Cohomological and geometric approaches to rationality problems: New perspectives

Progress in Mathematics Volume 282 Series Editors Hyman Bass Joseph Oesterl´e Alan Weinstein Cohomological and Geomet...

Author: Fedor Bogomolov | Yuri Tschinkel

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Progress in Mathematics Volume 282 Series Editors Hyman Bass Joseph Oesterl´e Alan Weinstein

Cohomological and Geometric Approaches to Rationality Problems New Perspectives

Fedor Bogomolov Yuri Tschinkel Editors

Birkh¨auser Boston • Basel • Berlin

Fedor Bogomolov Department of Mathematics New York University Courant Institute of Mathematical sciences New York, NY 10012 U.S.A [email protected]

Yuri Tschinkel Department of Mathematics New York University Courant Institute of Mathematical sciences New York, NY 10012 U.S.A [email protected]

ISBN 978-0-8176-4933-3 e-ISBN 978-0-8176-4934-0 DOI 10.1007/978-0-8176-4934-0 Library of Congress Control Number: 2009939069 Mathematics Subject Classification (2000): 11R32, 12F12, 13A50, 14D20,14E05, 14E08, 14F20, 14G05, 14G15, 14H10, 14H45, 14H60, 14J32, 14J35, 14L30 c Birkh¨auser Boston, a part of Springer Science+Business Media, LLC 2010

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkh¨auser Boston, c/o Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper. Birkh¨auser Boston is part of Springer Science+Business Media (www.birkhauser.com)

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

The Rationality of Certain Moduli Spaces of Curves of Genus 3 Ingrid Bauer and Fabrizio Catanese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

The Rationality of the Moduli Space of Curves of Genus 3 after P. Katsylo Christian B¨ ohning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

Unramified Cohomology of Finite Groups of Lie Type Fedor Bogomolov, Tihomir Petrov and Yuri Tschinkel . . . . . . . . . . . . . . . . .

55

Sextic Double Solids Ivan Cheltsov and Jihun Park . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Moduli Stacks of Vector Bundles on Curves and the King– Schofield Rationality Proof Norbert Hoﬀmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Noether’s Problem for Some p -Groups Shou-Jen Hu and Ming-chang Kang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Generalized Homological Mirror Symmetry and Rationality Questions Ludmil Katzarkov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 The Bogomolov Multiplier of Finite Simple Groups Boris Kunyavski˘ı . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

vi

Contents

Derived Categories of Cubic Fourfolds Alexander Kuznetsov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Fields of Invariants of Finite Linear Groups Yuri G. Prokhorov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 The Rationality Problem and Birational Rigidity Aleksandr V. Pukhlikov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Preface

The rationality problem links algebra and geometry. On the level of ﬁelds it can be formulated as follows: Let K be a ﬁeld of ﬁnite transcendence degree over a ground ﬁeld k, i.e., K is the ﬁeld of rational functions on an algebraic variety deﬁned over k. Decide whether or not K is a purely transcendental extension of k. It became apparent that, for k = C, the geometry of the corresponding variety is tightly linked with rationality of the ﬁeld K. The diﬃculty of the rationality problem depends on the transcendence degree of K over k or, geometrically, on the dimension of the variety. A major success of the 19th and the ﬁrst half of the 20th century algebraic geometry was a complete solution of the rationality problem in dimensions one and two over algebraically closed ground ﬁelds of characteristic zero. In these cases it suﬃces to consider ﬁnite, easily computable geometric invariants. For such ﬁelds in dimension two, the presence of suﬃciently ﬂexible rational curves suﬃces for the rationality of the ﬁeld. This geometric property (rational connectedness) deﬁnes a class of algebraic varieties (and ﬁelds) which in higher dimensions is substantially larger than the class of rational varieties: it is known that most of rationally connected varieties are not rational already in dimension three. It is still an open question whether or not they are unirational, i.e., the corresponding function ﬁelds are proper subﬁelds of rational ﬁelds. The papers in this collection are devoted to various aspects of rationality. Four articles address the rationality of quotient spaces. The classical preHilbert 19th century invariant theory dealt mostly with rings of invariants of linear actions of groups on complex linear spaces—many beautiful explicit constructions can be traced back to the works of Sylvester, Cayley, and Gordan. Emmy Noether, who was a student of Paul Gordan, one of the experts in the theory of explicit invariants, formulated a general question: are all such ﬁelds of invariants rational? The ﬁrst counterexamples, over nonclosed ground ﬁelds, were constructed by Swan, and over the complex numbers, by Saltman. This made it even more interesting to determine all rational ﬁelds of invariants.

viii

Preface

The article by Prokhorov reviews known results in case of four-dimensional complex representations of ﬁnite groups. It is shown that almost all of the arising quotients are rational. The paper by Hu and Kang contains a proof of rationality for quotients by small p-groups. Kunyavski˘ı proves triviality of the ﬁrst cohomological obstruction to rationality, for all ﬁnite simple groups and quasi-simple groups of Lie type. He also shows the existence of exceptions— there are quasi-simple groups which are central extensions of one particular simple group of Lie type with a nonvanishing obstruction. Remarkably, these are exactly the extensions which are not of Lie type themselves. The paper by Bogomolov, Petrov, and Tschinkel is devoted to more general cohomological invariants of quasi-simple groups of Lie type. It is shown that all obstructions to rationality coming from nonramiﬁed cohomology vanish for a large class of quasi-simple groups of Lie type, at least when the group of coeﬃcients has order coprime to the characteristic of the ﬁeld of deﬁnition of the group. Two papers treat rationality of the moduli space of curves of genus three. This was established 15 years ago by Katsylo but the proof was computationally very involved and diﬃcult to follow. C. B¨ ohning wrote a detailed and transparent version of this proof. Bauer and Catanese prove rationality of the related moduli space of curves of genus three with a ﬁxed nontrivial 3-torsion point in the Jacobian. They identify this moduli space with the moduli space of curves of bidegree (4, 4) ⊂ P1 × P1 , with simple singularities at the presecribed set of six points. Hoﬀmann’s paper is devoted to rationality of moduli spaces of vector bundles on curves. It contains a stack-theoretic version of the King– Schoﬁeld proof of the rationality of the moduli space of vector bundles on a projective curve having a ﬁxed determinant line bundle, with coprime rank and degree of the determinant. Rational ﬁelds have many automorphisms and the absence of these points to nonrationality. Pukhlikov surveys nonrationality results obtained via the study of rigidity properties of the projective models of the ﬁeld. First examples of rigid rationally connected varieties were discovered by Manin and Iskovskikh in the seventies. Cheltsov and Park consider an arithmetic aspect of rationality: a classical diophantine problem concerns the description of rational solutions of a system of polynomial equations, i.e., rational points on algebraic varieties. If the variety is rational or unirational, then there are many such points, at least over a ﬁnite extension of the ground ﬁeld. This potential density of rational points holds for all Fano varieties of dimension ≤ 3 with an exception of one class: double covers of the three-dimensional projective space ramiﬁed over a surface of degree 6. Cheltsov–Park show that potential density of rational points holds if the ramiﬁcation surface has at least one singular point. They also show that these varieties are superrigid. Katzarkov introduces a completely novel approach to the rationality problem, based on ideas of Mirror Symmetry. The Mirror Symmetry conjecture is still open but the ideas suggested by physics already highlight new interesting phenomena related to rationality. Applications of derived categories to rationality questions are discussed in the paper by Kuznetsov. He studies the

Preface

ix

derived category of coherent sheaves of four-dimensional cubic hypersurfaces. It is expected that a generic cubic fourfold is nonrational; this remains one of the challenging problems in the ﬁeld. Needless to say, the book does not cover the whole spectrum of modern approaches to rationality. Nevertheless, we hope that it provides a glimpse of the variety of related concrete problems, new methods and results.

Fedor Bogomolov Yuri Tschinkel April 2009

The Rationality of Certain Moduli Spaces of Curves of Genus 3 Ingrid Bauer and Fabrizio Catanese Mathematisches Institut Universität Bayreuth, NW II D-95440 Bayreuth, Germany [email protected], [email protected] Summary. We prove rationality of the moduli space of pairs of curves of genus three together with a point of order three in their Jacobian.

Key words: Rationality, moduli spaces of curves 2000 Mathematics Subject Classiﬁcation codes: 14E08, 14H10, 14H45

1 Introduction The aim of this paper is to give an explicit geometric description of the birational structure of the moduli space of pairs (C, η), where C is a general curve of genus 3 over an algebraically closed ﬁeld k of arbitrary characteristic and η ∈ P ic0 (C)3 is a nontrivial divisor class of 3-torsion on C. As was observed in [B-C04, Lemma (2.18)], if C is a general curve of genus 3 and η ∈ P ic0 (C)3 is a nontrivial 3-torsion divisor class, then we have a morphism ϕη := ϕ|KC +η| × ϕ|KC −η| : C → P1 × P1 , corresponding to the sum of the linear systems |KC + η| and |KC − η|, which is birational onto a curve Γ ⊂ P1 × P1 of bidegree (4, 4). Moreover, Γ has exactly six ordinary double points as singularities, located in the six points of the set S := {(x, y)|x = y, x, y ∈ {0, 1, ∞}}. In [B-C04] we only gave an outline of the proof (and there is also a minor inaccuracy). Therefore we dedicate the ﬁrst section of this article to a detailed geometrical description of such pairs (C, η), where C is a general curve of genus 3 and η ∈ P ic0 (C)3 \ {0}. The main result of the ﬁrst section is the following: Theorem 1.1. Let C be a general (in particular, nonhyperelliptic) curve of genus 3 over an algebraically closed ﬁeld k (of arbitrary characteristic) and η ∈ P ic0 (C)3 \ {0}.

F. Bogomolov, Y. Tschinkel (eds.), Cohomological and Geometric Approaches to Rationality Problems, Progress in Mathematics 282, DOI 10.1007/978-0-8176-4934-0_1, © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

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I. Bauer and F. Catanese

Then the rational map ϕη : C → P1 × P1 deﬁned by ϕη := ϕ|KC +η| × ϕ|KC −η| : C → P1 × P1 is a morphism, birational onto its image Γ , which is a curve of bidegree (4, 4) having exactly six ordinary double points as singularities. We can assume, up to composing ϕη with a transformation of P1 × P1 in PGL(2, k)2 , that the singular set of Γ is the set S := {(x, y) ∈ P1 × P1 |x = y ; x, y ∈ {0, 1, ∞}}. Conversely, if Γ is a curve of bidegree (4, 4) in P1 × P1 , whose singularities consist of exactly six ordinary double points at the points of S, its normalization C is a curve of genus 3, such that OC (H2 − H1 ) =: OC (η) (where H1 , H2 are the respective pullbacks of the rulings of P1 × P1 ) yields a nontrivial 3-torsion divisor class, and OC (H1 ) ∼ = OC (KC + η), OC (H2 ) ∼ = OC (KC − η). From Theorem 1.1 it follows that M3,η := {(C, η) : C is a general curve of genus 3, η ∈ P ic0 (C)3 \ {0}} is birational to P(V (4, 4, −S))/S3 , where V (4, 4, −S) := H 0 (OP1 ×P1 (4, 4)(−2

(a, b))).

a=b,a,b∈{∞,0,1}

In fact, the permutation action of the symmetric group S3 := S({∞, 0, 1}) extends to an action on P1 , so S3 is naturally a subgroup of PGL(2, k). We consider then the diagonal action of S3 on P1 × P1 , and observe that S3 is exactly the subgroup of PGL(2, k)2 leaving the set S invariant. The action of S3 on V (4, 4, −S) is naturally induced by the diagonal inclusion S3 ⊂ PGL(2, k)2 . On the other hand, if we consider only the subgroup of order three of P ic0 (C) generated by a nontrivial 3-torsion element η, we see from Theorem 1.1 that we have to allow the exchange of η with −η, which corresponds to exchanging the two factors of P1 × P1 . Therefore M3,η := {(C, η ) : C general curve of genus 3, η ∼ = Z/3Z ⊂ P ic0 (C)} is birational to P(V (4, 4, −S))/(S3 × Z/2), where the action of the generator σ (of Z/2Z) on V (4, 4, −S) is induced by the action on P1 × P1 obtained by exchanging the two coordinates. Our main result is the following: Theorem 1.2. Let k be an algebraically closed ﬁeld of arbitrary characteristic. We have: 1) the moduli space M3,η is rational; 2) the moduli space M3,η is rational.

Rationality of Moduli

3

One could obtain the above result abstractly from the method of Bogomolov and Katsylo (cf. [B-K85]), but we prefer to prove the theorem while explicitly calculating the ﬁeld of invariant functions. It mainly suﬃces to decompose the vector representation of S3 on V (4, 4, −S) into irreducible factors. Of course, if the characteristic of k equals two or three, it is no longer possible to decompose the S3 -module V (4, 4, −S) as a direct sum of irreducible submodules. Nevertheless, we can write down the ﬁeld of invariants and see that it is rational. Acknowledgment. The research of the authors was performed in the realm of the DFG Forschergruppe 790 “Classiﬁcation of algebraic surfaces and compact complex manifolds.”

2 The geometric description of pairs (C, η) In this section we give a geometric description of pairs (C, η), where C is a general curve of genus 3 and η is a nontrivial element of P ic0 (C)3 , and we prove Theorem 1.1. Let k be an algebraically closed ﬁeld of arbitrary characteristic. We recall the following observation from [B-C04, p. 374]. Lemma 2.1. Let C be a general curve of genus 3 and η ∈ P ic0 (C)3 a nontrivial divisor class (i.e., η is not linearly equivalent to 0). Then the linear system |KC + η| is base point free. This holds more precisely under the assumption that the canonical system |KC | does not contain two divisors of the form Q + 3P , Q + 3P , and where the 3-torsion divisor class P − P is the class of η. This condition for all such η is in turn equivalent to the fact that C is either hyperelliptic or it is nonhyperelliptic but the canonical image Σ of C does not admit two inﬂexional tangents meeting in a point Q of Σ. Proof. Note that P is a base point of the linear system |KC + η| if and only if H 0 (C, OC (KC + η)) = H 0 (C, OC (KC + η − P )). Since dim H 0 (C, OC (KC + η)) = 2 this is equivalent to dim H 1 (C, OC (KC + η − P )) = 1. Since H 1 (C, OC (KC + η − P )) ∼ = H 0 (C, OC (P − η))∗ , this is equivalent to the existence of a point P such that P − η ≡ P (note that we denote linear equivalence by the classical notation “≡”). Therefore 3P ≡ 3P and P = P , whence in particular H 0 (C, OC (3P )) ≥ 2. By Riemann–Roch we have dim H 0 (C, OC (KC − 3P )) = deg(KC − 3P ) + 1 − g(C) + dim H 0 (C, OC (3P )) ≥ 1.

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In particular, there is a point Q such that Q ≡ KC − 3P ≡ KC − 3P . Going backwards, we see that this condition is not only necessary, but suﬃcient. If C is hyperelliptic, then Q + 3P, Q + 3P ∈ |KC |, hence P, P are Weierstrass points, whence 2P ≡ 2P , hence P − P yields a divisor class η of 2-torsion, contradicting the nontriviality of η. Consider now the canonical embedding of C as a plane quartic Σ. Our condition means, geometrically, that C has two inﬂection points P , P , such that the tangent lines to these points intersect in Q ∈ C. We shall show now that the (nonhyperelliptic) curves of genus 3 whose canonical image is a quartic Σ with the above properties are contained in a ﬁve-dimensional family, whence are special in the moduli space M3 of curves of genus 3. Let now p, q, p be three noncollinear points in P2 . The quartics in P2 form a linear system of dimension 14. Imposing that a plane quartic contains the point q is one linear condition. Moreover, the condition that the line containing p and q has intersection multiplicity equal to 3 with the quartic in the point p gives three further linear conditions. Similarly for the point p , and it is easy to see that the above seven linear conditions are independent. Therefore the linear subsystem of quartics Σ having two inﬂection points p, p , such that the tangent lines to these points intersect in q ∈ Σ, has dimension 14 − 3 − 3 − 1 = 7. The group of automorphisms of P2 leaving the three points p, q, p ﬁxed has dimension 2 and therefore the above quartics give rise to a ﬁve-dimensional algebraic subset of M3 . Finally, if the points P, P , Q are not distinct, we have (w.l.o.g.) P = Q and a similar calculation shows that we have a family of dimension 7 − 3 = 4.

Consider now the morphism ϕη (:= ϕ|KC +η| × ϕ|KC −η| ) : C → P1 × P1 , and denote by Γ ⊂ P1 × P1 the image of C under ϕη . Remark 2.2. 1) Since η is nontrivial, either Γ is of bidegree (4, 4), or degϕη = 2 and Γ is of bidegree (2, 2). In fact, deg ϕη = 4 implies η ≡ −η. 2) We shall assume in the following that ϕη is birational, since otherwise C is either hyperelliptic (if Γ is singular) or C is a double cover of an elliptic curve Γ (branched in 4 points). In both cases C lies in a ﬁve-dimensional subfamily of the moduli space M3 of curves of genus 3. Let P1 , . . . , Pm be the (possibly inﬁnitely near) singular points of Γ , and let ri be the multiplicity in Pi of the proper transform of Γ . Then, denoting by H1 , respectively H2 , the divisors of a vertical, respectively of a horizontal

Rationality of Moduli

5

line in P1 × P1 , we have that Γ ∈ |4H1 + 4H2 − m ri Pi |. By adjunction, i=1 m the canonical system of Γ is cut out by |2H1 + 2H2 − i=1 (ri − 1)Pi |, and therefore 4 = degKC = Γ · (2H1 + 2H2 − Hence

m

i=1 ri (ri

m

(ri − 1)Pi ) = 16 −

i=1

m

ri (ri − 1).

i=1

− 1) = 12, and we have the following possibilities:

i) ii) iii) iv)

m (r1 , . . . , rm ) 1 (4) 2 (3,3) 4 (3,2,2,2) 6 (2,2,2,2,2,2)

We will show now that for a general curve only the last case occurs, i.e., Γ has exactly 6 singular points of multiplicity 2. We denote by S the blowup of P1 × P1 in P1 , . . . , Pm , and let Ei be the exceptional divisor of the ﬁrst kind, total transform of the point Pi . We shall ﬁrst show that the ﬁrst case (i.e., m = 1) corresponds to the case η ≡ 0. Proposition 2.3. Let Γ ⊂ P1 × P1 be a curve of bidegree (4, 4) having a point P of multiplicity 4, such that its normalization C ∈ |4H1 + 4H2 − 4E| has genus 3 (here, E is the exceptional divisor of the blowup of P1 × P1 in P ). Then OC (H1 ) ∼ = OC (H2 ) ∼ = OC (KC ). In particular, if Γ = ϕη (C) (i.e., we are in the case m = 1), then η ≡ 0. Remark 2.4. Let Γ be as in the proposition. Then the rational map P1 × P1 P2 given by |H1 + H2 − E| maps Γ to a plane quartic. Vice versa, given a plane quartic C , blowing up two points p1 , p2 ∈ (P1 × P1 ) \ C , and then contracting the strict transform of the line through p1 , p2 , yields a curve Γ of bidegree (4, 4) having a singular point of multiplicity 4. Proof (of the proposition). Let H1 be the full transform of a vertical line through P . Then there is an eﬀective divisor H1 on the blowup S of P1 × P1 in P such that H1 ≡ H1 + E. Since H1 · C = E · C = 4, H1 is disjoint from C, whence OC (H1 ) ∼ = OC (E). The same argument for a horizontal line through P obviously shows that OC (H2 ) ∼ = OC (E). If h0 (C, OC (H1 )) = 2, then the 1 two projections p1 , p2 : Γ → P induce the same linear series on C, thus ϕ|H1 | and ϕ|H2 | are related by a projectivity of P1 , hence Γ is the graph of a projectivity of P1 , contradicting the fact that the bidegree of Γ is (4, 4). Therefore we have a smooth curve of genus 3 and a divisor of degree 4 such that h0 (C, OC (H1 )) ≥ 3. Hence h0 (C, OC (KC − H1 )) ≥ 1, which implies

that KC ≡ H1 . Analogously, KC ≡ H2 .

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The next step is to show that for a general curve C of genus 3, cases ii) and iii) do not occur. In fact, we show: Lemma 2.5. Let C be a curve of genus 3 and η ∈ P ic0 (C)3 \{0} such that ϕη is birational and the image ϕη (C) = Γ has a singular point P of multiplicity 3. Then C belongs to an algebraic subset of M3 of dimension ≤ 5. Proof. Let S again be the blow up of P1 ×P1 in P , and denote by E the exceptional divisor. Then OC (E) has degree 3 and arguing as in Proposition 2.3, we see that there are points Q1 , Q2 on C such that OC (Hi ) ∼ = OC (Qi +E). Therefore OC (Q2 −Q1 ) ∼ = OC (H2 −H1 ) ∼ = OC (KC −η −(KC +η)) ∼ = OC (η), whence 3Q1 ≡ 3Q2 , Q1 = Q2 . This implies that there is a morphism f : C → P1 of degree 3, having double ramiﬁcation in Q1 and Q2 . By Hurwitz’ formula the degree of the ramiﬁcation divisor R is 10 and since R ≥ Q1 +Q2 f has at most eight branch points in P1 . Fixing three of these points to be ∞, 0, 1, we obtain (by Riemann’s existence theorem) a ﬁnite number of families of dimension at most 5.

