may be treated as a real 2n-dimensional manifold with the closed nondegenerate differential 2-form given by the first Chern class of L, !L D c1 .L/. Then the volume 1 n of X equals vol X D nŠ L . For two symplectic manifolds, .X1 ; !1 / and .X2 ; !2 / we define a symplectic embedding of X1 to X2 . Definition 3. We say that f W .X1 ; !1 / ! .X2 ; !2 / is a symplectic embedding if f is a C 1 -diffeomorphism onto the image and f  !2 D !1 : Consider the symplectic packing problem: Given a symplectic manifold .X; !/ find the maximal radius R such that there exists a symplectic embedding of a disjoint union of N euclidean balls of radius R into the given symplectic manifold .X; !/, fW

N a

.B 2n .R/; !0 / ! .X; !/:

iD1

Assume that the volume of X is finite. Then there is an obvious upper bound on R, N vol.B 2n .R//  vol.X /. However, it may happen that the volume bound is not the only obstacle for packing the balls into X , and even if the volume of X is infinite, there may be obstructions for packing balls into X . Let us recall here the Gromov Nonsqueezing Theorem (see [11]), which says that if there exists a symplectic embedding of a ball .B 2n .R/; !0 / into .B 2 ./  R2n2 ; !0 /, then R  . Let .X; !/ be a symplectic manifold and assume that vol X is finite. Packing constants (or packing numbers) measure how much of the volume of .X; !/ may be filled with the symplectic images of euclidean balls (see [3],[15]). Definition 4. Let .X; !/ be a symplectic manifold and let N be a natural number. A symplectic packing constant is defined as ³ ² N vol.B 2n .R// ; vN .X; !/ WD sup vol.X / where the supremum is taken over all R, such that there exists a symplectic packing ` 2n fW N iD1 .B .R/; !0 / ! .X; !/: If vN .X; !/ D 1 we say that full packing exists. By vN .X; !; P1 ; : : : ; PN / we will denote the analogously defined packing constant, with the images of the centers of the balls in P1 ; : : : ; PN . If .X; !/ are clear from the context, we will write vN instead of vN .X; !/. Following Lazarsfeld in [13] we define similar constants for embeddings being both symplectic and holomorphic:

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Definition 5. Let .X; !/ be a symplectic and holomorphic manifold and let N be a natural number. A symplectic and holomorphic packing constant is defined as ² ³ N vol.B 2n .R// h vN .X; !/ WD sup ; vol.X / where the supremum is taken over all R, such that there exists a symplectic and holo` 2n morphic packing f W N iD1 .B .R/; !0 / ! .X; !/: There are many interesting results about the constants vN .X; !/, cf. e.g. [3], [4], [15]. In his famous paper [4], Biran proved the following theorem (here quoted in the version restricted to algebraic surfaces with the symplectic form !L ): Theorem 6. Let .X; L/ be a smooth projective algebraic surface, treated as a 4-dimensional symplectic manifold with the symplectic form !L . Then there exists a number N0 , such that for any N  N0 there exists full packing, i.e. vN .X; !/ D 1. Moreover, this N0 can be taken equal k02 L2 where k0 is such that the linear system jk0 Lj contains a curve C of genus at least one.

4 Connection It seems that there exists a close connection between Seshadri constants and packing numbers. The possibility that such a connection exists was first observed in [15] and then in [3], [4], [13] and others. Consider the following examples: Example 7. Let X D P 2 with L D OP 2 .1/. For N D 1; : : : ; 9 we have ".X; L; N / D 6 1 1; 12 ; 12 ; 12 ; 25 ; 25 ; 38 ; 17 ; 3 respectively. In the same range of N , we have (see [3]): q L2 vN 24 63 288 vN D 1; 12 ; 34 ; 1; 20 ; ; ; ; 1, so ".X; L; N / D here. 25 25 64 289 N Example 8. For X D P 1 P 1 with the line bundle of type .1; 1/, we know ".X; L; N / 8 1 for N  8: ".X; L; N / D 1; 1; 23 ; 23 ; 35 ; 47 ; 15 ; 2 . From [3] we know that vN D 2 8 9 48 224 1 ; 1; 3 ; 9 ; 10 ; 49 ; 225 ; for N < 9 and 1 for all N  9. Thus here ".X; L; N / D 2 q L2 vN for N < 9. N The next set of examples is given by some surfaces with Picard number  D 1. Recently q Szemberg in [23] proved that for a surface X with Picard number one and 2 with La 2 N (where a is a positive integer and L is the ample generator of the Picard group of X ) the constant ".X; L; a/ is the maximal possible. Roé and Ross in [20] proved the following result. Theorem 9. Let X be a projective variety of dimension n with an ample line bundle L. Let r, s be integers. Then ".X; L; sr/  ".X; L; s/  ".P n ; OP n .1/; r/:

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Let now X be a projective q surface with  D 1 and let L be the ample generator 2 of the Picard group of X . Let LN D pq (where .p; q/ D 1). Then L2 D ap 2 and N D aq 2 for a positive integer a. We know that ".P 2 ; OP 2 .1/; q 2 / D q1 . Szemberg’s result gives that ".X; L; a/ D p. Thus, Theorem 9 implies that r L2 p  ".X; L; N /  ".X; L; a/".P 2 ; OP 2 .1/; q 2 / D ; N q so ".X; L; N / D

p : q

This gives us the following example. Example 10. Let X be a surface with Picard number  D 1. Let L be the ample generator of the Picard group of X . Assume also that L2 C LKX  0 – this implies that jLj contains a curve of genus at least one. Thus, from Theorem 6 it follows that 2 for these surfaces vN .X; !L / D 1, for q any N  L . So, we have that if N  L2 and

L2 N

2 Q, then

r ".X; L; N / D

L2 vN .X; !L / D N

r

L2 : N

Remark 11. We know by the results of Biran, [4, 3] that if N  9 for P 2 with L D OP 2 .1/ or N  8 for P 1  P 1 with L of type .1; 1/, then vN D 1. All the above speak in favour of the following Biran–Nagata–Szemberg Conjecture (see e.g. [22]): Conjecture 12. For any algebraic surface X, with an ample line q bundle L there exists

a number N0 , such that for any N  N0 we have ".X; L; N / D

L2 . N

We may consider also the following problem: Problem 13. For which complex projective surfaces X with an ample line bundle L and the symplectic form given by !L D c1 .L/ and for which natural N r L2 ‹ (1) ".X; L; N / D vN .X; !L / N We have seen that for X D P 2 ; L D OP 2 .1/ and N  9 the equality holds. It holds also for X D P 1  P 1 with the line bundle of type .1; 1/ and N  8 or for the surfaces as in Example 10 (and N D 1), but it is too much to expect it holds for any X, L and N . Consider the following example.

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Example 14. All abelian surfaces .X; L/ with the line bundle L of type .1; 1/ are symplectomorphic, so v1 .X; !L / is the same for them (however, unknown so far). On the other hand, if X D E  E, where E is an elliptic curve, then ".X; L; 1/ D 1 and for a generic X we have ".X; L; 1/ D 43 , see [21]. Anyway, it would be very interesting to understand when and why the equality in Problem 13 does or does not hold. There are also known results giving bounds for Seshadri constants by means of packing constants. In [5] Biran and Cieliebak proved that there is always an inequality in (1). Theorem 15. Let X be a projective n-dimensional manifold with an ample line bundle L. Then r Ln n vN .X; !L /  ".X; L; N /: N On the other hand, holomorphic and symplectic packing constants give the lower bound. Lazarsfeld in [13] proved the following result. Theorem 16. For a projective n-dimensional manifold X with an ample line bundle L we have r Ln n h ".X; L; N /  vN .X; !L / : N Remark 17. Lazarsfeld’s proof of this result is based on the construction of symplectic blowing up, cf. [15]. The theorem in [13] is actually stated for N D 1, but it can be generalized for N  1. For another proof of the result see [24].