From now on, we shall make the following Assumptions. C is a curve of genus 3, η ∈ P ic0 (C)3 \ {0}, and 1) |KC + η| and |KC − η| are base point free; 2) ϕη : C → Γ ⊂ P1 × P1 is birational; 3) Γ ∈ |4H1 + 4H2 | has only double points as singularities (possibly inﬁnitely near). Remark 2.6. By the considerations so far, we know that a general curve of genus 3 fulﬁlls the assumptions for any η ∈ P ic0 (C)3 \ {0}. We use the notation introduced above: we have π : S → P1 × P1 and 6 C ⊂ S, C ∈ |4H1 + 4H2 − 2 i=1 Ei |. Remark 2.7. Since S is a regular surface, we have an easy case of Ramanujam’s vanishing theorem: if D is an eﬀective divisor which is 1-connected (i.e., for every decomposition D = A + B with A, B > 0, we have A · B ≥ 1), then H 1 (S, OS (−D)) = 0. This follows immediately from Ramanujam’s lemma ensuring H 0 (D, OD ) = k, and from the long exact cohomology sequence associated to 0 → OS (−D) → OS → OD → 0. In most of our applications we shall show that D is linearly equivalent to a reduced and connected divisor (this is a stronger property than 1connectedness).

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7

We know now that OC (H1 + H2 ) ∼ = OC (2KC ), i.e., OC ∼ = OC (3H1 + 3H2 −

6

2Ei ).

i=1

Since h1 (S, OS (−H1 − H2 )) = 0, the exact sequence 0 → OS (−H1 − H2 ) → OS (3H1 + 3H2 −

6

2Ei )

i=1

→ OC (3H1 + 3H2 −

6

2Ei ) ∼ = OC → 0, (1)

i=1

is exact on global sections. 6 In particular, h0 (S, OS (3H1 + 3H2 − i=1 2Ei )) = 1. We denote by G the unique divisor in the linear system |3H1 + 3H2 − 6i=1 2Ei |. Note that C ∩ G = ∅ (since OC ∼ = OC (G)). ˜ on S such that G = G ˜ + Ei , Remark 2.8. There is no eﬀective divisor G ˜ · C = −2, contradicting that G ˜ and C have no common since otherwise G component. 6 This means that G + 2 i=1 Ei is the total transform of a curve G ⊂ P1 × P1 of bidegree (3,3). Lemma 2.9. h0 (G, OG ) = 3, h1 (G, OG ) = 0. Proof. Consider the exact sequence 0 → OS (KS ) → OS (KS + G) → OG (KG ) → 0. Since h0 (S, OS (KS )) = h1 (S, OS (KS )) = 0, we get h0 (S, OS (KS + G)) = h0 (G, OG (KG )). 6 Now, KS + G ≡ H1 + H2 − i=1 Ei , therefore (KS + G) · C = −4, whence h0 (G, OG (KG )) ∼ = h0 (S, OS (KS + G)) = 0. 1 Moreover, h (G, OG (KG )) = h1 (S, OS (KS + G)) + 1, and by Riemann– Roch we infer that, since h1 (S, OS (KS + G)) = h0 (S, OS (−G)) = 0, that

h1 (S, OS (KS + G)) = 2. We will show now that G is reduced, hence, by the above lemma, we shall obtain that G has exactly three connected components. Proposition 2.10. G is reduced.

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I. Bauer and F. Catanese

Proof. By Remark 2.8 it is suﬃcient to show that the image of G in P1 × P1 , which we denoted by G , is reduced. Assume that there is an eﬀective divisor A on P1 × P1 such that 3A ≤ G . We clearly have A ∩Γ = ∅ but, after blowing up the six points P1 , . . . , P6 , the strict transforms of A and of Γ are disjoint, whence A and G must intersect in one of the Pi ’s, contradicting Remark 2.8. If G is not reduced, we may uniquely write G = 2D1 + D2 with D1 , D2 reduced and having no common component. Up to exchanging the factors of P1 × P1 , we have the following two possibilities: i) D1 ∈ |H1 + H2 |; ii) D1 ∈ |H1 |. In the ﬁrst case also D2 ∈ |H1 + H2 | and its strict transform is disjoint from C. Remark 2.8 implies that D2 meets Γ in points which do not belong to D1 , whence D2 has double points where it intersects Γ . Since D2 · Γ = 8 we see that D2 has two points of multiplicity 2, a contradiction (D2 has bidegree (1, 1)). Assume now that D1 ∈ |H1 |. Then, since 2D1 · Γ = 8, D1 contains four of the Pi ’s and D2 passes through the other two, say P1 , P2 . This implies that ˆ 2 ≡ H1 + 3H2 − 2E1 − 2E2 , whence for the strict transform of D2 we have: D ˆ 2 · C = 8, a contradiction. D

We write now G = G1 + G2 + G3 as a sum of its connected components, and accordingly G = G1 + G2 + G3 . Lemma 2.11. The bidegree of Gj (j ∈ {1, 2, 3}) is (1, 1). Up to renumbering P1 , . . . , P6 we have G1 ∩ G2 = {P1 , P2 }, G1 ∩ G3 = {P3 , P4 } and G2 ∩ G3 = {P5 , P6 }. More precisely, G1 ∈ |H1 + H2 − E1 − E2 − E3 − E4 |, G2 ∈ |H1 + H2 − E1 − E2 − E5 − E6 |, G3 ∈ |H1 + H2 − E3 − E4 − E5 − E6 |. Proof. Assume for instance that G1 has bidegree (1, 0). Then there is a subset I ⊂ {1, . . . , 6} such that G1 = H1 − i∈I Ei . Since G1 · C = 0, it follows that |I| = 2. But then G1 · (G − G1 ) = 1, contradicting the fact that G1 is a connected component of G. 1 since areduced divisor of Let (aj , bj ) be the bidegree of Gj : then aj , bj ≥ bidegree (m, 0) is not connected for m ≥ 2. Since aj = bj = 3, it follows that aj = bj = 1. 6 Writing now Gj ≡ H1 + H2 − i=1 μ(j, i)Ei we obtain 3 j=1

μ(j, i) = 2,

6 i=1

μ(j, i) = 4,

6 i=1

μ(k, i)μ(j, i) = 2

Rationality of Moduli

9

since Gj · C = 0 and Gk · Gj = 0. We get the second claim of the lemma provided that we show: μ(j, i) = 1, ∀i, j. The ﬁrst formula shows that if μ(j, i) ≥ 2, then μ(j, i) = 2 and μ(h, i) = 0 for h = j. Hence the second formula shows that 6

μ(j, i)(μ(h, i) + μ(k, i)) ≤ 2,

h,k=j i=1

contradicting the third formula.

In the remaining part of the section we will show that each Gi consists of the union of a vertical and a horizontal line in P1 × P1 . Since OC (KC + η) ∼ = OC (H1 ) and OC (KC − η) ∼ = OC (H2 ) we get: OC (2H2 − H1 ) ∼ = OC (KC ) ∼ = OC (2H1 + 2H2 −

6

Ei ),

i=1

whence the exact sequence 0 → OS (−H1 − 4H2 +

6

Ei ) → OS (3H1 −

i=1

6

Ei )

i=1

→ OC (3H1 −

6

Ei ) ∼ = OC → 0. (2)

i=1

Proposition 2.12. H 1 (S, OS (−(H1 + 4H2 −

6 i=1

Ei ))) = 0.

Proof. The result follows immediately by Ramanujam’s vanishing theorem, but we can also give an elementary proof using Remark 2.7. 6 It suﬃces to show that the linear system |H1 + 4H2 − i=1 Ei | contains a reduced and connected divisor. 6 Note that G1 + |3H2 − E5 − E6 | ⊂ |H1 + 4H2 − i=1 Ei |, and that |3H2 − E5 − E6 | contains |H2 − E5 − E6 | + |2H2 |, if there is a line H2 containing P1 , P2 , else it contains |H2 − E5 | + |H2 − E6 | + |H2 |. Since G1 · H2 = G1 · (H2 − E5 ) = G1 · (H2 − E6 ) = G1 · (H2 − E5 − E6 ) = 1, we have obtained in both cases a reduced and connected divisor.

Remark 2.13. One can indeed show, using G2 + |3H2 − E3 − E4 | ⊂ |H1 + 4H2 −

6 i=1

Ei |,

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I. Bauer and F. Catanese

G3 + |3H2 − E1 − E2 | ⊂ |H1 + 4H2 −

6

Ei |,

i=1

that |H1 + 4H2 − 6i=1 Ei | has no ﬁxed part, and then by Bertini’s theorem, 6 since (H1 + 4H2 − i=1 Ei )2 = 8 − 6 = 2 > 0, a general curve in |H1 + 4H2 − 6 i=1 Ei | is irreducible. In view of Proposition 2.12 the above exact sequence (and the one where the roles of H1 , H2 are exchanged) yields the following: Corollary 2.14. For j ∈ {1, 2} there is exactly one divisor Nj ∈ |3Hj − 6 i=1 Ei |. By the uniqueness of G, we see that G = N1 + N2 . Denote by Nj the curve in P1 × P1 whose total transform is Nj + 6i=1 Ei . We have just seen that G is the strict transform of three vertical and three horizontal lines in P1 × P1 . Hence each connected component Gj splits into the strict transform of a vertical and a horizontal line. Since G is reduced, the lines are distinct (and there are no inﬁnitely near points). We can choose coordinates in P1 × P1 such that G1 = ({∞} × P1 ) ∪ (P1 × {∞}), G2 = ({0} × P1 ) ∪ (P1 × {0}), and G3 = ({1} × P1 ) ∪ (P1 × {1}). Remark 2.15. The points P1 , . . . , P6 are then the points of the set S previously deﬁned. Conversely, consider in P1 × P1 the set S := {P1 , . . . , P6 } = ({∞, 0, 1} × {∞, 0, 1}) \ {(∞, ∞), (0, 0), (1, 1)}. Let π : S → P1 × P1 be the blowup of the points P1 , . . . , P6 and suppose (denoting the exceptional divisor over Pi by Ei ) that C ∈ |4H 2Ei | 1 +4H2 − is a smooth curve. Then C has genus 3, OC (3H1 ) ∼ = OC ( Ei ) ∼ = OC (3H2 ). Setting OC (η) := OC (H2 − H1 ), we obtain therefore 3η ≡ 0. It remains to show that OC (η) is not isomorphic to OC . Lemma 2.16. η is not trivial. Proof. Assume η ≡ 0. Then OC (H1 ) ∼ = OC (H2 ) and, since Γ has bidegree (4, 4), we argue as in the proof of Proposition 2.3 that h0 (OC (Hi )) ≥ 3, whence OC (Hi ) ∼ = OC (KC ). The same argument shows that the two projections of Γ to P1 yield two diﬀerent pencils in the canonical system. It follows that the canonical map of C factors as the composition of C → Γ ⊂ P1 × P1 with the rational map ψ : P1 × P1 P2 which blows up one point and contracts the vertical and horizontal line through it. Since Γ has six singular points, the canonical map

sends C birationally onto a singular quartic curve in P2 , contradiction.

Rationality of Moduli

11

3 Rationality of the moduli spaces In this section we will use the geometric description of pairs (C, η), where C is a genus 3 curve and η a nontrivial 3-torsion divisor class, and study the birational structure of their moduli space. More precisely, we shall prove the following: Theorem 3.1. 1) The moduli space M3,η := {(C, η) : C a general curve of genus 3, η ∈ P ic0 (C)3 \ {0}} is rational. 2) The moduli space M3,η := {(C, η ) : C a general curve of genus 3, η ∼ = Z/3Z ⊂ P ic0 (C)} is rational. Remark 3.2. By the result of the previous section, and since any automorphism of P1 ×P1 which sends the set S to itself belongs to the group S3 ×Z/2Z, it follows immediately that, if we set V (4, 4, −S) := H 0 (OP1 ×P1 (4, 4)(−2 Pij )), i=j,i,j∈{∞,0,1}

then M3,η is birational to P(V (4, 4, −S))/S3 , while M3,η is birational to P(V (4, 4, −S))/(S3 ×Z/2Z), where the generator σ of Z/2Z acts by coordinate exchange on P1 × P1 , whence on V (4, 4, −S). In order to prove the above theorem we will explicitly calculate the respective subﬁelds of invariants of the function ﬁeld of P(V (4, 4, −S)) and show that they are generated by purely transcendental elements. Consider the following polynomials of V := V (4, 4, −S), which are invariant under the action of Z/2Z: f11 (x, y) := x20 x21 y02 y12 , f∞∞ (x, y) := x21 (x1 − x0 )2 y02 (y1 − y0 )2 , f00 (x, y) := x20 (x1 − x0 )2 y02 (y1 − y0 )2 .

Let ev : V → i=0,1,∞ k(i,i) =: W be the evaluation map at the three standard diagonal points, i.e., ev(f ) := (f (0, 0), f (1, 1), f (∞, ∞)). Since fii (j, j) = δi,j , we can decompose V ∼ = U ⊕ W, where U := ker(ev) and W is the subspace generated by the three above polynomials, which is easily shown to be an invariant subspace using the following formulae (∗):

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I. Bauer and F. Catanese

•

(1, 3) exchanges x0 with x1 , multiplies x1 − x0 by −1,

•

(1, 2) exchanges x1 − x0 with x1 , multiplies x0 by −1,

•

(2, 3) exchanges x0 − x1 with x0 , multiplies x1 by −1.

In fact, ‘the permutation’ representation W of the symmetric group splits (in characteristic = 3) as the direct sum of the trivial representation (generated by e1 + e2 + e3 ) and the standard representation, generated by x0 := e1 − e2 , x1 := −e2 + e3 , which is isomorphic to the representation on V (1) := H 0 (OP1 (1)). Note that U = x0 x1 (x1 − x0 )y0 y1 (y0 − y1 )H 0 (P1 × P1 , OP1 ×P1 (1, 1)). We write V (1, 1) := H 0 (P1 × P1 , OP1 ×P1 (1, 1)) = V (1) ⊗ V (1), where V (1) := H 0 (P1 , OP1 (1)), is as above the standard representation of S3 . Now V (1)⊗V (1) splits, in characteristic = 2, 3, as a sum of irreducible representations I ⊕ A ⊕ W , where the three factors are the trivial, the alternating and the standard representation of S3 . Explicitly, V (1) ⊗ V (1) ∼ = ∧2 (V (1)) ⊕ Sym2 (V (1)), and Sym2 (V (1)) is isomorphic to W, since it has the following basis: x0 y0 , x1 y1 , (x1 −x0 )(y1 −y0 ). We observe for further use that Z/2Z acts as the identity on Sym2 (V (1)), while it acts on ∧2 (V (1)), spanned by x1 y0 − x0 y1 via multiplication by −1. We have thus seen Lemma 3.3. If char(k) = 2, 3, then the S3 -module V splits as a sum of irreducible modules as follows: V∼ = 2(I ⊕ W ) ⊕ A. Choose now a basis (z1 , z2 , z3 , w1 , w2 , w3 , u) of V, such that the zi ’s and the wi ’s are respective bases of I ⊕ W consisting of eigenvectors of σ = (123), and u is a basis element of A. The eigenvalue of zi , wi with respect to σ = (123) is i−1 , u is σ-invariant and (12)(u) = −u. Note that if (v1 , v2 , v3 ) is a basis of I⊕W , such that S3 acts by permutation of the indices, then z1 = v1 + v2 + v3 , z2 = v1 + v2 + 2 v3 , z3 = v1 + 2 v2 + v3 , where is a primitive third root of unity. Remark 3.4. Since z1 , w1 are S3 -invariant, P(V (4, 4, −S))/S3 is birational to a product of the aﬃne line with Spec(k[z2 , z3 , w2 , w3 , u]S3 ), and therefore it suﬃces to compute k[z2 , z3 , w2 , w3 , u]S3 . Part 1 of the theorem follows now from the following Proposition 3.5. Let T := z2 z3 , S := z23 , A1 := z2 w3 + z3 w2 , A2 := z2 w3 − z3 w2 . Then

Rationality of Moduli

k(z2 , z3 , w2 , w3 , u)S3 ⊃ K := k(A1 , T, S +

13

T3 T3 T3 , u(S − ), A2 (S − )), S S S

and [k(z2 , z3 , w2 , w3 , u) : K] = 6, hence k(z2 , z3 , w2 , w3 , u)S3 = K. Proof. We ﬁrst calculate the invariants under the action of σ = (123), i.e., k(z2 , z3 , w2 , w3 , u)σ . Note that u, z2 z3 , z2 w3 , w2 w3 ,z23 are σ-invariant, and [k(z2 , z3 , w2 , w3 , u) : k(u, z2 z3 , z2 w3 , w2 w3 , z23 )] = 3. In particular, k(z2 , z3 , w2 , w3 , u)σ = k(u, z2 z3 , z2 w3 , w2 w3 , z23 ) =: L. Now, we calculate Lτ , with τ = (12). Observe that L = k(T, A1 , A2 , S, u). Since τ (z2 ) = z3 , τ (z3 ) = 2 z2 (and similarly for w2 , w3 ), we see that τ (A1 ) = 3 A1 and τ (T ) = T . On the other hand, τ (u) = −u, τ (A2 ) = −A2 , τ (S) = TS . Claim. Lτ = k(A1 , T, S +

T3 S , u(S

−

T3 S ), A2 (S

−

T3 S ))

=: E.

3

3

3

Proof of the Claim. Obviously A1 ,T ,S + TS ,u(S − TS ), A2 (S − TS ) are invariant under τ , whence E ⊂ Lτ . Since L = E(S), using the equation B · S = S 2 + T 3 3 for B := S + TS , we get that [E(S) : E] ≤ 2.

This proves the claim and the proposition.

It remains to show the second part of the theorem. We denote by τ the involution on k(z1 , z2 , z3 , w1 , w2 , w3 , u) induced by the involution (x, y) → (y, x) on P1 × P1 . It suﬃces to prove the following

Proposition 3.6. E τ = k(A1 , T, S + 3

T3 S , (u(S 3

−

T3 2 S )) , A2 (S

−

T3 S )).

3

Proof. Since [E : k(A1 , T, S + TS , (u(S − TS ))2 , A2 (S − TS ))] ≤ 2, it suﬃces 3 3 3 to show that the ﬁve generators A1 , T ,S + TS ,(u(S − TS ))2 , A2 (S − TS ) are τ -invariant. This will now be proven in Lemma 3.7.

Lemma 3.7. τ acts as the identity on (z1 , z2 , z3 , w1 , w2 , w3 ) and sends u → −u. Proof. We note ﬁrst that τ acts trivially on the subspace W generated by the polynomials fii . Since U = x0 x1 (x1 −x0 )y0 y1 (y1 −y0 )V (1, 1) and x0 x1 (x1 −x0 )y0 y1 (y1 −y0 ) is invariant under exchanging x and y, it suﬃces to recall that the action of τ on V (1, 1) = V (1) ⊗ V (1) is the identity on the subspace Sym2 (V (1)), while the action on the alternating S3 -submodule A sends the generator u to −u.

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I. Bauer and F. Catanese

3.1 Char(k) = 3 In order to prove Theorem 3.1 if the characteristic of k is equal to 3, we describe the S3 -module V as follows: V∼ = 2W ⊕ A, where W is the (three-dimensional) permutation representation of S3 . Let now z1 , z2 , z3 , w1 , w2 , w3 , u be a basis of V such that the action of S3 permutes z1 , z2 , z3 (resp. w1 , w2 , w3 ), and (123) : u → u, (12)u → −u. Then we have: Proposition 3.8. The S3 -invariant subﬁeld k(V)S3 of k(V) is rational. More precisely, the seven S3 -invariant functions σ1 = z1 + z2 + z3 , σ2 = z1 z2 + z1 z3 + z2 z3 , σ3 = z1 z2 z3 , σ4 = z1 w1 + z2 w2 + z3 w3 , σ5 = w1 z2 z3 + w2 z1 z3 + w3 z1 z2 , σ6 = w1 (z2 + z3 ) + w2 (z1 + z3 ) + w3 (z1 + z2 ), σ7 = u(z1 (w2 − w3 ) + z2 (w3 − w1 ) + z3 (w1 − w2 )) form a basis of the purely transcendental extension over k. Proof. σ1 , . . . , σ7 determine a morphism ψ : V → A7k . We will show that ψ induces a birational map ψ¯ : V/S3 → A7k , i.e., for a Zariski open set of V we have: ψ(x) = ψ(x ) if and only if there is a τ ∈ S3 such that x = τ (x ). By [Cat, Lemma 2.2] we can assume (after acting on x with a suitable τ ∈ S3 ) that xi = xi for 1 ≤ i ≤ 6, and we know that (setting u := x7 , u := x7 ) u(x1 (x5 − x6 ) + x2 (x6 − x4 ) + x3 (x4 − x5 )) = u (x1 (x5 − x6 ) + x2 (x6 − x4 ) + x3 (x4 − x5 )). Therefore, if B(x1 , . . . , x6 ) := x1 (x5 − x6 ) + x2 (x6 − x4 ) + x3 (x4 − x5 ) = 0, this implies that u = u .

Therefore, we have shown part 1 of Theorem 3.1. We denote again by τ the involution on k(z1 , z2 , z3 , w1 , w2 , w3 , u) induced by the involution (x, y) → (y, x) on P1 × P1 . In order to prove part 2 of Theorem 3.1, it suﬃces to observe that σ1 , . . . , σ6 , σ72 are invariant under τ and [k(σ1 , . . . , σ7 ) : k(σ1 , . . . , σ72 )] ≤ 2, whence (k(V)S3 )(Z/2Z) = k(σ1 , . . . , σ72 ). This proves Theorem 3.1.

Rationality of Moduli

15

3.2 Char(k) = 2 Let k be an algebraically closed ﬁeld of characteristic 2. Then we can describe the S3 -module V as follows: V∼ = W ⊕ V (1, 1), where W is the (three-dimensional) permutation representation of S3 . We denote a basis of W by z1 , z2 , z3 . As in the beginning of the chapter, V (1, 1) = H 0 (P1 × P1 , OP1 ×P1 (1, 1)). We choose the following basis of V (1, 1): w1 := x1 y1 , w2 := (x0 + x1 )(y0 + y1 ), w3 := x0 y0 , w := x0 y1 . Then S3 acts on w1 , w2 , w3 by permutation of the indices and (1, 2) : w → w + w3 , (1, 2, 3) : w → w + w2 + w3 . Let ∈ k be a nontrivial third root of unity. Then Theorem 3.1 (in characteristic 2) follows from the following result: Proposition 3.9. Let k be an algebraically closed ﬁeld of characteristic 2. Let σ1 , . . . , σ6 be as deﬁned in (3.6) and set v := (w + w2 )(w1 + w2 + 2 w3 ) + (w + w1 + w3 )(w1 + 2 w2 + w3 ), t := (w + w2 )(w + w1 + w3 ). Then 1) k(z1 , z2 , z3 , w1 , w2 , w3 , w)S3 = k(σ1 , . . . , σ6 , v); 2) k(z1 , z2 , z3 , w1 , w2 , w3 , w)S3 ×Z/2Z = k(σ1 , . . . , σ6 , t). In particular, the respective invariant subﬁelds of k(V) are generated by purely transcendental elements, and this proves Theorem 3.1. Proof (of Proposition 3.9). 2) We observe that Z/2Z (xi → yi ) acts trivially on z1 , z2 , z3 , w1 , w2 , w3 and maps w to w + w1 + w2 + w3 . It is now easy to see that t is invariant under the action of S3 × Z/2Z. Therefore k(σ1 , . . . , σ6 , t) ⊂ K := k(z1 , z2 , z3 , w1 , w2 , w3 , w)S3 ×Z/2Z . By [Cat, Lemma 2.8], [k(z1 , z2 , z3 , w1 , w2 , w3 , t) : k(σ1 , . . . , σ6 , t)] = 6, and obviously, [k(z1 , z2 , z3 , w1 , w2 , w3 , w) : k(z1 , z2 , z3 , w1 , w2 , w3 , t)] = 2. Therefore [k(z1 , z2 , z3 , w1 , w2 , w3 , w) : k(σ1 , . . . , σ6 , t)] = 12, whence K = k(σ1 , . . . , σ6 , t). 1) Note that for W2 := w1 + w2 + 2 w3 , W3 := w1 + 2 w2 + 3 w3 , we have: W23 and W33 are invariant under (1, 2, 3) and are exchanged under (1, 2). Therefore v is invariant under the action of S3 and we have seen that k(σ1 , . . . , σ6 , v) ⊂ L := k(z1 , z2 , z3 , w1 , w2 , w3 , w)S3 , in particular [k(z1 , z2 , z3 , w1 , w2 , w3 , w) : k(σ1 , . . . , σ6 , v)] ≥ 6. On the other hand, note that k(z1 , z2 , z3 , w1 , w2 , w3 , w) = k(z1 , z2 , z3 , w1 , w2 , w3 , v) (since v is linear in w) and again, by [Cat, Lemma 2.8], [k(zi , wi , v) : k(σ1 , . . . , σ6 , v)] = 6. This implies that L = k(σ1 , . . . , σ6 , v).

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References [B-C04] [B-K85] [Cat]

Bauer, I. and Catanese, F., Symmetry and variation of Hodge structures, Asian J. Math., vol.8, no.2, 363–390, (2004). Bogomolov, F. A. and Katsylo, P. I., Rationality of some quotient varieties, Mat. Sb. (N.S.), vol. 126 (168), no. 4, 584–589, (1985). Catanese, F., On the rationality of certain moduli spaces related to curves of genus 4, Algebraic geometry (Ann Arbor, Mich., 1981), 30–50, Lecture Notes in Math., 1008, Springer, Berlin (1983).

The Rationality of the Moduli Space of Curves of Genus 3 after P. Katsylo Christian Böhning Mathematisches Institut Bunsenstrasse 3-5 37073 Göttingen, Germany [email protected] Summary. This article is a survey of P. Katsylo’s proof that the moduli space M3 of smooth projective complex curves of genus 3 is rational. We hope to make the argument more comprehensible and transparent by emphasizing the underlying geometry in the proof and its key structural features.

Key words: Rationality, moduli spaces of curves 2000 Mathematics Subject Classiﬁcation codes: 14E08, 14H10, 14H45

1 Introduction The question whether or not M3 is a rational variety had been open for a long time until an aﬃrmative answer was ﬁnally given by P. Katsylo in 1996. There is a well-known transition in the behavior of the moduli spaces Mg of smooth projective complex curves of genus g from unirational for small g to general type for larger values of g; the moral reason that M3 should have a good chance to be rational is that it is birational to a quotient of a projective space by a connected linear algebraic group. No variety of this form has been proved to be irrational up to now. More precisely, M3 is birational to the moduli space of plane quartic curves for PGL3 C-equivalence. All the moduli spaces C(d) of plane curves of given degree d are conjectured to be rational (see [Dol2, p.162]; in fact, there it is conjectured that all the moduli spaces of hypersurfaces of given degree d in Pn for the PGLn+1 C-action are rational). There are heuristic reasons that the spaces C(d) should be rational at least for all large enough values for d. It should not be completely out of reach to prove this rigorously and we hope to return to this problem in the future. In any case, one might guess that irregular behavior of C(d) is most likely to be found for small values of d, and showing rationality for C(4) turned out to be exceptionally hard.

F. Bogomolov, Y. Tschinkel (eds.), Cohomological and Geometric Approaches to Rationality Problems, Progress in Mathematics 282, DOI 10.1007/978-0-8176-4934-0_2, © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

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C. Böhning

Katsylo’s proof is long and computational, and, due to the importance of the result, it seems desirable to give a more accessible and geometric treatment of the argument. This paper is divided into two main sections (Sections 2 and 3) which are further divided into subsections. Section 2 treats roughly the contents of Katsylo’s ﬁrst paper [Kat1] and Section 3 deals with his second paper [Kat2]. Acknowledgment. I would like to thank Professor Yuri Tschinkel for proposing the project and many useful discussions. Moreover, I am especially grateful to Professor Fedor Bogomolov with whom I discussed parts of the project and who provided a wealth of helpful ideas.

2 A remarkable (SL3 C, SO3 C)-section 2.1 (G, H)-sections and covariants A general, i.e., nonhyperelliptic, smooth projective curve C of genus 3 is realized as a smooth plane quartic curve via the canonical embedding, whence M3 is birational to the orbit space C(4) := P(H 0 (P2 , O(4)))/SL3 C. We remark that whenever one has an aﬃne algebraic group G acting on an irreducible variety X, then, according to a result of Rosenlicht, there exists a nonempty invariant open subset X0 ⊂ X such that there is a geometric quotient for the action of G on X0 (cf. [Po-Vi, Theorem 4.4]). In the following we denote by X/G any birational model of this quotient, i.e., any model of the ﬁeld C(X)G of invariant rational functions. The number of methods to prove rationality of quotients of projective spaces by connected reductive groups is quite limited (cf. [Dol1] for an excellent survey). The only approach which our problem is immediately amenable to seems to be the method of (G, H)-sections. (There are two other points of view I know of: The ﬁrst is based on the remark that if we have a nonsingular plane quartic curve C, the double cover of P2 branched along C is a Del Pezzo surface of degree 2, and conversely, given a Del Pezzo surface S of degree 2, then | − KS | is a regular map which exhibits S as a double cover of P2 branched along a plane quartic C; this sets up a birational isomorphism between M3 and DP(2), the moduli space of Del Pezzo surfaces of degree 2. We can obtain such an S by blowing up 7 points in P2 , and one can prove that DP(2) is birational to the quotient of an open subset of P27 := (P2 )7 /PGL3 C, the conﬁguration space of 7 points in P2 (which is visibly rational), modulo an action of the Weyl group W (E7 ) of the root system of type E7 by Cremona transformations (note that W (E7 ) coincides with the permutation group of the (−1)-curves on S that preserves the incidence relations between them). This group is a rather large ﬁnite group, in fact, it has order 210 · 34 · 5 · 7. This approach does not seem to have led to anything deﬁnite in the direction of proving rationality of M3 by now, but see [D-O] for more information.

Rationality of M3

19

The second alternative, pointed out by I. Dolgachev, is to remark that M3 is birational to Mev 3 , the moduli space of genus 3 curves together with an even theta-characteristic; this is the content of the classical theorem due to G. Scorza. The latter space is birational to the space of nets of quadrics in P3 modulo the action of SL4 C, i.e., Grass(3, Sym2 (C4 )∨ )/SL4 C. See [Dol3, 6.4.2], for more on this. Compare also [Kat0], where the rationality of the related space Grass(3, Sym2 (C5 )∨ )/SL5 C is proven; this proof, however, cannot be readily adapted to our situation, the diﬃculty seems to come down to that 4, in contrast to 5, is even. Definition 2.1.1. Let X be an irreducible variety with an action of a linear algebraic group G, H < G a subgroup. An irreducible subvariety Y ⊂ X is called a (G, H)-section of the action of G on X if (1) G · Y = X; (2) H · Y ⊂ Y ; (3) g ∈ G, gY ∩ Y = ∅ =⇒ g ∈ H. In this situation H is the normalizer NG (Y ) := {g ∈ G | gY ⊂ Y } of Y in G. The following proposition collects some properties of (G, H)-sections. Proposition 2.1.2. (1) The ﬁeld C(X)G is isomorphic to the ﬁeld C(Y )H via restriction of functions to Y . (2) Let Z and X be G-varieties, f : Z → X a dominant G-morphism, Y a (G, H)-section of X, and Y an irreducible component of f −1 (Y ) that is H-invariant and dominates Y . Then Y is a (G, H)-section of Z. Part (2) of the proposition suggests that, to simplify our problem of proving rationality of C(4), we should look at covariants Sym4 (C3 )∨ → Sym2 (C3 )∨ of low degree (C3 is the standard representation of SL3 C). The highest weight theory of Cartan–Killing allows us to decompose Symi (Sym4 (C3 )∨ ), i ∈ N, into irreducible subrepresentations (this is best done by a computer algebra system, e.g., Magma) and pick the smallest i such that Sym2 (C3 )∨ occurs as an irreducible summand. This turns out to be 5 and Sym2 (C3 )∨ occurs with multiplicity 2. For nonnegative integers a, b we denote by V (a, b) the irreducible SL3 Cmodule whose highest weight has numerical labels a, b. Let us now describe the two resulting independent covariants α1 , α2 : V (0, 4) → V (0, 2) of order 2 and degree 5 geometrically. We follow a classical geometric method of Clebsch to pass from invariants of binary forms to contravariants of ternary forms (see [G-Y, §215]). The covariants α1 , α2 are described in Salmon’s treatise [Sal, p. 261, and p. 259], cf. also [Dix, pp. 280–282]. We start by recalling the structure of the ring of SL2 C-invariants of binary quartics ([Muk, Section 1.3], [Po-Vi, Section 0.12]).

20

C. Böhning

2.2 Binary quartics Let f4 = ξ0 x40 + 4ξ1 x30 x1 + 6ξ2 x20 x21 + 4ξ3 x0 x31 + ξ4 x41

(1)

be a general binary quartic form. The invariant algebra R = C[ξ0 , . . . , ξ4 ]SL2 C is freely generated by two homogeneous invariants g2 and g3 (where subscripts indicate degrees): ξ ξ ξ ξ g2 (ξ) = det 0 2 − 4 det 1 2 , (2) ξ2 ξ4 ξ2 ξ3 ⎞ ⎛ ξ0 ξ1 ξ2 (3) g3 (ξ) = det ⎝ ξ1 ξ2 ξ3 ⎠ . ξ2 ξ3 ξ4 If we identify f4 with its zeros z1 , . . . , z4 ∈ P1 = C ∪ {∞} and write λ=

(z1 − z3 )(z2 − z4 ) (z1 − z4 )(z2 − z3 )

for the cross-ratio, then 1 , 2 2πi with ω = e 3 ,

g3 = 0 ⇐⇒ λ = −1, 2, or g2 = 0 ⇐⇒ λ = −ω or − ω 2

the ﬁrst case being commonly referred to as harmonic cross-ratio, the second as equi-anharmonic cross-ratio (see [Cle, p. 171]; the terminology varies a lot among diﬀerent authors, however). Clebsch’s construction is as follows: Let x, y, z be coordinates in P2 , and let u, v, w be coordinates in the dual projective plane (P2 )∨ . Let ϕ = ax4 + 4bx3 y + . . . be a general ternary quartic. We consider those lines in P2 such that their intersection with the associated quartic curve Cϕ is a set of points whose crossratio is harmonic resp. equi-anharmonic. Writing a line as ux + vy + wz = 0 and substituting in (2) resp. (3), we see that in the equi-anharmonic case we get a quartic in (P2 )∨ , and in the harmonic case a sextic. More precisely, this gives us two SL3 C-equivariant polynomial maps σ : V (0, 4) → V (0, 4)∨ , ψ : V (0, 4) → V (0, 6)∨ ,

(4) (5)

and σ is homogeneous of degree 2 in the coeﬃcients of ϕ whereas ψ is homogeneous of degree 3 in the coeﬃcients of ϕ (we say σ is a contravariant of

Rationality of M3

21

degree 2 on V (0, 4) with values in V (0, 4), and analogously for ψ). Finally we have the Hessian covariant of ϕ: Hess : V (0, 4) → V (0, 6)

(6)

which associates to ϕ the determinant of the matrix of second partial derivatives of ϕ. It is of degree 3 in the coeﬃcients of ϕ. We will now cook up α1 , α2 from ϕ, σ, ψ, Hess: Let ϕ operate on ψ; by this we mean that if ϕ = ax4 + 4bx3 y + . . . , then we act on ψ by the diﬀerential operator a

∂4 ∂4 + 4b 3 + . . . 4 ∂u ∂u ∂v

(i.e., we replace a coordinate by partial diﬀerentiation with respect to the dual coordinate). In this way we get a contravariant ρ of degree 4 on V (0, 4) with values in V (0, 2). If we operate with ρ on ϕ, we get α1 . We obtain α2 if we operate with σ on Hess. This is a geometric way to describe α1 , α2 . For every c = [c1 : c2 ] ∈ P1 we get in this way a rational map fc = c1 α1 + c2 α2 : P(V (0, 4)) P(V (0, 2)) .

(7)

For the special quartics ϕ = ax4 + by 4 + cz 4 + 6f y 2 z 2 + 6gz 2 x2 + 6hx2 y 2

(8)

the quantities α1 and α2 were calculated by Salmon in [Sal, p. 257 ﬀ]. We reproduce the results here for the reader’s convenience. Put L := abc , P := af 2 + bg 2 + ch2 , R := f gh. ;

(9)

Then α1 = (3L + 9P + 10R)(af x2 + bgy 2 + chz 2 ) 2

2

(10)

2

+(10L + 2P + 4R)(ghx + hf y + f gz ) −12(a2 f 3 x2 + b2 g 3 y 2 + c2 h3 z 2 ) ; α2 = (L + 3P + 30R)(af x2 + bgy 2 + chz 2 )

(11)

+(10L − 6P − 12R)(ghx + hf y + f gz ) 2

2

2

−4(a2 f 3 x2 + b2 g 3 y 2 + c2 h3 z 2 ) . 1 (α1 − 3α2 ) looks a little simpler. Note that the covariant conic − 20 Let us see explicitly, using (8)–(11), that fc is dominant for every c ∈ P1 ; for a = b = c = f = g = h = 1 we get α1 = 48(x2 + y 2 + z 2 ), α2 = 16(x2 + y 2 + z 2 ), so the image of ϕ under fc in this case is a nonsingular

22

C. Böhning

conic unless c = [−1 : 3]. But for a = 1, b = c = 0, f = g = h = 1 we obtain α1 = 13x2 + 6y 2 + 6z 2 , α2 = 11x2 − 18y 2 − 18z 2, and for these values −α1 + 3α2 deﬁnes a nonsingular conic. Let Lc be the linear system generated by six quintics which deﬁnes fc and let Bc be its base locus; thus Uc := P(V (0, 4))\B is an SL3 C-invariant open −1 (Ch0 ), where h0 deﬁnes a nonsingular set, and if fc,0 := fc |Uc , then Xc := fc,0 conic, is a good candidate for an (SL3 C, SO3 C)-section of Uc . We choose h0 = xz − y 2 . Proposition 2.2.1. The variety Xc is a smooth irreducible SO3 C-invariant variety, SL3 C · X = P(V (0, 4)), and the normalizer of Xc in SL3 C is exactly SO3 C. Xc is an (SL3 C, SO3 C)-section of Uc . Proof. The SO3 C-invariance of Xc follows from its construction. We show that the diﬀerential d(fc,0 )x is surjective for all x ∈ Xc : In fact, d(fc,0 )x (Tx Uc ) ⊃ d(fc,0 )x (sl3 (x)) = sl3 (fc,0 (x)) = TCh0 PV (0, 2). Here sl3 (x) denotes the tangent space to the SL3 C-orbit of x in Uc , i.e., if Ox : SL3 C → Uc is the map with Ox (g) = gx, then we get a map d(Ox )e : sl3 → Tx Uc , and sl3 (x) := {d(Ox )e (ξ) | ξ ∈ sl3 }. Hence Xc is smooth. Assume Xc were reducible, let X1 and X2 be two irreducible components. By Proposition 2.1.2 (2) and the irreducibility of the group SO3 C, X1 and X2 are (SL3 C, SO3 C)-sections of Uc , so we can ﬁnd g ∈ SL3 C, x1 ∈ X1 , x2 ∈ X2 , such that gx1 = x2 . But then, by the SL3 C-equivariance of fc,0 , g stabilizes Ch0 and is thus in SO3 C. But, again by the irreducibility of SO3 C, x2 is also a point of X1 , i.e., X1 and X2 meet. This contradicts the smoothness of Xc .

The trouble is that if X c is the closure of Xc in P(V (0, 4)), then X c is an irreducible component of the intersection of ﬁve quintics. To prove rationality, however, we would prefer to work with equations of lower degree. This can be done for special c. 2.3 From quintic to cubic equations If Γfc ⊂ PV (0, 4) × PV (0, 2) is the graph of fc , it is natural to look for SL3 Cequivariant maps ϑ : V (0, 4) × V (0, 2) → V where V is another SL3 C-representation, ϑ is a homogeneous polynomial map in both factors V (0, 4), V (0, 2), of low degree, say d, in the ﬁrst factor, linear in the second, and such that Γfc is an irreducible component of {(x, y) ∈ PV (0, 4) × PV (0, 2) | ϑ(x, y) = 0}. If V is irreducible, there is an easy way to tell if ϑ vanishes on Γfc for some c ∈ P1 : This will be the case if V occurs with multiplicity one in Symd+5 V (0, 4). Here is the result.

Rationality of M3

23

Definition 2.3.1. Let Ψ : V (0, 4) → V (2, 2) be the up to factor unique SL3 Cequivariant, homogeneous of degree 3 polynomial map with the indicated source and target spaces, and let Φ : V (2, 2) × V (0, 2) → V (2, 1) be the up to factor unique bilinear SL3 C-equivariant map. Deﬁne Θ : V (0, 4) × V (0, 2) → V (2, 1) by Θ(x, y) := Φ(Ψ (x), y). Remark 2.3.2. The existence and essential uniqueness of the maps of Deﬁnition 2.3.1 can be easily deduced from known (and implemented in Magma) decomposition laws for SL3 C-representations. That they are only determined up to a nonzero constant factor will never bother us, and we admit this ambiguity in notation. The explicit form of Ψ , Φ, Θ will be needed later for checking certain nondegeneracy conditions through explicit computation. They can be found in Appendix A, formulas (64), (65). Theorem 2.3.3. (1) The linear map Θ(f, ·) : V (0, 2) → V (2, 1) has onedimensional kernel for f in an open dense subset V0 of V (0, 4), and, in particular, ker Θ(h20 , ·) = Ch0 . (2) For some c0 ∈ P1 , Γfc0 is an irreducible component of {Θ(x, y) = 0} ⊂ PV (0, 4) × PV (0, 2). (3) Xc0 ⊂ PV (0, 4) coincides with the closure X in PV (0, 4) of the preimage X of h0 under the morphism from PV0 → PV (0, 2) given by f → ker Θ(f, ·), and is thus an irreducible component of the algebraic set {Cf | Φ(Ψ (f ), h0 ) = 0} ⊂ PV (0, 4) deﬁned by 15 cubic equations. (4) The rational map Ψ : PV (0, 4) Ψ PV (0, 4) ⊂ PV (2, 2) as well as its restriction to X are birational isomorphisms unto their images. Proof. (1): One checks that V (2, 1) occurs with multiplicity one in the decomposition of Sym8 V (0, 4). Thus for some c0 ∈ P1 , we have Θ(f, (c0,1 α1 + c0,2 α2 )(f )) = 0 for all f ∈ V (0, 4). The fact that ker Θ(h20 , ·) = Ch0 follows from a direct computation using the explicit form of Θ. Thus, by uppersemicontinuity, (1) follows. (2): We have seen in (1) that Γfc0 is contained in {Θ(x, y) = 0}. Again by (1), Γfc0 ∩ ((Uc0 ∩ PV0 ) × PV (0, 2)) = {Θ(x, y) = 0} ∩ ((Uc0 ∩ PV0 ) × PV (0, 2)) , and (2) follows. (3) follows from (2) and the deﬁnition of Xc0 . (4): Since X is an (SL3 C, SO3 C)-section of PV0 , it suﬃces to prove that the SL3 C-equivariant rational map Ψ : PV (0, 4) Ψ PV (0, 4) (deﬁned e.g. in the point Ch20 ) is birational. We will do this by writing down an explicit rational inverse. To do this, remark that V (a, b) sits as an SL3 C-invariant linear subspace inside Syma C3 ⊗ Symb (C3 )∨ (it has multiplicity one in the decomposition into irreducibles), thus elements of V (a, b) may be viewed as b b 3 tensors x = (xij11,...,i ,...,ja ) ∈ Ta C , covariant of order b and contravariant of

24

C. Böhning

order a, or of type ab . The inverse of the determinant tensor det−1 is thus in T30 C3 . For f ∈ V (0, 4) and g ∈ V (2, 2) one deﬁnes a bilinear SL3 C-equivariant map α : V (0, 4) × V (2, 2) → Sym2 C3 ⊗ Sym3 (C3 )∨ , (f, g) → α(f, g), as the contraction sij11 ij22 i3 := f i1 i2 i4 i5 gii56 ji31 det−1 j2 i4 i6 , followed by the symmetrization map. One checks that Sym2 C3 ⊗ Sym3 (C3 )∨ decomposes as V (2, 3) ⊕ V (1, 2) ⊕ V (0, 1), but Sym4 V (0, 4) does not contain these as subrepresentations (use Magma), so α(f, Ψ (f )) = 0 for all f ∈ V (0, 4). But the explicit forms of Ψ and α show that ker α(·, Ψ (h20 )) = Ch20 , whence, by upper-semicontinuity, the dimension of the kernel of α(·, Ψ (f )) is one for all f in a dense open subset of V (0, 4), and the rational map Ψ : PV (0, 4) Ψ PV (0, 4) ⊂ PV (2, 2) has the rational inverse Ψ (f ) → ker α(·, Ψ (f )).