5 Special case: toric manifolds In this part of the paper we consider the special case of toric varieties. On the one hand we have Seshadri constants in a fixed point of a toric variety, on the other hand we may consider a symplectic and equivariant embedding of a ball into the manifold and for such an embedding analogously define the packing constant v1 . It turns out that in this special case the equality (1) holds. 5.1 Seshadri constants on toric manifolds. This chapter is written on the base of Chapter 4 from [2]; for more about toric varieties see e.g. [10] or [7]. First, let us recall some basic facts about toric manifolds. Let X be a nonsingular compact toric variety, i.e. an n-dimensional smooth compact complex manifold with an action of a torus .C  /n , such that .C  /n is a Zariski open subset of X and the action of .C  /n on itself extends to the action of .C  /n on X . In what follows we assume that X is smooth and projective. We also assume that the standard volume form !0 is normalized in such a way that the area of the unit disc 2n is one. This implies that vol.B 2n .R// D RnŠ :

Seshadri and packing constants

495

X may be described by means of a fan,   M , where M Š Z2n is a lattice. In particular, prime torus invariant divisors correspond bijectively to 1-dimensional cones in . The toric variety X with an ample line bundle L may be describedPby a certain lattice polytope. Every line bundle on X may be written as L D OX . iD1 ai Di /, where  is the rank of PicX and Di are prime action invariant divisors on X . Denote by ni the lattice generators of the cones corresponding to Di . We define a polytope of .X; L/ as P .X; L/ WD fm 2 Mjhm; ni i  ai ; for any ni 2 M g; where M denotes the lattice dual to M . A polytope of .X; L/ is called a Delzant polytope if there are exactly n edges from each vertex and for each vertex, the first integer points on the edges (originating from the vertex) form the basis of the lattice. If X is nonsingular, then its polytope is Delzant, and in [8] it is proved that X as above is uniquely determined by its Delzant polytope. Let us denote by P .k/ the set of faces of P .X; L/ of dimension k. The elements of P .k/ correspond bijectively to the invariant (with respect to the action of .C  /n ) subvarieties of X of dimension k; so to each vertex of the polytope corresponds one fixed point of X , each edge corresponds to an invariant curve etc. For any edge e 2 P .1/ by l.e/ denote the length of e, i.e. the number of lattice points on e minus one. For a vertex w 2 P .0/ define s.P .X; L/; w/ WD minfl.e/jw 2 eg: In [2] the following theorem is proved: Theorem 18. Let x be a fixed point of X , corresponding to the vertex w 2 P .0/. Then ".X; L; x/ D s.P .X; L/; w/: 5.2 Packing one ball into a toric manifold. This chapter is based on the results proved by Pelayo and Schmidt in [18], [19]. Let .X; L/ be a smooth projective toric variety, as described in the previous subsection. Then X has a symplectic form !L D c1 .L/. It may be seen that the restriction of the .C  /n action to its real subgroup T n D .S 1 /n is effective and Hamiltonian, see [6]. Thus, X is a symplectic toric manifold (i.e. a compact connected symplectic manifold of real dimension 2n with an effective and Hamiltonian torus action of T n , see [6]). For such X we may consider a symplectic and equivariant packing of balls. Definition 19. Let .X; L/ be a symplectic toric manifold, with the torus action W T n  X ! X. Let ƒ 2 Aut.T n /: A subset B  X is a ƒ-equivariantly embedded ball (of radius r) if there exists a symplectic embedding f W B.r/ ! B D f .B.r//  X such