Remark 2.3.4. It would probably be illuminating to have a geometric interpretation of the covariant Ψ : V (0, 4) → V (2, 2) given above similar to the one for α1 , α2 in Subsection 2.2. Though there is a huge amount of classical projective geometry attached to plane quartics, I have been unable to ﬁnd such a geometric description. Clearly, Ψ vanishes on the cone of dominant vectors in V (0, 4), and one may check, using the explicit formula for Ψ in Appendix A (64), that Ψ also vanishes on the SL3 C-orbit of the degree 4 forms in two variables, x and y, say. However, this, together with the fact that Ψ is of degree 3, is not enough to characterize Ψ since the same holds also for, e.g., the Hessian covariant. 2.4 From cubic to quadratic equations We have to ﬁx some further notation. Definition 2.4.1. (1) Z is the aﬃne cone in V (2, 2) over Ψ (X) ⊂ PV (2, 2). (2) L is the linear subspace L := {g ∈ V (2, 2) | Φ(g, h0 ) = 0} ⊂ V (2, 2). (3) : V (0, 4) × V (0, 2) → V (2, 2) is the unique (up to a nonzero factor) nontrivial SL3 C-equivariant bilinear map with the indicated source and target spaces (the explicit form is in Appendix A (66)). (4) ζ : V (0, 4)×V (0, 2) → V (1, 1) is the unique (up to factor) nontrivial SL3 Cequivariant map with the property that it is homogeneous of degree 2 in both factors of its domain (cf. Appendix A (67) for the explicit description). We put Γ := ζ(·, h0 ) : V (0, 4) → V (1, 1). Let us state explicitly what we are heading toward: The aﬃne cone Z over the birational modiﬁcation Ψ (X) of our (SL3 C, SO3 C)-section X ⊂ PV0 ⊂ PV (0, 4) (whose closure in PV (0, 4) was seen to be an irreducible component of an algebraic set deﬁned by 15 cubic equations) has the following wonderful properties: Z lies

Rationality of M3

25

in L, the linear map (·, h0 ) : V (0, 4) → V (2, 2) restricts to an SO3 Cequivariant isomorphism between V (0, 4) and L, and if, via this isomorphism, we transport Z into V (0, 4) and call this Y , then the equations for Y are given by Γ ! More precisely, Y is the unique irreducible component of Γ −1 (0) passing through the point h20 , and Γ maps V (0, 4) into a ﬁve-dimensional SO3 C-invariant subspace of V (1, 1)! Thus, if we have carried out this program, Y (or Z) will be proven to be an irreducible component of an algebraic set deﬁned by ﬁve quadratic equations! This seems quite miraculous, but a satisfactory explanation why this happens probably requires an answer to the problem raised in Remark 2.3.4. We start with some preliminary observations: It is clear that Z ⊂ L and ∗ C(PV (0, 4))SL3 C C(Z)SO3 C×C , C∗ acting by homotheties. In the following, we need the decomposition into irreducibles of SL3 C-modules such as V (2, 2), V (2, 1), V (1, 1), and V (0, 4) as SO3 C-modules. The patterns according to which irreducible representations of a complex semi-simple algebraic group decompose when restricted to a smaller semi-simple subgroup are generally known as branching rules. In our case the answer is V (2, 2) = V (2, 2)8 ⊕ V (2, 2)6 ⊕ V (2, 2)4 ⊕ V (2, 2)4 ⊕ V (2, 2)0 ,

(12)

V (2, 1) = V (2, 1)6 ⊕ V (2, 1)4 ⊕ V (2, 1)2 , V (1, 1) = V (1, 1)4 ⊕ V (1, 1)2 ,

(13) (14)

V (0, 4) = V (0, 4)8 ⊕ V (0, 4)4 ⊕ V (0, 4)0 .

(15)

Here the subscripts indicate the numerical label of the highest weight of the respective SO3 C-submodule of the ambient SL3 C-module under consideration. Note also that SO3 C PSL2 C, so we are really back in the much classically studied theory of binary forms. It is not diﬃcult (and fun) to check (12), (13), (14), (15) by hand; let us brieﬂy digress on how this can be done (cf. [Fu-Ha]): We ﬁx the following notation. Let ﬁrst n = 2l + 1 be an odd integer, g = sln C the Lie algebra of SLn C, and let tg be its standard torus of diagonal matrices of trace 0, and deﬁne the standard weights i ∈ t∨ g , i = 1, . . . , n, by

i (diag(x1 , . . . , xn )) := xi . Inside g we ﬁnd h := son C deﬁned by ⎧⎛ ⎞ X Y U ⎨ h := ⎝ Z −X t V ⎠ | X, Y, Z ∈ gll C, Y t = −Y t , ⎩ −V t −U t 0 Z = −Z t , U, V ∈ Cl . Then th := {diag(x1 , . . . , xl , −x1 , . . . , −xl , 0) | xi ∈ C}; by abuse of notation we denote the restrictions of the functions i to th by the same letters. The fundamental weights of g are πi := 1 +· · ·+ i , i = 1, . . . , n−1, the fundamental weights of h are ωi := 1 +· · ·+ i (1 ≤ i ≤ l −1) and ωl := ( 1 +· · ·+ l )/2. Let + Λg and Λh be the corresponding weight lattices. Λ+ g and Λh are the dominant

26

C. Böhning

weights. For g (and similarly for h) an irreducible representation V (λ) for λ ∈ Λ+ g comes with its formal character chλ :=

mλ (μ)e(μ) ∈ Z[Λg ] ,

μ∈Π(λ)

an element of the group algebra Z[Λg ] generated by the symbols e(λ) for λ ∈ Λg , where Π(λ) means the weights of V (λ), and mλ (μ) is the dimension of the weight space corresponding to μ in V (λ). We have a formal character chV for any ﬁnite-dimensional g-module V = V (λ1 ) ⊕ · · · ⊕ V (λt ), λ1 , . . . , λt ∈ Λ+ g deﬁned by chV :=

t

chλi .

i=1

The important point is that V (i.e., its irreducible constituents) can be recovered from the formal character chV , meaning that in Z[Λg ] we can write chV uniquely as a Z-linear combination of characters corresponding to dominant weights λ ∈ Λ+ g. We go back to the case l = 1, n = 3. We have h = so3 C = sl2 C. The character chV (a) of the irreducible so3 C-module V (a) := V (aω1 ) is not hard: The weights of V (a) are −aω1 , (−a + 2)ω1 , . . . , (a − 2)ω1 , aω1 (all multiplicities are 1). It remains to understand the weights and their multiplicities in the irreducible g = sl3 C-module V (a, b) := V (aπ1 + bπ2 ). In fact, noting that π1 restricted to the diagonal torus of so3 C above is 2ω1 , and the restriction of π2 is 0, we see that, once we know the formal character of V (a, b) as sl3 C-module, we simply substitute 2ω1 for π1 and 0 for π2 in the result and obtain in this way the formal character of the so3 C-module V (a, b), and hence its decomposition into irreducible constituents as so3 C-module. Let us assume a ≥ b (otherwise pass to the dual representation); we describe the weights and their multiplicities of the sl3 C-module V (a, b) following [Fu-Ha, p. 175ﬀ.]: Imagine a plane with a chosen origin from which we draw two vectors of unit length, representing π1 and π2 , such that the angle measured counterclockwise from π1 to π2 is 60◦ . Thus the points of the lattice spanned by π1 , π2 are the vertices of a set of equilateral congruent triangles which gives a tiling of the plane. The weights of V (a, b) are the lattice points which lie on the edges of a sequence of b (not necessarily regular) hexagons Hi with vertices at lattice points, i = 0, . . . , b − 1, and a sequence of [(a − b)/3] + 1 triangles Tj , j = 0, . . . , [(a − b)/3]. The Hi and Tj are concentric around the origin, and Hi has one vertex at (a − i)π1 + (b − i)π2 , Tj has one vertex at the point (a − b − 3j)π1 , and Hi and Tj are otherwise determined by the condition that the

Rationality of M3

27

lines through π1 , π2 , π2 − π1 are axes of symmetry for them, i.e., they are preserved by the reﬂections in these lines (one should make a picture now). The multiplicities of the weights obtained in this way are as follows: Weights lying on Hi have multiplicity i + 1, and weights lying on one of the Tj have multiplicity b. This completely determines the formal character of V (a, b). Let us look at V (2, 2) for example. Here we get three concentric regular hexagons (one of them is degenerate and consists of the origin alone). The weights are thus 2π1 + 2π2 , 3π2 , −2π1 + 4π2 , −3π1 + 3π2 , −4π1 + 2π2 , −3π1 , −2π1 − 2π2 , −3π2 , 2π1 − 4π2 , 3π1 − 3π2 , 4π1 − 2π2 , 3π1 (these are the ones on the outer hexagon, read counterclockwise, and have multiplicity 1), π1 + π2 , −π1 + 2π2 , −2π1 + π2 , −π1 − π2 , π1 − 2π2 , 2π1 − π2 (these lie on the middle hexagon and have multiplicity 2), and ﬁnally there is 0 with multiplicity 3 corresponding to the origin. Consequently, the formal character of V (2, 2) as a representation of so3 C is e(−8ω1 ) + 2e(−6ω1) + 4e(−4ω1) + 4e(−2ω1 ) + 5e(0ω1 ) , +4e(2ω1) + 4e(4ω1 ) + 2e(6ω1) + e(8ω1 ) which is equal to chV (8) + chV (6) + 2chV (4) + chV (0) . This proves (12), and (13), (14), and (15) are similar. We resume the discussion of the main content of Subsection 2.4. Before stating the main theorem, we collect some preliminary facts in the following lemma. Lemma 2.4.2. (1) The following decomposition of L ⊂ V (2, 2) as SO3 C-subspace of V (2, 2) holds (possibly after interchanging the roles of V (2, 2)4 and V (2, 2)4 ): L = V (2, 2)8 ⊕ V (2, 2)4 ⊕ V (2, 2)0 . (2) The map (·, h0 ) : V (0, 4) → V (2, 2) is an SO3 -equivariant isomorphism onto L. (3) Putting Y := (·, h0 )−1 (Z) ⊂ V (0, 4), we have h20 ∈ Y . (4) One has Γ (V (0, 4)) ⊂ V (1, 1)4 ⊂ V (1, 1), and the inclusion Y ⊂ Γ −1 (0) holds. Proof. (1): Using the explicit form of Φ one calculates that the dimension of the image of the SO3 C-equivariant map Φ(·, h0 ) : V (2, 2) → V (2, 1) is 12. Thus, in view of the decomposition (13) of V (2, 1) as SO3 C-representation, we must have Φ(V (2, 2), h0 ) = V (2, 1)6 ⊕ V (2, 1)4 . Since

28

C. Böhning

dim V (a, b) =

1 (a + 1)(b + 1)(a + b + 2) , 2

(16)

the dimension of V (2, 2) is 27 and the kernel L of Φ(·, h0 ) has dimension 15; in fact, V (2, 2)8 , V (2, 2)0 , and (after possibly exchanging V (2, 2)4 and V (2, 2)4 ) V (2, 2)4 must all be in the kernel, since these representations do not appear in the decomposition of the image. (2): Using the explicit form of given in Appendix A (66), one calculates that the dimension of the image of (·, h0 ) is 15 whence this linear map is injective. Moreover, its image is contained in L, hence equals L, because the map V (0, 4) × V (0, 2) → V (2, 1) given by (f, g) → Φ( (f, g), g) is identically zero since there is no V (2, 1) in the decomposition of V (0, 4) ⊗ Sym2 V (0, 2). (3): As we saw in Theorem 2.3.3 (1), Ch20 ∈ X, and we have 0 = Ψ (h20 ) ∈ Z. From the decomposition (12), we get, Ψ (h20 ) being invariant, Ψ (h20 ) C = LSO3 C . By the decomposition (15), we get that the preimage under (·, h0 ) of Ψ (h20 ) spans the SO3 C-invariants V (0, 4)0 which are thus in Y . So in particular, h20 ∈ Y . (4): The ﬁrst part is straightforward: Just decompose Sym2 V (0, 4) as SO3 C-module by the methods explained above, and check that it does not contain any SO3 C-submodule the highest weight of which has numerical label 2 (this suﬃces by (14)). The second statement of (4) follows from the observation that the map ζ : V (0, 4) × V (0, 2) → V (1, 1) (Deﬁnition 2.4.1 (4)) factors: c · ζ = γ˜ ◦ , c ∈ C∗ , where γ˜ : V (2, 2) → V (1, 1) is the unique (up to nonzero scalar) nontrivial SL3 C-equivariant map which is homogeneous of degree 2. This is because V (1, 1) occurs in the decomposition of Sym2 V (0, 4) ⊗ Sym2 V (0, 2) with multiplicity 1, and γ˜ ◦ is not identically zero, as follows from the explicit form of these maps (cf. Appendix A, (66), (68)). Thus, deﬁning Γ˜ : V (0, 4) → V (1, 1) by Γ˜ (·) := (˜ γ ◦ )(·, h0 ) (which thus diﬀers from Γ just by a nonzero scalar), we must show Γ˜ (Y ) = 0. But recalling the deﬁnitions of Y , Γ˜ , and Z (Deﬁnition 2.4.1 (1)), it suﬃces to show that γ˜ ◦ Ψ is identically zero; the latter is true since it is an SL3 C-equivariant map from V (0, 4) to V (1, 1), homogeneous of

degree 6, but Sym6 V (0, 4) does not contain V (1, 1). This proves (4). Let us now pass from SO3 C to the PSL2 C-picture and denote by V (d) the space of binary forms of degree d in the variables z1 , z2 . This is of course consistent with our previous notation since, under the isomorphism so3 C sl2 C, V (d) is just the irreducible so3 C-module the highest weight of which has numerical label d; since we consider PSL2 C-representations, d is always even. We will ﬁx a covering SL2 C → SO3 C and thus an isomorphism PSL2 C SO3 C, and we will ﬁx isomorphisms δ1 : V (0) ⊕ V (4) ⊕ V (8) → V (0, 4) and δ2 : V (4) → V (1, 1)4 such that (1, 0, 0) maps to h20 under δ1 and both δ1

Rationality of M3

29

and δ2 are equivariant with respect to the isomorphism PSL2 C SO3 C; we will discuss in a moment how this is done, but for now this is not important. Look at the diagram

h20

δ1

Y ⊂ Γ −1 (0) ⊂ V (0, 4)

U := δ1−1 (Y ) ∩ V (0) ⊕ V (4) ⊕ V (8)

(1, 0, 0)

δ1

Γ |Γ −1 (0)

δ := δ2−1 ◦ Γ ◦ δ1

Γ ? ? 0 ∈ V (1, 1)4

δ2

? V (4)

∩ V (1, 1) V (1, 1)4 ⊕ V (1, 1)2 By part (4) of Lemma 2.4.2, we have δ −1 (0) ⊃ U , and by part (3) of the same lemma, (1, 0, 0) ∈ U . Moreover, recalling our construction of X in Theorem 2.3.3, we see that dim X = dim P V (0, 4)−dim P V (0, 2) = 14−5 = 9, whence, chasing through the deﬁnitions of Z, Y , U , we get dim U = 10. But the explicit form of δ (we will see this in a moment) allows us to conclude, by explicit calculation of the rank of the diﬀerential of δ at the invariant point (1, 0, 0), that dim T(1,0,0) U = 10, whence T(1,0,0) U = V (0) ⊕ V (8). Therefore, as U is irreducible, it is the unique component of the (possibly reducible) variety δ −1 (0) passing through (1, 0, 0). Moreover, it is clear the condition {δ = 0} amounts to ﬁve quadratic equations! We have proven Theorem 2.4.3. There is an isomorphism C(P V (0, 4))SL3 C C(U )PSL2 C×C

∗

(17)

where δ : V (0) ⊕ V (4) ⊕ V (8) → V (4) is PSL2 -equivariant and homogeneous of degree 2, and U is the unique irreducible component of δ −1 (0) passing through (1, 0, 0). Moreover, dim U = 10 and T(1,0,0) U = V (0) ⊕ V (8). We close this section by describing the explicit form of the covering SL2 C → SO3 C and the maps δ1 , δ2 , and by making some remarks on transvectants and the ﬁnal formula for the map δ. Let e1 , e2 , e3 be the standard basis in C3 , and denote by x1 , x2 , x3 the dual basis in (C3 )∨ . In this notation, h20 = x1 x3 − x22 . We may view the x’s as coordinates on C3 and identify C3 with the Lie algebra sl2 C by assigning to (x1 , x2 , x3 ) the matrix

30

C. Böhning

X=

x2 −x1 x3 −x2

∈ sl2 C .

Consider the adjoint representation Ad of SL2 C on sl2 C. Clearly, for X ∈ sl2 C, A ∈ SL2 C, the map Ad(A) : X → AXA−1 preserves the determinant of X, which is just our h0 ; the kernel of Ad is the center {±1} of SL2 C, and since SL2 C is connected, the image of Ad is SO3 C. This is how we ﬁx the isomorphism PSL2 C SO3 C explicitly, and how we view SO3 C as a subgroup of SL3 C. Note that the induced isomorphism sl2 C → so3 C on the Lie algebra level can be described as follows: ⎛ ⎞ 020 01 e := → ⎝ 0 0 1 ⎠ , (18) 00 000 ⎛ ⎞ 000 00 f := → ⎝ 1 0 0 ⎠ , 10 020 ⎛ ⎞ 20 0 1 0 h := → ⎝ 0 0 0 ⎠ 0 −1 0 0 −2 (where we view so3 C as a subalgebra of sl3 C in a way consistent with the inclusion on the group level described above). For example, 01 01 x2 −x1 ad (X) = 00 00 x3 −x2 x2 −x1 01 x3 −2x1 − = , x3 −x2 0 −x3 00 so

⎛

⎞ ⎛ x1 02 ⎝ x2 ⎠ → ⎝ 0 0 x3 00

⎞⎛ ⎞ 0 x1 1 ⎠ ⎝ x2 ⎠ . x3 0

To give the isomorphism δ1 : V (0) ⊕ V (4) ⊕ V (8) → V (0, 4) explicitly, we just have to ﬁnd highest weight vectors inside V (0), V (4), V (8) and corresponding highest weight vectors inside V (0, 4). For example, h acts on z24 ∈ V (4) by multiplication by 4, and z24 is killed by e, so this is a highest weight vector inside V (4). But if we compute h · (x1 x33 − x22 x23 ) = (h · x1 )x33 + 3x1 (h · x3 )x23 − 2(h · x2 )x2 x23 − 2x22 (h · x3 )x3 = (−2x1 )x33 + 3x1 (2x3 )x23 − 2 · 0 · x2 x23 − 2x22 (2x3 )x3 = 4(x1 x33 − x22 x23 ), e · (x1 x33 − x22 x23 ) = (e · x1 )x33 + 3x1 (e · x3 )x23 − 2(e · x2 )x2 x23 − 2x22 , (e · x3 )x3 = (−2x2 ) · x33 + 3x1 · 0 · x23 − 2(−x3 )x2 x23 − 2x22 · 0 · x3 = 0

Rationality of M3

31

(use (18) and remark that the x’s are dual variables, so we have to use the dual action), then we ﬁnd that a corresponding highest weight vector for the submodule of V (0, 4) isomorphic to V (4) is x1 x33 − x22 x23 . Proceeding in this way, we see that we can deﬁne δ1 uniquely by the requirements: δ1 : 1 → h20 , z24 → x1 x33 − x22 x23 , z28 → x43 ,

(19)

and using the Lie algebra action and linearity, we can compute the values of δ1 on a set of basis vectors in V (0) ⊕ V (4) ⊕ V (8). To write down δ2 explicitly, remark that V (1, 1) may be viewed as the SL3 C-submodule of C3 ⊗(C3 )∨ consisting of those tensors that are annihilated by Δ :=

∂ ∂ ∂ ∂ ∂ ∂ ⊗ + ⊗ + ⊗ . ∂e1 ∂x1 ∂e2 ∂x2 ∂e3 ∂x3

We take again our highest weight vector z24 ∈ V (4), and all we have to do is to ﬁnd a vector in C3 ⊗ (C3 )∨ on which h acts by multiplication by 4 and which is annihilated by e and Δ. Indeed, e1 x3 is one such. Thus we deﬁne δ2 by δ2 : z24 → e1 x3 . Then it is easy to compute the values of δ2 on basis elements of V (4) in the same way as for δ1 . Let us recall the classical notion of transvectants (“Überschiebung” in German). Let d1 , d2 , n be nonnegative integers such that 0 ≤ n ≤ min(d1 , d2 ). For f ∈ V (d1 ) and g ∈ V (d2 ) one puts ψn (f, g) :=

n n ∂nf ∂ng (d1 − n)! (d2 − n)! (−1)i n−i i i i ∂z1 ∂z2 ∂z1 ∂z2n−i d1 ! d2 ! i=0

(20)

(cf. [B-S, p. 122]). The map (f, g) → ψn (f, g) is a bilinear and SL2 Cequivariant map from V (d1 ) × V (d2 ) onto V (d1 + d2 − 2n). The map

min(d1 ,d2 )

V (d1 ) ⊗ V (d2 ) →

V (d1 + d2 − 2n)

n=0

min(d1 ,d2 )

(f, g) →

ψn (f, g)

n=0

is an isomorphism of SL2 C-modules (“Clebsch–Gordan decomposition”). Thus transvectants make the decomposition of V (d1 ) ⊗ V (d2 ) into irreducibles explicit; a similar result for SL3 C-representations would be very important in several areas of computational invariant theory and also for the rationality question for moduli spaces of plane curves, but is apparently unknown. The explicit form of δ that results from the computations is then

32

C. Böhning

6 1 ψ6 (f8 , f8 ) + ψ4 (f8 , f4 ) 1225 840 11 7 + ψ2 (f4 , f4 ) − f4 f0 , 54 36

δ(f0 , f4 , f8 ) = −

(21)

where (f0 , f4 , f8 ) ∈ V (0) ⊕ V (4) ⊕ V (8). Note that the fact that δ turns out to be such a linear combination of transvectants is no surprise in view of the Clebsch–Gordan decomposition: In fact, δ may be viewed as a map δ : (V (0) ⊕ V (4) ⊕ V (8)) ⊗ (V (0) ⊕ V (4) ⊕ V (8)) → V (4) and using the fact that δ is symmetric and collecting only those tensor products in the preceding formula for which V (4) is a subrepresentation, we see that δ comes from a map δ : (V (0) ⊗ V (4)) ⊕ (V (4) ⊗ V (4)) ⊕(V (8) ⊗ V (4)) ⊕ (V (8) ⊗ V (8)) → V (4) . Thus it is clear from the beginning that δ will be a linear combination of ψ6 , ψ4 , ψ2 , ψ0 as in formula (21), and the actual coeﬃcients are easily calculated once we know δ explicitly! In fact, the next lemma shows that the actual coeﬃcients of the transvectants ψi ’s occurring in δ are not very important. Lemma 2.4.4. For λ := (λ0 , λ2 , λ4 , λ6 ) ∈ C4 consider the homogeneous of degree 2 PSL2 -equivariant map δλ : V (8) ⊕ V (0) ⊕ V (4) → V (4) f8 + f0 + f4 → λ6 ψ6 (f8 , f8 ) + 2λ4 ψ4 (f8 , f4 ) + λ2 ψ2 (f4 , f4 ) + 2λ0 f4 f0 . Suppose that λ0 = 0. Then: (1) One has 1 ∈ δλ−1 (0) and T1 δλ−1 (0) = V (8) ⊕ V (0); thus there is a unique irreducible component Uλ of δλ−1 (0) passing through 1 on which 1 is a smooth point. (2) If furthermore λ ∈ (C∗ )4 , then PUλ is PSL2 C-equivariantly isomorphic to PU(1,6 ,1,6) for some = 0 (depending on λ). Proof. Part (1) is a straightforward calculation, and for part (2) we choose complex numbers μ0 , μ4 , μ8 with the properties 6μ28 = λ6 , μ4 μ8 = λ4 , μ0 μ4 = λ0 , and compute from 6 μ24 = λ2 . Then the map from PUλ to PU(1,6 ,1,6) given by sending [f0 + f4 + f8 ] to [μ0 f0 + μ4 f4 + μ8 f8 ] gives the desired isomorphism.

In the next section we will see that for any = 0, the PSL2 C-quotient of PU(1,6 ,1,6) is rational, and so the same holds for PUλ for any λ ∈ (C∗ )4 ; note, however, that the reduction step in Lemma 2.4.4 (2) just simpliﬁes the subsequent calculations, but is otherwise not substantial.

Rationality of M3

33

3 Further sections and inner projections 3.1 Binary quartics again and a (PSL2 C, S4 )-section All the subsequent constructions and calculations depend very much on the geometry of the PSL2 C-action on the module V (4). In fact, the ﬁrst main point in the proof that PUλ /PSL2 C is rational will be the construction of a (PSL2 C, S4 )-section of this variety (S4 being the group of permutations of four elements); this is done by using Proposition 2.1.2 (2) for the projection of V (8) ⊕ V (0) ⊕ V (4) to V (4) and producing such a section for V (4) via the concept of stabilizer in general position which we recall next. Definition 3.1.1. Let G be a linear algebraic group G acting on an irreducible variety X. A stabilizer in general position (s.g.p.) for the action of G on X is a subgroup H of G such that the stabilizer of a general point in X is conjugate to H in G. An s.g.p. (if it exists) is well-deﬁned to within conjugacy, but it need not exist in general; however, for the action of a reductive group G on an irreducible smooth aﬃne variety, an s.g.p. always exists by results of Richardson and Luna (cf. [Po-Vi, §7]). Proposition 3.1.2. For the action of PSL2 C on V (4), an s.g.p. is given by the subgroup H generated by 0 1 i 0 ω := and ρ := . −1 0 0 −i H is isomorphic to the Klein four-group V4 Z/2Z⊕Z/2Z and its normalizer N (H) in PSL2 C is isomorphic to S4 ; one has N (H)/H S3 . More explicitly, N (H) = τ, σ , where, putting θ := exp(2πi/8), one has −1 3 7 1 θ 0 θ θ τ := , σ := √ . 5 5 0 θ 2 θ θ Proof. We will give a geometric proof due to Bogomolov [Bog, p. 18]. A general homogeneous degree 4 binary form f ∈ V (4) determines a set of 4 points Σ ⊂ P1 ; the double cover of P1 with branch points Σ is an elliptic curve; it is acted on by its subgroup of 2-torsion points Hf Z/2Z ⊕ Z/2Z, and this action commutes with the sheet exchange map, hence descends to an action of Hf on P1 which preserves the point set Σ and thus the polynomial f ; in general Hf will be the full automorphism group of the point set Σ since a general elliptic curve does not have complex multiplication. Let us see that Hf is conjugate to H: Hf is generated by two commuting reﬂections γ1 , γ2 acting on the Riemann sphere P1 (with two ﬁxed points each). By applying a suitable projectivity, we see that Hf is conjugate to ω, γ2 where γ2 is another reﬂection commuting with ω; thus ω interchanges the

34

C. Böhning

ﬁxed points of γ2 and also the ﬁxed points of ρ: Thus if we change coordinates via a suitable dilation (a projectivity preserving the ﬁxed points of ω), γ2 goes over to ρ, and thus Hf is conjugate to H. One computes that σ and τ normalize H; in fact, σ −1 ωσ = ρ, σ −1 ρσ = ωρ, and τ −1 ωτ = ωρ, τ −1 ρτ = ρ. Moreover, τ has order 4 and σ order 3, (τ σ)2 = 1, thus one has the relations τ 4 = σ 3 = (τ σ)2 = 1 . It is known that S4 is the group on generators R, S with relations R4 = S 2 = (RS)3 = 1; mapping R → τ −1 , S → τ σ, we see that the group τ, σ < N (H) is a quotient of S4 ; since τ, σ contains elements of order 4 and order 3, its order is at least 12, but since there are no normal subgroups of order 2 in S4 , S4 = τ, σ . To ﬁnish the proof, it therefore suﬃces to note that the order of N (H) is at most 24: For this one just has to show that the centralizer of H in PSL2 C is just H, for then N (H)/H is a subgroup of the group of permutations of the three nontrivial elements H − {1} in H (in fact equal to it). Elements in PGL2 C commuting with ω must be of the form a b a b or , −b a b −a and if these commute also with ρ, the elements 1, ω, ρ, ωρ are the only possibilities.

Corollary 3.1.3. The variety (V (4)H )0 ⊂ V (4) consisting of those points whose stabilizer in PSL2 C is exactly H is a (PSL2 C, N (H))-section of V (4). Proof. The fact that the orbit PSL2 C · (V (4)H )0 is dense in V (4) follows since a general point in V (4) has stabilizer conjugate to H; the assertion ∀g ∈ PGL2 C, ∀x ∈ (V (4)H )0 : gx ∈ (V (4)H )0 =⇒ g ∈ N (H) is clear by deﬁnition.

Let us recall the representation theory of N (H) = S4 viewed as the group of permutations of four letters {a, b, c, d}; the character table is as follows (cf. [Ser]). χ0

θ ψ

ψ

1 (ab) (ab)(cd) (abc) (abcd) 1 1 1 1 1 1 −1 1 1 −1 2 0 2 −1 0 3 1 −1 0 −1 3 −1 −1 0 1

Vχ0 is the trivial one-dimensional representation, V is the one-dimensional representation where (g) is the sign of the permutation g; S4 = N (H) being the semidirect product of N (H)/H = S3 by the normal subgroup H, Vθ is the

Rationality of M3

35

irreducible two-dimensional representation induced from the representation of S3 acting on the elements of C3 which satisfy x + y + z = 0 by permutation of coordinates. Vψ is the extension to C3 of the natural representation of S4 on R3 as the group of rigid motions stabilizing a regular tetrahedron; ﬁnally, V ψ = V ⊗ Vψ . We want to decompose V (8) ⊕ V (0) ⊕ V (4) as N (H)-module; we ﬁx the notation: a0 := 1;

a1 := z14 + z24 , a2 := 6z12 z22 , a3 := z14 − z24 ,

(22)

a4 := 4(z13 z2 − z1 z23 ), a5 := 4(z13 z2 + z1 z23 ); e1 := 28(z16 z22 − z12 z26 ), e2 := 56(z17 z2 + z15 z23 − z13 z25 − z1 z27 ), e3 := 56(z17z2 − z15 z23 − z13 z25 + z1 z27 ), e4 := z18 − z28 , e5 := 8(z17 z2 − 7z15 z23 + 7z13 z25 − z1 z27 ), e6 := 8(z17 z2 + 7z15 z23 + 7z13 z25 + z1 z27 ), e7 := z18 + z28 , e8 := 28(z16 z22 + z12 z26 ), e9 := 70z14z24 . Lemma 3.1.4. One has the following decompositions as N (H)-modules: V (0) = Vχ0 , V (4) = Vψ ⊕ Vθ , V (8) = V ψ ⊕ Vψ ⊕ Vθ ⊕ Vχ0 .

(23)

More explicitly, V (0) = a0 , V (4) = a3 , a4 , a5 ⊕ a1 , a2 ,

(24)

V (8) = e4 , e5 , e6 ⊕ e1 , e2 , e3 ⊕ e8 , 7e7 − e9 ⊕ 5e7 + e9 . Here e4 , e5 , e6 corresponds to V ψ and e1 , e2 , e3 corresponds to Vψ . Moreover, V (0)H = a0 , V (4)H = a1 , a2 , V (8)H = e7 , e8 , e9 .

(25)

Proof. We will prove (25) ﬁrst; one observes that quite generally for k ≥ 0, V (2k)H = (V (2k)ρ )ω (ρ and ω commute) and that the monomials z1j z22k−j , j = 0, . . . , 2k, are invariant under ρ if j + k is even, and otherwise anti-invariant, so if k = 2s, dim V (2k)ρ = 2s+1, and if k = 2s+1, dim V (2k)ρ = 2s+1. Since ω is also a reﬂection, we have 2 dim(V (2k)ρ )ω − dim V (2k)ρ = tr(ω|V (2k)ρ ), and the trace is 1 for k = 2s, and −1 for k = 2s + 1, thus dim V (2k)H = s + 1, k = 2s,

dim V (2k)H = s, k = 2s + 1 .

In particular, the H-invariants in V (0), V (4), V (8) have the dimensions as claimed in (25), and one checks that the elements given there are indeed invariant. To prove (23), we use the Clebsch–Gordan formula V (2k) ⊗ V (2) = V (2k + 2) ⊕ V (2k) ⊕ V (2k − 2)

36

C. Böhning

(cf. (20)) iteratively together with the fact that the character of the tensor product of two representations of a ﬁnite group is the product of the characters of each of the factors; since V (2) has dimension 3 and dim V (2)H = 0, V (2) is irreducible; the value of the character of the N (H)-module V (2) on τ is 1, so V (2) = V ψ . Now V (2) ⊗ V (2) = V (4) ⊕ V (2) ⊕ V (0), and looking at the character table, one checks that ( ψ)2 = χ0 + ( ψ) + (ψ) + (θ) . This proves the decomposition in (23) for V (4). The decomposition for V (8) is proven similarly (one proves V (6) = Vψ ⊕ V ψ ⊕ V ﬁrst). The proof of (24) now amounts to checking that the given spaces are invariant under σ and τ ; ﬁnally note that V ψ corresponds to e4 , e5 , e6 since the value of the character on τ is 1.

Recall from Lemma 2.4.4 that we want to prove the rationality of the quotient (P Uλ )/PSL2 C, and we can and will always assume in the sequel that λ = (1, 6 , 1, 6) for = 0. In view of Lemma 3.1.4 it will be convenient for subsequent calculations to write the map δλ : V (8) ⊕ V (0) ⊕ V (4) → V (4) in terms of the basis (e1 , . . . , e9 , a0 , a1 , . . . , a5 ) in the source and the basis (a1 , . . . , a5 ) in the target. Denote coordinates in V (8) ⊕ V (0) ⊕ V (4) with respect to the chosen basis by (x1 , . . . , x9 , s0 , s1 , . . . , s5 ) =: (x, s). Then one may write ⎞ ⎛ Q1 (x, s) ⎟ ⎜ .. (26) δλ (x, s) = ⎝ ⎠ . Q5 (x, s) with Q1 (x, s), . . . , Q5 (x, s) quadratic in (x, s); their values may be computed using formulas (20), (22), and the deﬁnition of δλ in Lemma 2.4.4, and they can be found in Appendix B. ˜ λ ⊂ V (8) ⊕ V (0) ⊕ V (4) be the subvariety deﬁned by Theorem 3.1.5. Let Q the equations Q1 = · · · = Q5 = 0, s3 = s4 = s5 = 0. There is exactly ˜ λ passing through the one seven-dimensional irreducible component Qλ of Q N (H)-invariant point 5e7 + e9 in V (8); Qλ is N (H)-invariant and C(P Uλ )PSL2 C = C(P Qλ )N (H) .

(27)

Proof. We want to use Proposition 2.1.2 (2). Note that 5e7 + e9 ∈ Uλ : In fact, δλ maps the N (H)-invariants in V (8) ⊕ V (0) ⊕ V (4) to the N (H)-invariants in V (4) which are 0. Since Uλ is the unique irreducible component of δλ−1 (0) passing through a0 = 1, Uλ contains the whole plane of invariants a0 , 5e7 + e9 . If we denote by p : V (8) ⊕ V (0) ⊕ V (4) → V (4) the projection, then ˜ λ = p−1 (V (4)H )∩δ −1 (0). Clearly, Q ˜ λ is N (H)-invariant, and one only has to Q λ

Rationality of M3

37

check that 5e7 +e9 is a nonsingular point on it with tangent space of dimension 7 by direct calculation: Then there is a unique seven-dimensional irreducible ˜ λ passing through 5e7 + e9 which is N (H)-invariant (since component Qλ of Q ˜ λ ). 5e7 + e9 is an invariant point on it and this point is nonsingular on Q −1 It remains to prove (27): Qλ is an irreducible component of p (V (4)H ) ∩ Uλ and Q0λ = Qλ ∩ p−1 ((V (4)H )0 ) is a dense N (H)-invariant open subset of Qλ dominating (V (4)H )0 . Thus by Proposition 2.1.2 (2), C(P Uλ )PSL2 C C(P Q0λ )N (H) C(P Qλ )N (H) .

3.2 Dividing by the action of H Next we would like to "divide out" the action by H, so that we are left with an invariant theory problem for the group N (H)/H = S3 . Look back at the action of N (H) on M := {s3 = s4 = s5 = 0} ⊂ V (8) ⊕ V (0) ⊕ V (4) which is explained in formulas (23), (24); we will adopt the notational convention to denote the irreducible N (H)-submodule of V (8) isomorphic to Vψ by V (8)(ψ) and so forth; thus M = V (0)(χ0 ) ⊕ V (4)(θ) ⊕ V (8)(χ0 ) ⊕ V (8)(θ) ⊕ V (8)(ψ) ⊕ V (8)( ψ) ,

(28)

and looking at the character table of S4 , we see that the action of H is nontrivial only on V (8)(ψ) ⊕V (8)( ψ) = e1 , e2 , e3 ⊕e4 , e5 , e6 where x1 , x2 , x3 and x4 , x5 , x6 are coordinates; in terms of these, we have (ω)(x1 , . . . , x6 ) = (−x1 , x2 , −x3 , −x4 , x5 , −x6 ) ,

(29)

(ρ)(x1 , . . . , x6 ) = (x1 , −x2 , −x3 , x4 , −x5 , −x6 ) , (ωρ)(x1 , . . . , x6 ) = (−x1 , −x2 , x3 , −x4 , −x5 , x6 ) , and τ (x1 , . . . , x6 ) = (−x1 , −ix3 , −ix2 , x4 , −ix6 , −ix5 ) , i i σ(x1 , . . . , x6 ) = 4x3 , − x1 , ix2 , −8x6 , − x4 , −ix5 . 4 8

(30)

Thus we see that the map P(V (8)(ψ) ⊕ V (8)( ψ) ) − {x1 x2 x3 = 0} → R × P2 , x4 x5 x6 1 1 1 (x1 , . . . , x6 ) → , , : : , , x1 x2 x3 x21 x22 x23 where R = C3 , is dominant with ﬁbers H-orbits, and furthermore N (H)equivariant for a suitable action of N (H) on R × P2 : In fact, we will agree to write x2 x3 x3 x1 x1 x2 1 1 1 : : : : = x21 x22 x23 x1 x2 x3

38

C. Böhning

and remark that the subspaces x4 x5 x6 x2 x3 x3 x1 x1 x2 R= , , , , , T := x1 x2 x3 x1 x2 x3 of the ﬁeld of fractions of C V (8)(ψ) ⊕ V (8)( ψ) are invariant under σ and τ (thus P2 = P(T )). If we denote the coordinates with respect to the basis vectors in R resp. T given above by r1 , r2 , r3 resp. y1 , y2 , y3 , then the actions of τ and σ are described by τ (r1 , r2 , r3 ) = (−r1 , r3 , r2 ) , σ(r1 , r2 , r3 ) = (−2r3 , r1 /2, −r2 ), τ (y1 , y2 , y3 ) = (y1 , −y3 , −y2 ) , σ(y1 , y2 , y3 ) = ((1/16)y3 , −16y1 , −y2 ) . Thus the only N (H)-invariant lines in R resp. T are the ones spanned by (2, 1, −1) resp. (−1, 16, −16) on which τ acts by multiplication by −1 resp. by +1 and hence R = R( ) ⊕ R(θ) , T = T(χ0 ) ⊕ T(θ) .

(31)

We see that the morphism π : P(M ) − {x1 x2 x3 = 0}

(32)

→ R × P(T ⊕ V (8)(χ0 ) ⊕ V (8)(θ) ⊕ V (0)(χ0 ) ⊕ V (4)(θ) ) R × P , x4 x5 x6 x2 x3 x3 x1 x1 x2 π(x, s) := , , : : , : x7 : x8 : x9 : s0 : s1 : s2 x1 x2 x3 x1 x2 x3 8

is N (H)-equivariant, dominant, and all ﬁbers are H-orbits. If we consider (x7 , x8 , x9 , s0 , s1 , s2 ) as coordinates in V (8)(χ0 ) ⊕ V (8)(θ) ⊕ V (0)(χ0 ) ⊕ V (4)(θ) in the target of the map π (as we do in formula (32)), we denote them by (y7 , y8 , y9 , y10 , y11 , y12 ) to achieve consistency with [Kat2]. How do we get equations which deﬁne the image ˜ λ ∩ {x1 x2 x3 = 0}) ⊂ R × (P8 − {y1 y2 y3 = 0}) π(P Q in P8 − {y1 y2 y3 = 0} from the quadrics Q1 (x, s), . . . , Q5 (x, s) in formula (26)? ¯ 1, . . . , Q ¯ 5 for We can set s3 = s4 = s5 = 0 in Q1 , . . . , Q5 to obtain equations Q ˜ λ in P(M ); the point is now that the quantities PQ ¯ ¯ ¯ ¯ 1, Q ¯ 2 , Q3 , Q4 , Q4 Q x1 x2 x3 are H-invariant (as one sees from the equations in Appendix B). Moreover, the map π : P(M ) − {x1 x2 x3 = 0} → R × (P8 − {y1 y2 y3 = 0}) is a geometric quotient for the action of H on the source (by [Po-Vi, Theorem 4.2]), so we can write

Rationality of M3

39

¯ ¯ 2 = q2 (r1 , . . . , y12 ), Q3 = q3 (r1 , . . . , y12 ), ¯ 1 = q1 (r1 , . . . , y12 ), Q Q x1 ¯4 ¯4 Q Q = q4 (r1 , . . . , y12 ), = q5 (r1 , . . . , y12 ) x2 x3 where q1 , . . . , q5 are polynomials in (r1 , r2 , r3 ), (y1 , y2 , y3 , y7 , . . . , y12 ) which one may ﬁnd written out in Appendix B. Here we just want to emphasize their structural properties which will be most important for the subsequent arguments: (1) The polynomials q1 , q2 are homogeneous of degree 2 in the set of variables (y1 , . . . , y12 ); the coeﬃcients of the monomials in the y’s are (inhomogeneous) polynomials of degrees ≤ 2 in r1 , r2 , r3 . For r1 = r2 = r3 = 0, q1 , q2 do not vanish identically. (2) The polynomials q3 , q4 , q5 are homogeneous linear in (y1 , . . . , y12 ); the coeﬃcients of the monomials in the y’s are (inhomogeneous) polynomials of degrees ≤ 2 in r1 , r2 , r3 . For r1 = r2 = r3 = 0, q3 , q4 , q5 do not vanish identically. Theorem 3.2.1. Let Y˜λ be the subvariety of R × P8 deﬁned by the equations q1 = q2 = q3 = q4 = q5 = 0. There is an irreducible N (H)-invariant component Yλ of Y˜λ with π([x0 ]) ∈ Yλ , where x0 := 13i(5e7 + e9 ) + 5(4e1 − ie2 + e3 ), such that C(P Qλ )N (H) C(Yλ )N (H) .