496

H. Tutaj-Gasi´nska

that T n  B.r/

ƒf

J

Rot

 B.r/

f

/ Tn  X  /X

where the action Rot W T n  B.r/ ! B.r/ is given by .1 ; : : : ; n /.z1 ; : : : ; zn / ! .1 z1 ; : : : ; n zn /. B is called an equivariantly embedded ball if there exists ƒ 2 Aut.T n /, such that B is a ƒ-equivariantly embedded ball. Let e1 ; : : : ; en be the standard basis of Rn . Definition 20. 1. By .r/ we denote the set of points in Rn , belonging to the convex hull of f0; r 2 e1 ; : : : ; r 2 en g but not to the convex hull of fr 2 e1 ; : : : ; r 2 en g. 2. Let  be a Delzant polytope. A subset w of  is called an p admissible simplex . r/ by an element of of radius r with center at the vertex w, if w is the image of p AGL(n; Z) which takes the origin to w and the edges of . r/ to the edges of w meeting w. (AGL(n; Z) is the special affine group of Rn with integer coefficients). We define rw WD maxfr > 0 W there exists an admissible simplex w of radius rg: Lemma 21 ([19], Lemma 2.10). rw D minfl.e/jw 2 eg: Thus, keeping the notation of the previous subsection, rw D s.P .X; L/; w/. Remark 22. .r/ is the image of B.r/  C n by the momentum map (for the action T n  B.r/ ! B.r/ given by .1 ; : : : ; n /  .z1 ; : : : ; zn / D .1 z1 ; : : : ; n zn /). We have the following facts. Theorem 23 ([19], Lemma 2.10). Let .X; L/ be a symplectic toric manifold with Delzant polytope P .X; L/. Let w 2 P .0/ be a vertex of P .X; L/ Then, there is an admissible simplex w  P .X; L/ of radius r if and only if 0  r  rw . Theorem 24 ([19], Lemma 2.13). Let .X; L/ be as above, let W T n  X ! X and let the momentum map of be  W X ! Rn . Let B  X be a symplectically and equivariantly embedded ball of radius r and the center mapped to a fixed point x 2 X . Then .B/ is an admissible simplex of radius r 2 , with center .x/. Conversely, having an admissible simplex w of radius p r, there exists a symplectically and equivariantly embedded ball B 2 X of radius r, with .B/ D w .

Seshadri and packing constants

497

Summarizing the above, we see that for any symplectically and equivariantly embedded ball (with the image of the center in a fixed point x of X , corresponding to the vertex w) we have an admissible simplex (and vice versa). Let x 2 X be a fixed point and w the corresponding vertex. From Theorem 23, it follows that if we pack symplectically and equivariantly a ball into X , so that x is the image of the center of the ball, then the radius of the ball must be such that r 2  rw . Denote by vol B 2n .r/ ; v1 .X; !L ; w/ WD sup vol X where supremum is taken over r such that there exists symplectic and equivariant embedding of B 2n .r/ into X , with the image of the center in x. 5.3 Problem 13 on toric manifolds. From Theorem 18 and from the discussion above we have the following result. Proposition 25. For a symplectic toric variety .X; L/ with a fixed point x corresponding to the vertex w 2 P .0/, we have ".X; L; x/ D

p n Ln v1 .X; !L ; w/:

Proof. From Theorem 18 we know that ".X; L; x/ D s.P .X; L/; w/: Taking R being the supremum of radii of all balls, such that there exists a symplectic and equivariant embedding of B 2n .r/ into X , with the image of the center in w, we have vol B 2n .R/ : v1 .X; !L ; w/ D Ln =nŠ On the other hand, we know from Theorems 23 and 24 and from the observation above that R2 D rw D s.P .X; L/; w/. Thus, v1 .X; !L ; w/ D So,

and finally

rwn =nŠ s.P .X; L/; w/n : D Ln =nŠ Ln

p n Ln v1 .X; !L ; w/ D s.P .X; L/; w/; p n Ln v1 .X; !L ; w/ D ".X; L; w/:



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References [1] M. Audin and J. Lafontaine, Holomorphic curves in symplectic geometry. Progr. Math. 117, Birkhäuser, Basel 1994. [2] Th. Bauer, S. Di Rocco, B. Harbourne, M. Kapustka, A. L. Knutsen, W. Syzdek, and T. Szemberg, A primer on Seshadri constants. In Interactions of classical and numerical algebraic geometry, Contemp. Math. 496, Amer. Math. Soc., Providence, RI, 2009, 33–70. [3] P. Biran, Symplectic packing in dimension 4. Geom. Funct. Anal. 7 (1997), 420–437. [4] P. Biran, A stability property of symplectic packing. Invent. Math. 136 (1999), 123–155. [5] P. Biran and K. Cieliebak, Symplectic topology on subcritical manifolds. Comment. Math. Helv. 76 (2002), 712–753 [6] A. Cannas da Silva, Symplectic toric manifolds. Lecture notes for the CRM (Barcelona) short course delivered in July of 2001; http://www.math.princeton.edu/~acannas/. [7] D. Cox, Lectures on toric varieties. http://www.cs.amherst.edu/~dac/lectures/coxcimpa.pdf. [8] T. Delzant, Hamiltoniens p´eriodiques et images convexes de l’application moment. Bull. Soc. Math. France 116 (1988), 315–339. [9] J.-P. Demailly, Singular Hermitian metrics on positive line bundles. In Complex algebraic varieties, Lecture Notes in Math. 1507, Springer-Verlag, Berlin 1992, 87–104. [10] W. Fulton, Introduction to toric varieties. Ann. of Math. Stud. 131, Princeton University Press, Princeton 1993. [11] M. Gromov, Pseudo-holomorphic curves in symplectic manifolds. Invent. Math. 82 (1985), 307–347. [12] B. Harbourne, On Nagata’s conjecture. J. Algebra 236 (2001), 692–702. [13] R. Lazarsfeld, Lengths of periods and Seshadri constants of abelian varieties. Math. Res. Lett 3 (1996), 439–449. [14] R. Lazarsfeld, Positivity in algebraic geometry I, II. Ergeb. Math. Grenzgeb. (3) 48, 49, Springer-Verlag, Berlin 2004. [15] D. McDuff and L. Polterovich, Symplectic packing and algebraic geometry. Invent. Math. 115 (1994), 405–429. [16] M. Nagata, On the 14-th problem of Hilbert. Amer. J. Math. 81 (1959), 766–772. [17] K. Oguiso, Seshadri constants in a family of surfaces. Math. Ann. 323 (2002), 625–631. [18] A. Pelayo, Toric symplectic ball packing. Topology Appl. 157 (2006), 3633–3644. [19] A. Pelayo and B. Schmidt, Maximal ball packing of symplectic toric manifolds. Internat Math. Res. Notices (2008), 24p ID rnm 139. [20] J. Roé and J. Ross, An inequality between multipoint Seshadri constants. Geom. Dedicata 140 (2009), 175–181. [21] A. Steffens, Remarks on Seshadri constants. Math. Z. 227 (1998), 505—510. [22] T. Szemberg, Global and local positivity of line bundles. Habilitationsschrift Essen, 2001. [23] T. Szemberg, Bounds on Seshadri constants on surfaces with Picard number 1. Comm. Alg., to appear; arXiv:1104.1198v1 [math.AG].

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[24] H. Tutaj-Gasi´nska, A short note on Seshadri constants and packing numbers. Ann. Polon. Math. 103 (2012), no. 1, 59–65. [25] G. Xu, Ample line bundles on smooth surfaces. J. Reine Angew. Math. 469 (1995), 199–209. Halszka Tutaj-Gasi´nska, Jagiellonian University, Institute of Mathematics, Łojasiewicza 6, 30348 Kraków, Poland E-mail: [email protected]

List of contributors Klaus Altmann, Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany E-mail: [email protected] Dave Anderson, Department of Mathematics, University of Washington, Seattle, WA 98195, U.S.A. E-mail: [email protected] Thomas Bauer, Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein-Straße, 35032 Marburg, Germany E-mail: [email protected] Gergely Bérczi, Mathematical Institute, University of Oxford, 24–29 St Giles, OX1 3LB Oxford, UK E-mail: [email protected] Piotr Blass, Ulam University, 113 West Tara Lakes Drive, Boynton Beach, FL 33436-6763, U.S.A. E-mail: [email protected] Cristiano Bocci, Dipartimento di Scienze Matematiche e Informatiche “R. Magari”, Università di Siena, Pian dei Mantellini 44, 53100 Siena, Italy E-mail: [email protected] Paolo Cascini, Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK E-mail: [email protected] Susan Cooper, Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, U.S.A. E-mail: [email protected] Sławomir Cynk, Institute of Mathematics, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland; and Institute of Mathematics of the Polish Academy of ´ Sciences, ul. Sniadeckich 8, 00-956 Warszawa, Poland E-mail: [email protected] Sandra Di Rocco, Department of Mathematics, Royal Institute of Technology (KTH), 10044 Stockholm, Sweden E-mail: [email protected] Marcin Dumnicki, Institute of Mathematics, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland E-mail: [email protected]