(33)

Proof. The variety Yλ will be the closure of the image π(P Qλ ∩{x1 x2 x3 = 0}) in R × P8 . It remains to see that x0 ∈ Qλ . Recall from Theorem 3.1.5 that Qλ is ˜ λ passing through the N (H)-invariant the unique irreducible component of Q ˜ λ ; thus, if we point 5e7 + e9 , and that this point is a nonsingular point on Q ˜ can ﬁnd an irreducible subvariety of Qλ which contains both 5e7 + e9 and x0 , ˜ λ ∩ V (8)σ , where V (8)σ are the we are done. The sought-for subvariety is Q elements in V (8) invariant under σ ∈ N (H). One sees that x0 and 5e7 + e9 lie on it, and computing V (8)σ = 5e7 + e9 , 8e4 − ie5 − e6 , 4e1 − ie2 + e3 , V (4)σ = 2(z14 − z24 ) + 4(z13 z2 + z1 z23 ) + 4i(z13 z2 − z1 z23 ) , ˜ λ ∩ V (8)σ is a quadric in V (8)σ and using δλ (V (8)σ ) ⊂ V (4)σ , we ﬁnd that Q which is easily checked to be irreducible.

Thus it remains to prove the rationality of Yλ /N (H) = Yλ /S3 .

40

C. Böhning

3.3 Inner projections and the "no-name" method The variety Y˜λ comes with the two projections p 8 Y˜λ −−−P−→ P8 ⏐ ⏐ pR

R Recall from (32) that N := P(V (8)θ ⊕ V (4)θ ) ⊂ P8 is an N (H)-invariant three-dimensional projective subspace of P8 . We will show C(Yλ )N (H) C(R × N )N (H) via the following theorem. Theorem 3.3.1. There is an open N (H)-invariant subset R0 ⊂ R containing 0 ∈ R with the following properties: ˜ (1) For all r ∈ R0 the ﬁber p−1 R (r) ⊂ Yλ is irreducible of dimension 3, and −1 pR (R0 ) is an open N (H)-invariant subset of Yλ . (2) There exist N (H)-sections σ1 , σ2 of the N (H)-equivariant projection R0 × P8 → R0 such that N (r) ⊂ P8 deﬁned by σ1 (r), σ2 (r), (1 : 0 : 0 : · · · : 0), (0 : 1 : 0 : · · · : 0), (0 : 0 : 1 : 0 : · · · : 0) , r ∈ R0 , is an N (H)-invariant family of four-dimensional projective subspaces in P8 with the properties: (i) N (r) is disjoint from N for all r ∈ R0 . 8 (ii)The ﬁber pP8 (p−1 R (r)) ⊂ P contains the line σ1 (r), σ2 (r) ⊂ N (r) for all r ∈ R0 . (iii)The projection πr : P8 N from N (r) to N maps the ﬁber 8 pP8 (p−1 R (r)) ⊂ P dominantly onto N for all r ∈ R0 .

Before turning to the proof, let us note the following corollary. Corollary 3.3.2. One has the ﬁeld isomorphism C(Yλ )N (H) C(R × N )N (H) , and the latter ﬁeld is rational. Hence M3 is rational.

Rationality of M3

41

Proof. (of corollary) The N (H)-invariant set p−1 R (R0 ) is an open subset of Yλ . Let us see that the projection πr : Fr := pP8 (p−1 R (r)) N is birational. In fact, Fr is of dimension 3 and irreducible and the intersection of a threecodimensional linear subspace and two quadrics in P8 . Moreover, Fr ∩ N (r) contains a line Lr by Theorem 3.3.1 (2), (ii). Thus for a general point P in N , Fr ∩ Lr , P consists of Lr and a single point (namely, the point of intersection of the two lines which are the residual intersections of each of the two quadrics deﬁning Fr with Lr , P , the other component being Lr itself). Thus πr is generically one-to-one whence birational. Thus one has a birational N (H)-isomorphism p−1 R (R0 ) R0 × N , given by sending (r, [y]) to (r, πr ([y])). Thus one gets the ﬁeld isomorphism in Corollary 3.3.2. By the no-name lemma (cf., e.g., [Dol1, Section 4]), C(R × N )N (H) C(N )N (H) (T1 , T2 , T3 ), where T1 , T2 , T3 are indeterminates, thus it suﬃces to show that the quotient of N by N (H) is stably rational of level ≤ 3. This in turn follows from the same lemma, since clearly, if we take the representation of S3 in C3 by permutation of coordinates, the quotient of P(C3 ) by S3 , a unirational surface, is rational.

Proof. (of theorem) The proof will be given in several steps. Step 1. (Irreducibility of the ﬁber over 0) We have to show that the variety 8 pP8 (p−1 R (0)) ⊂ P is irreducible and three-dimensional. We have explicit equations for it (namely the ones that arise if we substitute r1 = r2 = r3 = 0 in q1 , . . . , q5 , which are thus three linear and two quadratic equations); the assertions can then be checked with a computer algebra system such as Macaulay 2. Recall from Theorem 3.2.1 that Yλ contains π([x0 ]). In fact, 5 0 π([x ]) = (0, 0, 0), − : 20 : −20 : 65 : 0 : 13 : 0 : 0 : 0 , (34) 4 as follows from the deﬁnition of x0 in Theorem 3.2.1 and the deﬁnition of π in (32). Thus π([x0 ]) lies in the ﬁber over 0 of p−1 R and thus, since there is an open subset around 0 in R over which the ﬁbers are irreducible and 3-dimensional, assertion (1) of Theorem 3.3.1 is established. Step 2. (Construction of σ1 ) To obtain σ1 , we just assign to r ∈ R the point (r, σ1 (r)) with σ1 (r) = (0 : 0 : 0 : 0 : 0 : 0 : 1 : 0 : 0), i.e., y10 = 1, the other y’s being 0. This always is in the ﬁber pP8 (p−1 R (r)) as one sees on substituting in the equations q1 , . . . , q5 . Moreover, this is an N (H)-section, since y10 is a coordinate in the space V (0)χ0 in formula (32). Step 3. (Construction of σ2 ; decomposition of V := P(δλ−1 (0) ∩ V (8)) ) The construction of a section σ2 ,

(1)

(9)

σ2 (r) = (σ2 (r) : · · · : σ2 (r)),

involves a little more work. Let us look back at the construction of Yλ in Subsection 3.2 for this, especially the deﬁnition of the projection π in formula

42

C. Böhning

(32), and the decomposition of the linear subspace M ⊂ V (8) ⊕ V (0) ⊕ V (4). By deﬁnition of R, the family of codimension 3 linear subspaces L(r) := {[(x, s)] | x4 = r1 x1 , x5 = r2 x2 , x6 = r3 x3 } ⊂ P(M ) ,

(35)

r = (r1 , r2 , r3 ) ∈ R, is N (H)-invariant, i.e., gL(r) = L(gr), for g ∈ N (H). It is natural to intersect this family with P(δλ−1 (0)∩V (8)) which, as we will see, has dimension 3 and look for an H-orbit Or in the intersection of P(δλ−1 (0)∩V (8)) with the open set of L(r) where x1 x2 x3 = 0. Moreover, we will see that for r = 0, the point [x0 ] is in this intersection. Thus passing to the quotient we may put (r, σ2 (r)) := π(Or )

(36)

to obtain a σ2 with the required properties. Indeed, note that we will have (7) (8) (9) σ2 (r) = σ2 = σ2 = 0 which ensures that σ2 and σ1 span a line. Moreover, 5 (37) σ2 (0) = − : 20 : −20 : 65 : 0 : 13 : 0 : 0 : 0 , 4 by formula (34), which allows us to check assertions (2), (i) and (iii) of Theorem 3.3.1, which are open properties on the base R, by explicit computation for the ﬁber over 0. Property (2), (ii) stated in the theorem is clear by construction. Let us now carry out this program. We will start by explicitly decomposing V := P(δλ−1 (0) ∩ V (8)) into irreducible components. To guess what V might be, note that according to the deﬁnition of δλ in Lemma 2.4.4, δλ vanishes on f8 ∈ V (8) if for the transvectant ψ6 one has ψ6 (f8 , f8 ) = 0; but looking back at the deﬁnition of transvectants in formula (20), we see that ψ6 : V (8) × V (8) → V (4) vanishes if f8 is a linear combination of z18 , z17 z2 , and z16 z22 (since we diﬀerentiate at least three times with respect to z2 in one factor in the summands in formula (20)). Thus X1 := PSL2 C · z18 , z17 z2 , z16 z22 , the variety of forms of degree 8 with a sixfold zero, is contained in V , and one computes that the diﬀerential of δλ |V (8) in z16 z22 is surjective, so that X1 is an irreducible component of V . The dimension of X1 is clearly three. Weyman, in [Wey, Corollary 4], computed the Hilbert function of Xp,g , the variety of binary forms of degree g having a root of multiplicity ≥ p which is g−p+d g−p+d−1 H(Xp,g , d) = (dp + 1) − (d(p + 1) − 1) . g−p g−p−1 For d = 6, g = 8, the leading term in d in this expression is 3d3 , which shows deg X1 = 18 .

(38)

Moreover, we know already that 5e7 + e9 is in V from the proof of Theorem 3.1.5; thus set X2 := PSL2 C · 5e7 + e9 . We know that the stabilizer of 5e7 +

Rationality of M3

43

e9 in PSL2 C contains N (H) because 5e7 + e9 = 5z18 + 5z28 + 70z14 z24 spans the N (H)-invariants in V (8) by Lemma 3.1.4. The claim is that the stabilizer is not larger. An easy way to check this is to use the beautiful theory developed in [Olv, p. 188 ﬀ.], using diﬀerential invariants and signature curves, which allows the explicit determination of the order of the symmetry group of a complex binary form. More precisely we have (cf. [Olv, Corollary 8.68]): Theorem 3.3.3. Let Q(p) be a binary form of degree n (written in terms of the inhomogeneous coordinate p = z1 /z2 ) which is not equivalent to a monomial. Then the cardinality k of the symmetry group of Q(p) satisﬁes k ≤ 4n − 8 , provided that U is not a constant multiple of H 2 , where U and H are the following polynomials in p: H := (1/2)(Q, Q)(2) , T := (Q, H)(1) , U := (Q, T )(1) where, if Q1 is a binary form of degree n1 , and Q2 is a binary form of degree n2 , we put (Q1 , Q2 )(1) := n2 Q1 Q2 − n1 Q1 Q2 , (Q1 , Q2 )(2) := n2 (n2 − 1)Q1 Q2 − 2(n2 − 1)(n1 − 1)Q1 Q2 +n1 (n1 − 1)Q1 Q2 (these are certain transvectants). Applying this result in our case, we ﬁnd the upper bound 24 for the symmetry group of 5e7 + e9 , which is indeed the order of N (H) = S4 . X2 is irreducible of dimension 3, and computing that the diﬀerential of δλ |V (8) is surjective in 5e7 + e9 , we get that X2 is another irreducible component of V . But let us intersect X2 with the codimension 3 linear subspace in V (8) consisting of forms with zeros ζ1 , ζ2 , ζ3 ∈ P1 ; there is a unique projectivity carrying these to three roots of 5e7 + e9 , which are all distinct, thus there are 8 · 7 · 6 such projectivities, and deg X2 ≥ (8 · 7 · 6)/|N (H)|. But one checks easily that V itself has dimension 3 and is the intersection of ﬁve quadrics in P(V (8)), thus has degree ≤ 32. Thus we must have deg X2 = 14, V = X1 ∪ X2 , deg V = 32.

(39)

[x0 ] ∈ X2 ∩ L(0) .

(40)

Note also that

In fact, from the proof of Theorem 3.2.1, we know [x0 ] ∈ V , and [x0 ] ∈ L(0) being clear, we just check that x0 has no root of multiplicity ≥ 6. Step 4. (Construction of σ2 ; intersecting V with a family of linear spaces in P(M )) Let L0 (r) be the open subset of L(r) ⊂ P(M ) where x1 x2 x3 = 0. According to the strategy outlined at the beginning of Step 3, we would like to compute the cardinalities

44

C. Böhning

|L0 (r) ∩ X1 |,

|L0 (r) ∩ X2 |,

for r varying in a small neighborhood of 0 in R. It is, however, easier from a computational point of view to determine the number of intersection points of X1 resp. X2 with certain boundary components of L0 (r) in L(r) ﬁrst; the preceding cardinalities will afterwards fall out as the residual quantities needed to have deg X1 = 18, deg X2 = 14. Thus let us introduce the following additional strata of L(r)\L0 (r): L0 := {[(x, s)] | x1 = x2 = x3 = x4 = x5 = x6 = 0}, L1 (r) := {[(x, s)] | x1 = 0, x4 = r1 x1 , x2 = x3 = x5 = x6 = 0},

(41)

L2 (r) := {[(x, s)] | x2 = 0, x5 = r2 x2 , x1 = x3 = x4 = x6 = 0}, L3 (r) := {[(x, s)] | x3 = 0, x6 = r3 x3 , x1 = x2 = x4 = x5 = 0}, ˜ 1 (r) := {[(x, s)] | x2 x3 = 0, x5 = r2 x2 , x6 = r3 x3 , x1 = x4 = 0} L ˜ 2 (r) := {[(x, s)] | x1 x3 = L 0, x4 = r1 x1 , x6 = r3 x3 , x2 = x5 = 0} ˜ 3 (r) := {[(x, s)] | x1 x2 = L 0, x4 = r1 x1 , x5 = r2 x2 , x3 = x6 = 0}. L(r) is the disjoint union of these and L0 (r). From the equations describing δλ one sees that V is deﬁned in P(V (8)) with coordinates x1 , . . . , x9 by −192x26 − 192x3 x6 + 384x23 − 192x25 − 192x2 x5 + 384x22 −12x1 x4 + 12x7 x8 + 180x8 x9 = 0,

(42)

64x26 − 192x3 x6 − 128x23 − 64x25 + 192x2 x5 + 128x22

(43)

−2x24

+

16x21

+

2x27

−

16x28

−

50x29

= 0,

96x5 x6 − 672x3 x5 − 672x2 x6 + 1248x2 x3 −12x1 x7 + 12x4 x8 + 180x1 x9 = 0,

(44)

6x4 x6 + 42x3 x4 + 84x1 x6 + 156x1 x3 −6x5 x7 − 42x2 x7 + 24x5 x8 − 264x2 x8 + 30x5 x9 − 30x2 x9 = 0,

(45)

−6x4 x5 − 42x2 x4 + 84x1 x5 + 156x1 x2 +6x6 x7 + 42x3 x7 + 24x6 x8 − 264x3 x8 − 20x6 x9 + 30x3 x9 = 0,

(46)

and thus ˜ i (r) ∩ V = ∅ ∀i = 1, 2, 3 L

(47)

˜ 1 (r) consider equation for r in a Zariski open neighborhood of 0 ∈ R (for L ˜ 2 (r) we see that (45) (44) and assume 96r2 r3 − 672r2 − 672r3 + 1248 = 0, for L ˜ cannot hold if 6r1 r3 + 42r1 + 84r3 + 156 = 0, and for L3 (r) equation (46) is impossible provided that −6r1 r2 − 42r1 + 84r2 + 156 = 0). Let us consider the intersection V ∩ L0 . We have to solve the equations 12x7 x8 + 180x8 x9 = 0,

2x27 − 16x28 − 50x29 = 0,

Rationality of M3

45

which have the four distinct solutions (x7 , x8 , x9 ) = (5, 0, ±1), (x7 , x8 , x9 ) = (15, ±5, −1), whence L0 ∩ V = {[5e7 ± e9 ], [15e7 ± 5e8 − e9 ]} .

(48)

We will also have to determine the intersection V ∩ L1 (r) explicitly. We have to solve the equations −12r1 x21 + 12x7 x8 + 180x8 x9 = 0, −2r12 x21 + 16x21 + 2x27 − 16x28 − 50x29 = 0, −12x1 x7 + 12r1 x1 x8 + 180x1 x9 = 0, in the variables x1 , x7 , x8 , x9 . We can check (e.g., with Macaulay 2) that the subscheme they deﬁne has dimension 0 (and degree 8) for r1 = 0. We already know four solutions with x1 = 0, namely, the ones given in formula (48). Then it suﬃces to check that (x1 , x7 , x8 , x9 ) = (±1, r1 , 1, 0), (x1 , x7 , x8 , x9 ) = (±a, (90 − 5r12 ), −5r1 , 6), where a is a square root of 25(r12 − 36), are also solutions (with x1 = 0 in a neighborhood of 0 in R, and obviously all distinct there). Thus L1 (r) ∩ V = {[±(e1 + r1 e4 ) + r1 e7 + e8 ], [±(ae1 + r1 ae4 ) + (90 −

5r12 )e7

(49)

− 5r1 e8 + 6e9 ]} .

We still have to see how the intersection points L0 ∩ V and L1 (r) ∩ V are distributed among X1 and X2 : Suppose f ∈ V (8) is a binary octic such that [f ] ∈ L0 ∩ P(V (8)) or [f ] ∈ L1 (r) ∩ P(V (8)); then f is a linear combination of the binary octics e1 , e4 , e7 , e8 , e9 deﬁned in (22), which involve only even powers of z1 and z2 ; thus if (a : b) ∈ P1 is a root of one of them, so is its negative (a : −b) whence [f ] lies in X1 if and only if (1 : 0) or (0 : 1) is a root of multiplicity ≥ 6. Applying this criterion, we get, using (48) and (49) L0 ∩ X1 = ∅, L0 ∩ X2 = {[5e7 ± e9 ], [15e7 ± 5e8 − e9 ]}, L1 (r) ∩ X1 = {[±(e1 + r1 e4 ) + r1 e7 + e8 ]},

(50)

L1 (r) ∩ X2 = {[±(ae1 + r1 ae4 ) + (90 − 5r12 )e7 − 5r1 e8 + 6e9 ]} . The reader may be glad to hear now that we do not have to repeat this entire procedure for L2 (r) and L3 (r); in fact, L1 (r), L2 (r), L3 (r) are permuted by N (H) in the following way: For the element σ ∈ N (H) we have σ · L1 (r) = L2 (σ · r),

σ · L2 (r) = L3 (σ · r),

σ · L3 (r) = L1 (σ · r) ,

which follows from (30) (and (28)) and the deﬁnition of R. Thus we get that generally for i = 1, 2, 3,

46

C. Böhning

Li (r) ∩ X1 = {P1 (r), P2 (r)}, Li (r) ∩ X2 = {Q1 (r), Q2 (r)}

(51)

where P1 (r), P2 (r), Q1 (r), Q2 (r) are mutually distinct points, and this is valid in a Zariski open N (H)-invariant neighborhood of 0 ∈ R. It remains to check that L(0) ∩ V consists of 32 reduced points. We check (with Macaulay 2) that if we substitute x4 = x5 = x6 = 0 in equations (42)–(46), they deﬁne a zero-dimensional reduced subscheme of degree 32 in the projective space with coordinates x1 , x2 , x3 , x7 , x8 , x9 . Taking into account (47), (50), (51), we see that all the intersections in equations (50), (51) are free of multiplicities in an open N (H)-invariant neighborhood of 0 ∈ R and moreover, since deg X1 = 18, deg X2 = 14, we must have there L0 (r) ∩ X1 consists of 12 reduced points, and L0 (r) ∩ X2 consists of 4 reduced points. Now these 4 points make up the H-orbit Or we wanted to ﬁnd in Step 3: Clearly L0 (r) ∩ X2 is H-invariant, and H acts with trivial stabilizers in L0 (r) (as is clear from (29)). Thus we have completed the program outlined at the beginning of Step 3. It just remains to notice that [x0 ] ∈ X2 ∩ L0 (0). This is clear since [x0 ] ∈ V , but x0 does not have a root of multiplicity ≥ 6. Step 5. (Veriﬁcation of the properties of N (r)) For the completion of the proof of Theorem 3.3.1, it remains to verify the properties of the subspace N (r) in parts (2), (i) and (iii) of that theorem. First of all, it is clear that N (r) = σ1 (r), σ2 (r), (1 : 0 : 0 : · · · : 0), (0 : 1 : 0 : · · · : 0), (0 : 0 : 1 : · · · : 0) is N (H)-invariant in the sense that g · N (r) = N (g · r) for g ∈ N (H) by the construction of σ1 , σ2 and because the last three vectors in the preceding formula are a basis in the invariant subspace P(T ) ⊂ P8 (where by (31) T = T(χ0 ) ⊕ T(θ) ). Moreover, by the deﬁnition of σ1 in Step 2, and the formula (37) for σ2 (0), one has dim N (0) = 4, which thus holds also for r ∈ R suﬃciently close to 0. Recall that N was deﬁned to be N := P(V (8)(θ) ⊕ V (4)(θ) ) ⊂ P8 , and as such can be described in terms of the coordinates (y1 : y2 : y3 : y7 : y8 : · · · : y12 ) in P8 as N = {y1 = y2 = y3 = y7 + 7y9 = y10 = 0} (cf. (24)). Thus we get that N (0) ∩ N = ∅, and the same holds in an open N (H)-invariant neighborhood of 0 in R. For Theorem 3.3.1, (2), (iii), it suﬃces to check that π0 maps the ﬁber pP8 (p−1 R (0)) dominantly onto N , which can be done by direct calculation. This concludes the proof.

Rationality of M3

47

A Formulas for Section 2 We start with some remarks on how to calculate equivariant projections, and then we give explicit formulas for the equivariant maps in Section 2. Let a, b be nonnegative integers, m := min(a, b), and let G := SL3 C. We denote the irreducible G-module whose highest weight has numerical labels a, b by V (a, b). For k = 0, . . . , m we deﬁne V k := Syma−k C3 ⊗ Symb−k (C3 )∨ . Let e1 , e2 , e3 be the standard basis in C3 and x1 , x2 , x3 the dual basis in (C3 )∨ . There are G-equivariant linear maps Δk : V k → V k+1 for k = 0, . . . , m − 1 and δ k : V k → V k−1 for k = 1, . . . , m given by

Δk :=

3 ∂ ∂ ⊗ , ∂ei ∂xi i=1

δ k :=

3

ei ⊗ xi .

(52)

i=1

(The superscript k thus only serves as a means to remember the sources and targets of the respective maps.) If for some positive integers α, β the G-module V k contains a G-submodule isomorphic to V (α, β) we will denote it by V k (α, β) to indicate the ambient module (this is unambiguous because it is known that all such modules occur with multiplicity one). It is clear that Δk is surjective and δ k injective; one knows that ker(Δk ) = k V (a − k, b − k) whence

k

V =

m

V k (a − i, b − i) .

(53)

i=k

We want to ﬁnd a formula for the G-equivariant projection of V 0 = Syma C3 ⊗ Symb (C3 )∨ onto the subspace V 0 (a − i, b − i) for i = 0, . . . , m. We call this i linear map πa,b . We remark that, by (53), one can decompose each vector v ∈ V 0 as v = v0 + · · · + vm where vi ∈ V 0 (a − i, b − i), and this decomposition is unique. Note that δ 1 . . . δ i (ker Δi ) = V 0 (a − i, b − i) so that V 0 = ker Δ0 ⊕ δ 1 (ker Δ1 ) ⊕ δ 1 δ 2 (ker Δ2 ) ⊕ · · · ⊕ δ 1 . . . δ i (ker Δi ) ⊕ · · · ⊕ δ 1 . . . δ m (V m ) . i Of course, πa,b (v) = vi . It will be convenient to put

(54)

48

C. Böhning

Li := δ 1 ◦ δ 2 ◦ · · · ◦ δ i ◦ Δi−1 ◦ · · · ◦ Δ1 ◦ Δ0 ,

i = 0, . . . , m

(55)

i = 0, . . . , m

(56)

(whence L0 is the identity) and U i := Δi−1 ◦ Δi−2 ◦ · · · ◦ Δ0 ◦ δ 1 ◦ · · · ◦ δ i−1 ◦ δ i ,

(U 0 being again the identity). By Schur’s lemma, we have U i |V i (a−i,b−i) = ci · idV i (a−i,b−i) for some nonzero rational number ci ∈ Q∗ . This is easy to calculate: For ⊗ xb−i ∈ ker Δi = V i (a − i, b − i), we have that ci is the example, since ea−i 1 2 unique number such that a−i ⊗ xb−i ⊗ xb−i . U i (ea−i 1 2 ) = ci · e 1 2

(57)

m−l for l = 0, . . . , m by induction on l; the case We will now calculate πa,b l = 0 can be dealt with as follows: Write v = v1 + · · · + vm ∈ V 0 as before. Then vm = δ 1 δ 2 . . . δ m (um ) for some um ∈ V m . Now

Lm (v) = Lm (vm ) = Lm (δ 1 δ 2 . . . δ m (um )) = δ 1 δ 2 . . . δ m ◦ U m (um ) = cm vm so we set m := πa,b

1 m L . cm

(58)

m−l m−l+1 m , πa,b , . . . , πa,b have already been Now assume, by induction, that πa,b m−l−1 determined. We show how to calculate πa,b . Now, by (54), vm−l−1 ∈ δ 1 . . . δ m−l−1 (ker Δm−l−1 ). We write vm−l−1 = 1 δ . . . δ m−l−1 (um−l−1 ), for some um−l−1 ∈ ker Δm−l−1 = V m−l−1 (a − (m − l − 1), b − (m − l − 1)), and using (57) we get

L

m−l−1

=

v−

l

m−i πa,b (v)

i=0 m−l−1 L (vm−l−1 )

= Lm−l−1 (v0 + v1 + · · · + vm−l−1 )

= Lm−l−1 (δ 1 . . . δ m−l−1 (um−l−1 ))

= δ 1 . . . δ m−l−1 ◦ Δm−l−2 . . . Δ0 ◦ δ 1 . . . δ m−l−1 (um−l−1 ) = δ 1 . . . δ m−l−1 ◦ U m−l−1 (um−l−1 ) = cm−l−1 vm−l−1 .

Rationality of M3

49

So we put

m−l−1 πa,b

:=

1 cm−l−1

L

m−l−1

idV 0 −

l

m−i πa,b

.

(59)

i=0

Formulas (52), (55), (56), (57), (58), (59) contain the algorithm to compute the G-equivariant linear projection i πa,b : V 0 → V 0 (a − i, b − i) ⊂ V 0

and thus to compute the associated G-equivariant bilinear map i : V (a, 0) × V (0, b) → V (a − i, b − i) βa,b

in suitable bases in source and target (remark that V (a, 0) = Syma C3 and V (0, b) = Symb (C3 )∨ ). In particular, we obtain for a = 2, b = 1 the map 0 π2,1 : V 0 = Sym2 C3 ⊗ (C3 )∨ → V (2, 1) ⊂ V 0 1 0 = id − δ 1 Δ0 , π2,1 4

(60)

for a = b = 2 the map 0 : V 0 = Sym2 C3 ⊗ Sym2 (C3 )∨ → V (2, 2) ⊂ V 0 π2,2 1 1 0 = id − δ 1 Δ0 + δ 1 δ 2 Δ1 Δ0 , π2,2 5 40

(61)

and for a = b = 1 the map 0 π1,1 : V 0 = C3 ⊗ (C3 )∨ → V (1, 1) ⊂ V 0 1 0 = id − δ 1 Δ0 . π1,1 3

(62)

In the following, we will often view elements x ∈ V (a, b) as tensors x = b 3 ⊗a ⊗ (C3∨ )⊗b =: Tab C3 (the indices ranging from 1 to 3) (xij11,...,i ,...,ja ) ∈ (C ) which are covariant of order b and contravariant of order a via the natural inclusions V (a, b) ⊂ Syma C3 ⊗ Symb (C3 )∨ ⊂ Tab C3 (the ﬁrst inclusion arises since V (a, b) is the kernel of Δ0 , the second is a tensor product of symmetrization maps). In particular, we have the determinant tensor det ∈ T03 C3 and its inverse det−1 ∈ T30 C3 . In formulas involving several

50

C. Böhning

tensors, we will also adopt the summation convention throughout. Finally, we deﬁne can : Tab C3 → Syma C3 ⊗ Symb (C3 )∨ , ej1 ⊗ · · · ⊗ eja ⊗ xi1 ⊗ · · · ⊗ xib → ej1 · · · · · eja ⊗ xi1 · · · · · xib

(63)

as the canonical projection. We write down the explicit formulas for the equivariant maps in section 2. The map Ψ : V (0, 4) → V (2, 2) (degree 3) is given by 0 Ψ (f ) := π2,2 (can(g)) ,

gji11 ij22

:= f

i1 i2 i3 i4

f

i5 i6 i7 i8

f

i9 i10 i11 i12

(64)

−1 −1 −1 det−1 i3 i5 i9 deti4 i6 i10 detj1 i7 i11 detj2 i8 i12

.

The map Φ : V (2, 2) × V (0, 2) → V (2, 1) (bilinear) is given by 0 Φ(g, h) := π2,1 (can(r)) ,

rji11 j2

:=

gji11 ii23 hi3 i4 det−1 i2 i4 j2

(65) .

The map : V (0, 4) × V (0, 2) → V (2, 2) (bilinear) is −1

(f, h) := can(g), gji11 ij22 := f i3 i4 i1 i2 hi5 i6 det−1 i3 j1 i5 deti4 j2 i6 .

(66)

The map ζ : V (0, 4) × V (0, 2) → V (1, 1) (homogeneous of degree 2 in both factors) is given by 0 ζ(f, h) := π1,1 (a) ,

aij11

:=

−1 −1 −1 hi1 i2 hi3 i4 f i5 i6 i7 i8 f i9 i10 i11 i12 det−1 i5 i9 j1 deti6 i10 i2 deti7 i11 i3 deti8 i12 i4

(67) .

The map γ˜ : V (2, 2) → V (1, 1) (homogeneous of degree 2) is given by 0 γ˜ := π1,1 (u) , uij11 := gii31ii42 gji31 ii42 .

(68)

B Formulas for Section 3 In Section 3.1, we saw (formula (26)) that δλ = Q1 (x, s)a1 + Q2 (x, s)a2 + Q3 (x, s)a3 + Q4 (x, s)a4 + Q5 (x, s)a5 . (69) We collect here the explicit values of the Qi (x, s) (recall λ = (1, 6 , 1, 6),

= 0):

Rationality of M3

ˆ 1 (x) + 2x7 s1 + 12x8 s2 + 2x9 s1 + (12s1 s2 ) + 2s0 s1 Q1 (x, s) = Q

51

(70)

+48x2 s4 − 48x3 s5 − 2x4 s3 + 16x5 s4 − 16x6 s5 + (−12s24 − 12s25 ) , ˆ 2 (x) + 4x8 s1 + 12x9 s2 + (2s21 − 6s22 ) + 2s0 s2 Q2 (x, s) = Q (71) 2 2 2 −4x1 s3 + 16x2 s4 + 16x3 s5 − 16x5 s4 − 16x6 s5 + (−2s3 − 4s4 + 4s5 ) , ˆ 3 (x) + 2x4 s1 + 12x1 s2 + 64x2 s5 + 64x3 s4 Q3 (x, s) = Q

(72)

−2x7 s3 + 2x9 s3 + (12s2 s3 − 24s4 s5 ) + 2s0 s3 , ˆ 4 (x) + 4x5 s1 + 12x2 s1 − 12x5 s2 + 12x2 s2 Q4 (x, s) = Q

(73) −8x1 s5 − 16x3 s3 + 8x8 s4 − 8x9 s4 + (−6s1 s4 − 6s2 s4 + 6s3 s5 ) + 2s0 s4 , ˆ 5 (x) + 4x6 s1 + 12x3 s1 + 12x6 s2 − 12x3 s2 (74) Q5 (x, s) = Q +8x1 s4 − 16x2 s3 − 8x8 s5 − 8x9 s5 + (6s1 s5 − 6s2 s5 − 6s3 s4 ) + 2s0 s5 , where ˆ 1 (x) = −192x26 − 192x3 x6 + 384x23 − 192x25 − 192x2 x5 + 384x22 Q −12x1 x4 + 12x7 x8 + 180x8 x9 , 2 ˆ 2 (x) = 64x − 192x3 x6 − 128x2 − 64x2 + 192x2 x5 + 128x2 Q 6

−2x24

3

+

16x21

+

2x27

−

5

16x28

−

2

(75) (76)

50x29 ,

ˆ 3 (x) = 96x5 x6 − 672x3 x5 − 672x2 x6 + 1248x2 x3 Q −12x1 x7 + 12x4 x8 + 180x1 x9 , ˆ 4 (x) = 6x4 x6 + 42x3 x4 + 84x1 x6 + 156x1 x3 Q −6x5 x7 − 42x2 x7 + 24x5 x8 − 264x2 x8 + 30x5 x9 − 30x2 x9 , ˆ 5 (x) = −6x4 x5 − 42x2 x4 + 84x1 x5 + 156x1 x2 Q

(77) (78) (79)

+6x6 x7 + 42x3 x7 + 24x6 x8 − 264x3 x8 − 20x6 x9 + 30x3 x9 . The polynomials q1 , . . . , q5 deﬁning Y˜λ ⊂ R × P8 (cf. Theorem 3.2.1) are:

52

C. Böhning

q1 = (−192r32 − 192r3 + 384)y1 y2 + (−192r22 − 192r2 + 384)y1 y3

(80)

+(−12r1 )y2 y3 + 12y7 y8 + 180y8 y9 + 2y7 y11 + 12y8 y12 +2y9 y1 1 + (12y11 y12 ) + 2y10 y11 , q2 = (64r32 − 192r3 − 128)y1y2 + (−64r22 + 192r2 + 128)y1 y3 +(−2r12

+

16)y2 y3 + 2y72 − 16y82 2 2 + (2y11 − 6y12 )

−

50y92

(81)

+ 4y8 y11 + 12y9 y12

+ 2y10 y12 , q3 = (96r2 r3 − 672r2 − 672r3 + 1248)y1

(82)

−12y7 + 12r1 y8 + 180y9 + 2r1 y11 + 12y12 , q4 = (6r1 r3 + 42r1 + 84r3 + 156)y2 + (−6r2 − 42)y7 + (24r2 − 264)y8 (83) +(30r2 − 30)y9 + (4r2 + 12)y11 + (−12r2 + 12)y12 , q5 = (−6r1 r2 − 42r1 + 84r2 + 156)y3 + (6r1 + 42)y7 + (24r3 − 264)y8 (84) +(−30r3 + 30)y9 + (4r3 + 12)y11 + (12r3 − 12)y12 .

References [Bog]

Bogomolov, F.A., Rationality of the moduli of hyperelliptic curves of arbitrary genus, in: Proceedings of the 1984 Vancouver Conference in Algebraic Geometry, CMS Conference Proceedings, vol. 6, American Math. Society (1986), 17–37. [B-S] Brion, M. & Schwarz, G.W., Théorie des invariants & Géométrie des variétés quotients, Collection Travaux en Cours, Hermann Éditeurs des Sciences et des Arts, Paris (2000). [Cle] Clebsch, A., Theorie der binären algebraischen Formen, B. G. Teubner, Leipzig (1872). [C-L] Clebsch, A. & Lindemann, F., Vorlesungen über Geometrie, Erster Band. Geometrie der Ebene. B. G. Teubner, Leipzig (1876). [Dix] Dixmier, J., On the Projective Invariants of Quartic Plane Curves, Adv. in Math. 64 (1987), 279–304. [Dol1] Dolgachev, I., Rationality of ﬁelds of invariants, in Algebraic Geometry, Bowdoin, Proc. Symp. Pure Math. vol. 46 (1987), 3–16. [Dol2] Dolgachev, I., Lectures on Invariant Theory, London Mathematical Society Lecture Note Series 296, Cambridge University Press (2003). [Dol3] Dolgachev, I., Topics in Classical Algebraic Geometry. Part I, available at www.math.lsa.umich.edu/˜ idolga/lecturenotes.html. [D-O] Dolgachev, I. & Ortland, D., Point Sets in Projective Space and Theta Functions, Astérisque, 165 (1989). [Fu-Ha] Fulton, W. & Harris, J., Representation Theory. A First Course, Springer GTM 129, Springer-Verlag (1991). [G-Y] Grace, J.H. & Young, W.H., The Algebra of Invariants, Cambridge University Press (1903); reprinted by Chelsea Publ. Co., New York (1965). [Kat0] Katsylo, P. I., Rationality of the moduli variety of curves of genus 5, Math. USSR Sb. 72 (1992), no. 2, 439–445.

Rationality of M3

53

[Kat1] Katsylo, P.I., On the birational geometry of the space of ternary quartics, Adv. in Soviet Math. 8 (1992), 95–103. [Kat2] Katsylo, P.I., Rationality of the moduli variety of curves of genus 3, Comment. Math. Helvetici 71 (1996), 507–524. [Muk] Mukai, S., An Introduction to Invariants and Moduli, Cambridge studies in advanced mathematics 81, Cambridge University Press (2003). [Mum] Mumford, D., Fogarty, J., & Kirwan, F., Geometric Invariant Theory, Third Enlarged Edition, Ergebnisse der Mathematik und ihrer Grenzgebiete 34, Springer-Verlag (1994). [Olv] Olver, P. J., Classical Invariant Theory, London Mathematical Society Student Texts 44, Cambridge University Press (1999). [Po-Vi] Popov, V.L. & Vinberg, E.B., Invariant Theory, in: Algebraic Geometry IV, A.N. Parshin, I.R. Shafarevich (eds.), Encyclopedia of Mathematical Sciences 55, Springer-Verlag (1994). [Pro] Procesi, C., Lie Groups. An Approach through Invariants and Representations, Springer Universitext, Springer-Verlag (2007). [Sal] Salmon, G., A Treatise on the Higher Plane Curves, Hodges, Foster and Figgis, (1879), reprinted by Chelsea Publ. Co. (1960). [Ser] Serre, J.-P., Linear Representations of Finite Groups, Springer GTM 42, Springer-Verlag (1977). [Wey] Weyman, J., The Equations of Strata for Binary Forms, Journal of Algebra 122 (1989), 244–249.

Unramified Cohomology of Finite Groups of Lie Type Fedor Bogomolov1, Tihomir Petrov2 , and Yuri Tschinkel1 1

2

Courant Institute of Mathematical Sciences 251 Mercer Street New York, NY 10012–1185, USA [email protected] [email protected] Department of Mathematics California State University, Northridge 18111 Nordhoﬀ Street Northridge, CA 91330-8313, USA [email protected]

Summary. We prove vanishing results for unramiﬁed stable cohomology of ﬁnite groups of Lie type.

Key words: Rationality, cohomological invariants, ﬁnite groups 2000 Mathematics Subject Classiﬁcation codes: 14E08, 14L30, 14F20

1 Introduction Let k be an algebraically closed ﬁeld, G a ﬁnite group and V a faithful representation of G over k. In this note we compute cohomological obstructions to stable rationality of quotients of V by G introduced by Saltman [13] and [4] and studied in [8], [11], [5]. Let K = k(V )G be the function ﬁeld of the quotient variety and s : GK → G the natural homomorphism from the absolute Galois group of K to G. We have an induced map on cohomology with coeﬃcients in the torsion group Z/, with trivial G-action, s∗i : H i (G, Z/) → H i (GK , Z/). Note that s∗i depends on the ground ﬁeld k, but not on the choice of the faithful representation V over that ﬁeld. The groups

F. Bogomolov, Y. Tschinkel (eds.), Cohomological and Geometric Approaches to Rationality Problems, Progress in Mathematics 282, DOI 10.1007/978-0-8176-4934-0_3, © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

56

F. Bogomolov, T. Petrov, and Y. Tschinkel i Hk,s (G, Z/) := H i (G, Z/)/ Ker(s∗i )

are called stable cohomology groups over k. They form a ﬁnite ring. We may consider them as subgroups of H i (GK , Z/). Every divisorial valuation ν ∈ ValK of K deﬁnes a residue map ∂ν : H i (GK , Z/) → H i−1 (GK ν , Z/), where K ν is the residue ﬁeld of ν. The groups i i Hk,un (G, Z/) := Ker(∂ν ◦ s∗i ) ⊂ Hk,s (G, Z/) ν∈ValK ∗ (G, Z/). A basic fact is that if there exists a faithful form a subring of Hk,s representation V of G over k and a unirational parametrization of the quotient V /G whose degree is prime to , then i (G, Z/) = 0, for all i > 0. Hk,un

In particular, these cohomology groups vanish if this quotient is stably rational. For example, the rings of invariants of ﬁnite groups generated by pseudoreﬂections are polynomial, the corresponding quotient varieties rational, and the cohomological invariants trivial. In particular, all Weyl groups W of semisimple Lie groups have i (W, Z/) = 0, for all i > 0, and all k. Hk,un

Conjecture 1.1. Let G be a ﬁnite simple group. Then i (G, Z/) = 0, for all i > 0, all k, and all . Hk,un

The i = 2 case of this conjecture was proved for G = PSLn (Fq ) and k = C in [5] and for simple and quasi-simple groups of Lie type in [10]. Examples of function ﬁelds with vanishing second and nonvanishing third unramiﬁed cohomology were given in [12]. Here we prove that many of these cohomology ¯ q . In fact, we prove stable groups vanish for ﬁnite groups of Lie type, for k = F rationality of many associated quotient spaces. Our main theorem is: Theorem 1.2. Let G be one of the following groups: SLn (Fq ),

Sp2n (Fq ), or 2 SLn (Fq ).

¯ p . Then the quotient of V Let V be a faithful representation of G over k = F by G is stably rational over k. In particular, Conjecture 1.1 holds in these cases for q. Our main tool is a theorem of Lang which proves the rationality of certain quotient spaces over ¯p. F

Unramiﬁed Cohomology of Finite Groups

57

Theorem 1.3. Let G be a semi-simple simply-connected Lie group deﬁned over a ﬁnite ﬁeld Fq . Then the image i H i (G(Fq ), Z/) → Hk,s (G/G(Fq ), Z/)

¯ q , all i > 0 and q. is zero, for k = F Combining this with results of Tits [15] we obtain the following: Theorem 1.4. Let G be a ﬁnite quasi-simple group of Lie type over a ﬁnite ﬁeld of characteristic p. Put ⎧ if G is > SLn (Fq ), 2 SLn (Fq ), Spn (Fq ), SOn (Fq ), or G2 (Fq ); ⎨∅ if G is of type F4 , E6 , E7 ; d(G) := {2, 3} ⎩ {2, 3, 5} if G is of type E8 . ¯ p , one has Then, for k = F i Hk,un (G, Z/) = 0, for all i > 0, and all ∈ / d(G).

Here is the roadmap of the paper. In Section 2 we study the birational type of quotients G\G/H, where G is an algebraic group over an algebraically closed ﬁeld k and G, H ⊂ G(k) are ﬁnite subgroups, acting on G by translations on the left, resp. on the right. In Section 3 we study the classical groups. In Section 4 we introduce stable and unramiﬁed cohomology over arbitrary algebraically closed ﬁelds and prove their basic properties. In Section 7 we establish general vanishing results, applying theorems of Lang and Tits. In Section 8 we sketch another approach which is using the structure of Sylow subgroups of quasi-simple groups of Lie type. As an example we prove the triviality of unramiﬁed cohomology over C for all groups GLn (Fq ) and coprime to q. Acknowledgment. We are grateful to J.-P. Serre for his comments. The ﬁrst author was supported by NSF grant DMS-0701578 and the third author by NSF grants DMS-0554280 and DMS-0602333.