502

List of contributors

Gavril Farkas, Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany E-mail: [email protected] Letterio Gatto, Dipartimento di Matematica, Politecnico di Torino, 10129 Torino, Italy E-mail: [email protected] Brian Harbourne, Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, U.S.A. E-mail: [email protected] Andreas Hochenegger, Mathematisches Institut der Freien Universität Berlin, Arnimallee 3, 14195 Berlin, Germany E-mail: [email protected] Kevin Hutchinson, School of Mathematical Sciences, Belfield Office Park 9/10, Belfield, Dublin 4, Ireland E-mail: [email protected] Nathan Owen Ilten, Max Planck Institut für Mathematik, PF 7280, 53072 Bonn, Germany E-mail: [email protected] Kelly Jabbusch, Department of Mathematics, Royal Institute of Technology (KTH), 10044 Stockholm, Sweden E-mail: [email protected] Clemens Jörder, Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstraße 1, 79104 Freiburg, Germany E-mail: [email protected] Michał Kapustka, Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland, and Institute of Mathematics, Jagiellonian University of Kraków, ul.Łojasiewicza 6, 30-348 Kraków, Poland E-mail: [email protected] Stefan Kebekus, Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstraße 1, 79104 Freiburg, Germany E-mail: [email protected] Andreas Leopold Knutsen, Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, 5008 Bergen, Norway E-mail: [email protected] Alex Küronya, Budapest University of Technology and Economics, Mathematical Institute, Department of Algebra, Pf. 91, 1521 Budapest, Hungary; current address: Albert-Ludwigs-Universität Freiburg, Mathematisches Institut, Eckerstraße 1, 79104 Freiburg, Germany E-mail: [email protected]

List of contributors

503

Vladimir Lazi´c, Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany E-mail: [email protected] Mateusz Michałek, Faculty of Mathematics and Computer Science, Jagiellonian University, 30-387 Kraków, Poland E-mail: [email protected] Rick Miranda, Department of Mathematics, Colorado State University, Fort Collins, CO 80523, U.S.A. E-mail: [email protected] Shigeru Mukai, Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan E-mail: [email protected] Mircea Musta¸ta˘ , Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, U.S.A. E-mail: [email protected] Özer Öztürk, Department of Mathematics of Mimar Sinan Fine Arts University, Çıra˘gan C., Çi˘gdem S., No. 1, 34349, Be¸sikta¸s, Istanbul, Turkey E-mail: [email protected] Lars Petersen, Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany E-mail: [email protected] ´ Piotr Pragacz, Institute of Mathematics of Polish Academy of Sciences, Sniadeckich 8, 00-956 Warszawa, Poland E-mail: [email protected] Sławomir Rams, Institute of Mathematics, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland E-mail: [email protected] Joaquim Roé, Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain E-mail: [email protected] Hal Schenck, Mathematics Department, University Illinois, Urbana, IL 61801, U.S.A. E-mail: [email protected] Inna Scherbak, School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel E-mail: [email protected]

504

List of contributors

Hendrik Süß, Institut für Mathematik, LS Algebra und Geometrie, Brandenburgische Technische Universität Cottbus, PF 10 13 44, 03013 Cottbus, Germany E-mail: [email protected] Tomasz Szemberg, Instytut Matematyki UP, PL-30-084 Kraków, Poland; current address: Albert-Ludwigs-Universität Freiburg, Mathematisches Institut, Eckerstraße 1, 79104 Freiburg, Germany E-mail: [email protected] Marek Szyjewski, ul. Mieszka I 15/97, 40-877 Katowice, Poland E-mail: [email protected] Halszka Tutaj-Gasi´nska, Jagiellonian University, Institute of Mathematics, Łojasiewicza 6, 30348 Kraków, Poland E-mail: [email protected] Masha Vlasenko, School of Mathematics, Trinity College, Dublin 2, Ireland E-mail: [email protected] Robert Vollmert, Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany E-mail: [email protected] Frederik Witt, Mathematisches Institut der Universität Münster, Einsteinstraße 62, 48149 Münster, Germany E-mail: [email protected] Zach Teitler, Department of Mathematics, Boise State University, 1910 University Drive, Boise, ID 83725-1555, U.S.A. E-mail: [email protected]