2 Equivariant birational geometry We work over an algebraically closed ﬁeld k. We say that k-varieties X and Y are stably birational, and write X ∼ Y , if X × An is birational to Y × Am , for some n, m ∈ N. Let G be an algebraic group and X an algebraic variety over k, with a G-action λ : G × X → X.

58

F. Bogomolov, T. Petrov, and Y. Tschinkel

We will sometimes consider diﬀerent actions of the same group. To emphasize the action we will write λ(G)\X for the quotient of X by the λ-action of G; we write G\X, when the action is clear from the context. We say that the action of G is almost free if there exists a Zariski open subset X ◦ ⊂ X on which the action is free. In particular, the quotient map X → G\X is separable. Example 2.1. Let V be a faithful complex representation of G. Then G acts almost freely on V . Lemma 2.2. Let G be a ﬁnite group and V a faithful representation of G over an algebraically closed ﬁeld k. Let Y be an aﬃne variety over k, with a free G-action, and y ∈ Y (k) a point. For every Zariski open U ⊂ V there exist a G-equivariant k-morphism φU : Y → V and a Zariski open G-invariant subset Y ◦ ⊂ Y such that • •

y ∈ Y ◦ (k); φU (Y ◦ ) ⊂ U . |G|

Proof. It suﬃces to consider V := Ak , the aﬃne k-space, with the induced faithful G-action. For any divisor D ⊂ V there exists a Zariski open subset U ⊂ V such that for every point v ∈ U (k) its G-orbit G · v ∈ / D. For every Zariski open U ⊂ V there exists a G-equivariant k-morphism φU : Y → V such that φU (y) ∈ U (k) (functions separate points). This implies the existence of a Zariski open G-invariant subset Y ◦ ⊂ Y with the claimed properties.

A G-variety X is called G-aﬃne, and the corresponding action aﬃne, if there exists a G-equivariant birational isomorphism between X and a faithful representation of G. Let V and X be aﬃne G-varieties. A G-morphism π : V → X is called an aﬃne G-bundle if it is an aﬃne bundle over some open subset X ◦ ⊂ X and the G-action is compatible with this structure of an aﬃne bundle. By Hilbert 90, an aﬃne G-bundle V → X is G-birational to a ﬁnite dimensional G-representation over the function ﬁeld of K = k(X), compatible with the given G-action on K. A morphism ρ : X → B of G-varieties will be called a G-ruling (and X - G-ruled) over B if there exists a ﬁnite set of aﬃne G-varieties Xn = X, Bn−1 , Xn−1 , Bn−2 , Xn−2 , . . . , X1 , B0 = B such that Xi → Bi−1 is an aﬃne G-bundle and Bi ⊂ Xi a G-stable Zariski open subset, for i = 1, . . . , n. Lemma 2.3. Assume that ρ : X → B is a G-ruling over B and that the action of G on X is almost free. Then X is G-aﬃne. Proof. Follows from Hilbert 90.

Unramiﬁed Cohomology of Finite Groups

59

Let X, Y be smooth varieties with an almost free action of G. We write X Y if there exist a G-representation V , a Zariski open G-stable subset X ◦ ⊂ X, and a G-morphism (not necessarily dominant) β : X ◦ × V → Y . G

G

G

We write X Y , and say that the G-actions are equivalent, if X Y and G Y X. G

Lemma 2.4. If X Y , then the morphism β : G\(X × Y ) → G\X has a rational section. Proof. Consider the morphism β : G\(X × V × Y ) → G\X, where G acts diagonally. The graph of the map X → (Y × V ) is G-stable and gives a section of β . The projection of this section to G\(X × Y ) is a section of β.

Lemma 2.5. Let G be a Lie group over an algebraically closed ﬁeld k. Let G ⊂ G(k) be a ﬁnite subgroup. Let X be an algebraic variety over k with an G

almost free action of G. Assume that X G, where G is considered as a G-variety, with a left action. Then G\(X × G) ∼ G\X. Proof. By Lemma 2.4, there is a Zariski open G-stable subset X ◦ ⊂ X so that the G-morphism (projection to the ﬁrst factor) β : G\(X × G) → (G\X) has a section. We also have a right action of G, which preserves the ﬁbration structure given by β. Thus it is a principal homogeneous space over G\X ◦ , for some G-stable Zariski open X ◦ ⊂ G, with a section. Hence it is birational to (G\X) × G. It suﬃces to recall that G is rational over k.

Let G be a connected algebraic group and F ∈ Autk (G) a k-automorphism of G. Let G ⊂ G(k) be a ﬁnite subgroup, with a natural left action λ : G×G → G (γ, g) → γ · g. We also have an F -twisted right action ρF : G × G → G (γ, g) → g · F (γ −1 ) and an F -conjugation κF : G × G → G (γ, g) → γ · g · F (γ −1 ).

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Lemma 2.6. Assume that G ⊂ G(k) has the following properties: G

(1) there exists a faithful G-representation V such that V G, where G acts on G via λ; (2) the twisted action ρF on G is almost free. Then the quotient of G by the F -twisted conjugation κF of G is stably birational to the quotient of G by λ. Proof. Consider the diagonal action of G on G × G: (λ,ρF )

G×G G × G × G −→ (γ, g, g ) → (γ · g, g · F (γ −1 )). Let ΔF := {(g, F −1 (g))} ⊂ G × G be the F -twisted (anti)diagonal. Then ΔF is preserved under the (λ, ρF )-action of G and descends to a section of the principal (right) G-ﬁbration (λ, ρF )(G)\(G × G) → ρF (G)\G, projection to the second factor. It follows that ρF (G)\G ∼ (λ, ρF )(G)\(G × G). Observe that ρF (G)\G ∼ λ(G)\G. Now we show that κF (G)\G ∼ (λ, ρF )(G)\(G × G).

(1)

Let V be a faithful representation of G as in (1) and V ◦ ⊂ V a G-stable Zariski subset admitting a G-map into G, considered with the λ-action of G. We know that there exists a G-morphism ξ : G → V , where G is considered with the κF -action of G, such that ξ(G) ∩ V ◦ = ∅ (see Lemma 2.2). It follows G

that G G, where the source carries the κF -action of G and the image the λ-action of G. Equation (1) now follows from Lemma 2.4.

Corollary 2.7. Let G ⊂ G(k) by a ﬁnite subgroup satisfying Assumption (1) of Lemma 2.6. Let V be a faithful representation of G over k. Then G\G ∼ G\V. More generally, for any X with an almost free action of G we have G\(X × V ) ∼ G\X. Proof. Note that G\(G × V ) is a vector bundle over G\G, and hence stably birational to it. On the other hand, it is a right G-ﬁbration over G\V with section deﬁned by the G-equivariant map V → G. To prove the statement for X it suﬃces to notice that a dense Zariski open

G-stable subset X ◦ ⊂ X admits a nontrivial G-morphism to V .

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3 Equivariant birational geometry of classical groups In this section, k is an algebraically closed ﬁeld, of any characteristic. Lemma 3.1. The conjugation action κ : SL2 → SL2 is equivalent to a linear action. Proof. Realize SL2 as a nonsingular quadric in A4 = M2 . The conjugation action is linear on M2 and has a ﬁxed point corresponding to the identity. The projection of SL2 from the identity to the locus of trace zero matrices is equivariant and has degree 1. Hence the conjugation action is rationally equivalent to the action on trace zero matrices.

Lemma 3.2. Let G = (Z/2)n and X be a G-aﬃne variety over k. Then G\X is rational. Lemma 3.3. Let G ⊂ PGL2 (k) SO3 (k) ⊂ M2 (k) be a ﬁnite subgroup. Then the action of G on PGL2 by conjugation is equivalent to the left (linear) action of G on trace zero matrices M02 ⊂ M2 . In particular, G\PGL2 is stably rational. Lemma 3.4. The left action of (Z/2)2 ⊂ SO3 "→ SO4 is linear. Proof. Consider the subgroup (Z/2)3 ⊂ SO4 . It contains a central subgroup Z/2 and a complementary subgroup H2 = (Z/2)2 ⊂ SO3 = PGL2 . By Lemma 3.1, the conjugation action H2 ⊂ PGL2 is linear. This is a subgroup of the diagonal subgroup SO3 = PGL2 ⊂ SL2 × SL2 /(Z/2). Note that SO4 is a product of SO3 = (h, h) and Spin3 = (g, 1), and that conjugation by elements in (Z/2)2 respects this decomposition. Thus the action is a vector bundle over SL2 /(Q8 )conj = SL2 /(Z/2)2 (where Q8 are the quaternions). Hence the action on SO4 is linear and the automorphism F is the identity

on the diagonal SO3 . The same holds for the twisted F -action. Lemma 3.5. Let G be an algebraic group over k. Assume that G admits an aﬃne action on itself, e.g., G = GLn , SLn , Spn . Let G ⊂ G(k) be a ﬁnite subgroup which has trivial intersection with the center of G. Then the conjugation action of G on G is stably birationally equivalent to a linear action. Proof. By Corollary 2.7, the diagonal left translation action of G on G × G is equivalent to the action on a principal G-bundle over G, with G acting on the base by conjugation. This proves the equivalence.

Corollary 3.6. Let G ⊂ G(k) be a ﬁnite subgroup as in Lemma 3.5. Assume that the action of G on G by left translations is aﬃne. Then both the left and the conjugation action of G on G are stably birational to a linear action.

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Proposition 3.7. Let G be a classical simply-connected Lie group of type A or C, i.e., G = SLn or G = Sp2n over k. Let G ⊂ G(k) be a ﬁnite subgroup. Then G is a G-aﬃne variety, for the standard left action of G. Proof. Note ﬁrst that any ﬁnite subgroup G ⊂ GLn (k) induces a G-aﬃne structure on GLn . Indeed, GLn ⊂ Mn×n = ⊕ni=1 V (i) , a direct sum of n copies of the standard representation V of GLn . Put B0 = 0, a point, and X1 := V (1) = V . We have a canonical projection X1 → B0 . Deﬁne Xj ⊂ ⊕ji=1 V (i) as the (i) set of those vectors, whose projections to ⊕j−1 are linearly independent, i=1 V and Bj ⊂ Xj as the subset of vectors in Xj , which are linearly independent in ⊕ji=1 V (i) , under the standard identiﬁcation V = V (i) . For any G ⊂ GLn this deﬁnes the structure of a G-ruling on GLn over a point. For G = SLn , and G ⊂ G we have a similar G-ruling: For j = 1, . . . , n − 1 it is the same as above. For j = n, put Xn := SLn . The map Xn → Bn−1 is the restriction of the map above. Explicitly, it is the projection to the ﬁrst (n − 1)-vectors (v1 , . . . , vn−1 ), with ﬁber an aﬃne subspace F(v1 ,...,vn−1 ) ⊂ V (n) = V , given by the aﬃne equation in the coordinates of the last vector det(v1 , . . . , vn ) = 1. Now we can apply Lemma 2.3 to conclude that SLn is G-aﬃne. Note that by Tsen’s theorem, the morphism GLn → GLn /SLn has a section. This gives a G-equivariant birational isomorphism SLn × Gm → GLn . The group G = Sp2n has a canonical embedding into M2n×2n , deﬁned by the equations ω(vi , vi ) = δi ,n+i , for i < i (2) (ω is the standard bilinear form and δ is the delta function). The system of projections is induced from the one above: Xj = {(v1 , . . . , vj )} ⊂ ⊕ji=1 V (i) , satisfying equations (2), for indices 1 ≤ i < i ≤ j, and the property that the vectors v1 , . . . , vj−1 are linearly independent in V , under the identiﬁcations V (i) = V . The subvariety Bj ⊂ Xj is given as the locus where v1 , . . . , vj are linearly independent. Each map Xj → Bj−1 is an aﬃne G-bundle, for any ﬁnite subgroup G ⊂ G(k)—its ﬁbers are given by a system of linear equations

on the coordinates of vj . Proposition 3.8. Let G = SOn and G ⊂ G(k) be a ﬁnite subgroup. Then there exist a G-ruling X, a variety Y with trivial G-action, and a G-equivariant ﬁnite morphism π : G × Y → X. Moreover, deg(π) | 2n−1 . Proof. Keep the notations in the proof of Proposition 3.7: G ⊂ Mn .

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Assume that char(k) = 2. Every quadratic form can be diagonalized over k. Let Xn ⊂ Mn be the subvariety given by (vi , vi ) = δi ,n+i , for i < i .

(3)

The system of projections is the same as above: Xj ⊂ ⊕ji=1 V (i) is the subset of vectors satisfying equations (3), for indices 1 ≤ i < i ≤ j, and the condition that v1 , . . . , vj−1 are linearly independent. The subvariety Bj ⊂ Xj corresponds to j-tuples (v1 , . . . , vj ) which are linearly independent (as vectors in V = V (i) ). Each map Xj → Bj−1 is a G-equivariant vector bundle, for any ﬁnite subgroup G ⊂ G(k). Each Xj carries the action of a j-dimensional torus Gjm , over k, commuting with the action of G. The action of G × Gnm on Xn is transitive, and the stabilizer of a general k-point has order 2n−1 . The claim follows, for Y := Gnm . Assume that char(k) = 2. In this case, SO2n+1 Sp2n and we can apply Proposition 3.7. We also have SO2n ⊂ Sp2n , where Sp2n (k) is the set of elements of GL2n (k) which preserve a symplectic bilinear form ω, and SO2n (k) the set of those elements which in addition preserve a quadratic form f . The forms are related by the condition f (x + y) = f (x) + f (y) + ω(x, y). We may identify a general element γ ∈ Sp2n (k) with a choice of an orthogonal basis {v1 , . . . , v2n }. Observe that the map Sp2n → Sp2n /SO2n ∼ A2n {v1 , . . . , v2n } → (f (v1 ), . . . , f (v2n )). Indeed, γ ∈ SO2n (k) if and only if f (γx) = f (x), for all x ∈ V . We have 2n 2n f( ai vi ) = a2i f (vi ) + ai aj (vi , vj ) i=1

i=1

=

2n

i=j

a2i f (γvi ) +

i=1

ai aj ω(γvi , γvj )

i=j

2n = f (γ( ai vi )), i=1

since f (γvi ) = f (vi ) and γ preserves ω. We claim that the bundle Sp2n → Sp2n /SO2n ∼ A2n admits a multisection of degree 22n . Explicitly, it can be constructed as follows: ﬁx an orthogonal basis {v1 , . . . , v2n } such that f (vi ) = 0, for i = 1, . . . , 2n. We have an action of the aﬃne group B = Gm Ga ⊂ SL2 given by (xi , xn+i ) → (λxi , μxi + λ−1 xn+i ), for i = 1, . . . , n.

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We claim that this gives a generically surjective map: SO2n × Bn → Sp2n of degree 22n . The image of Bn · {f (v1 ), . . . , f (v2n )} is dense in A2n . Consider the intersection Bn ∩ SO2n : f (vi ) = λ2 f (vi ), and f (vi+n ) = μ2 f (vi ) + λμω(vi , vi+n ) + λ−2 f (vi+n ). These equations can be solved in k, for each i = 1, . . . , n, and we have a dominant map SO2n × B → A2i of degree 4, for each i. This concludes the proof.

4 Stable cohomology In this section we collect background material on stable cohomology of ﬁnite groups, developing the theory over arbitrary algebraically closed ﬁelds k. We will omit k from the notation when the ﬁeld is clear from the context. For every ﬁnite group G and a G-module M we have the notion of group cohomology, as the derived functor M → M G , the G-invariants, or, topologically, as the cohomology of the classifying space BG = X/G, where X is a contractible space with a ﬁxed point free action of G. Passing to algebraic geometry, let X be an algebraic variety over k, with an almost free action of a G. Let X ◦ ⊂ X be the locus where the action is ˜ := X ◦ /G. This gives free. Let M be a ﬁnite G-module. It deﬁnes a sheaf on X ˜ a homomorphism from group cohomology of G to étale cohomology of X: i ˜ H i (G, M ) → Het (X, M ).

Composing with restriction to the generic point we get a homomorphism σi∗ : H i (G, M ) → H i (GK , M ), ˜ There where GK is the absolute Galois group of the function ﬁeld K = k(X). are canonical isomorphisms H i (GK , M ) = lim H i (X \ D, M ), −→ D

where the limit is taken over divisors of D. We can interpret elements in the ˜ ⊂ X. ˜ kernel of σ ∗ as classes vanishing on some Zariski open subvariety U Remark 4.1. Note that for ﬁxed G and M , the groups σi∗ (H i (G, M )) = 0, for all i > dim(X), while the usual group cohomology need not vanish.

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˜ and a sequence Proposition 4.2. There exist a ﬁnite group G ρ σ ˜ ˜ −→ GK −→ G G

of homomorphisms σ ˜ and ρ such that for all 0 ≤ i ≤ dim(X) one has Ker(σi∗ ) ⊂ Ker(ρ∗i ), ˜ M ) → H i (G, M ) is the induced map on group cohomology. where ρ∗i : H i (G, Proof. The cohomology classes in H i (GK , Z/) are represented by continuous cocycles (in the natural topology on GK ). Any element is induced from a ﬁnite ˜ of GK and the group H. If it vanishes, it also vanishes on a ﬁnite quotient G ˜ → G are continuous. Since the initial group H i (G, Z/) is maps GK → G ˜ where all elements from H i (G, Z/), which vanish on ﬁnite there exists a G GK , are killed.

A special case of the above construction arises as follows: let # : G → V be a faithful representation over an algebraically closed ﬁeld k and let K = k(V )G be the function ﬁeld of the quotient. We have induced maps s∗i : H i (G, M ) → H i (GK , M ) and we can deﬁne the stable cohomology groups over k: i Hk,s (G, M ) := H i (G, M )/ Ker(s∗i ),

which we will often identify with their image in H i (GK , M ). i (G, M ) Proposition 4.3. The cohomology groups Hk,s

(1) do not depend on the representation; (2) are functorial in G; (3) are universal for G-actions: for any G-variety X over k the homomorphism i (G, M ); H i (G, M ) → H i (Gk(X) , M ) factors through Hk,s (4) if M is an -torsion module, then i i (G, M ) = Hk,s (Syl , M )N Hk,s

where Syl = Syl (G) is an -Sylow subgroup of G and N = N (G) its normalizer in G. Proof. We apply Lemma 2.2. Choosing an appropriate Zariski open Ginvariant subvariety X ◦ ⊂ X we can reduce to the aﬃne case, with free G-action. Let V ◦ ⊂ V be a Zariski open subset where the action of G is free. ˜ := G\X ◦ and V˜ := G\V ◦ . We need to show that a class α ∈ H i (G, M ) Put X whose image in H i (Gk(V˜ ) , M ) is zero also vanishes in H i (Gk(X) ˜ , M ). Such a i ˜ ˜ ˜ class vanishes in Het (U , M ), where U ⊂ V is an aﬃne Zariski open subset. The

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˜ in V is a nonempty G-invariant aﬃne Zariski open subset. preimage U of U Thus there exist an aﬃne nonempty G-invariant Zariski open subset UX ⊂ X ◦ and a G-morphism φU : X ◦ → V such that φU (UX ) ⊂ U . This descends to a ˜ ⊂ V˜ . The image of α under the composition ˜ ⊃U ˜X → U morphism X i ˜ i ˜ i (U , M ) → Het (UX , M ) → Het (Gk(X) H i (G, M ) → Het ˜ , M)

is zero. This proves (3). Applying this to X = V , for another faithful representation, we get (1). Property (2) is proved as follows: First, let H ⊂ G be a subgroup and V a faithful G-representation. Consider the morphism H\V → G\V . A class vanishing on a Zariski open subset of G\V also vanishes on a Zariski open subset of H\V . Next, let G → H be a surjective homomorphism and VG , resp. VH , a faithful representation of G, resp. H. Then WG := VH ⊕ VG is a faithful representation of G and we have a commutative diagram Gk(G\WG ) → Gk(H\VH ) ↓ ↓ G → H, giving natural maps on cohomology. We proceed with the proof of Property (4). Since and the cardinality of G/Syl are coprime, the map Syl \V → G\V induces an invertible map on cohomology of the open subvarieties of G\V . The group N (G)/Syl (G) has order prime to . The action of N (G)/Syl (G) on M decomposes the module into a direct sum; so that H i (Syl (G), M ) = H i (Syl (G), M )N (G) ⊕ R, so that the restriction of the trace map is zero on the module R. We have ∼

H i (G, M ) −→ H i (Syl (G), M ) ⊂ H i (Syl (G), M )N(Syl (G)) . ∗ (Syl (G), M ). We get a direct decomposiConsider the image of r ∈ R in Hk,s i (G, M ) surjects tion Hsi (Syl (G), M )N(Syl (G)) ⊕ Rs , with Tr(r) = 0. Thus Hk,s i N(Syl (G)) , and the map is an isomorphism. onto H (Syl (G), M )

Lemma 4.4. Let V be a representation space for a faithful representation of group G over an algebraically closed ﬁeld k. Assume that G\V is isomorphic to aﬃne space. Then any nontrivial element α ∈ Hk,s (G, Z/) has nontrivial restriction to the stable cohomology of a centralizer of a quasi-reﬂection in G. ∗ Proof. If α ∈ Hk,s (k(An ), Z/) is nontrivial, then the residue of α is nontrivial on some irreducible divisor D ⊂ An = G\V (see [6]). The preimage of D in V is a union of irreducible divisors D1 , . . . , Dr . For each i, there exists a nontrivial γi ∈ G acting trivially on all points of Di . Thus each Di is a hyperplane in

V . Hence γi is a quasi-reﬂection.

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Corollary 4.5. Let W be a Weyl group. Then i Hk,s (W, Z/) = 0,

for all i > 0 and all = 2.

We have i i Hk,s (W, Z/2) "→ ⊕τ Hk,s (τ, (Z/2)rτ ),

where τ runs over the set of 2-elementary abelian subgroups of W, modulo conjugation, and rτ ∈ N. Proof. The quasi-reﬂections in the standard faithful representation of W have order 2. Their centralizers are products of powers of Z/2 with smaller Weyl groups. It suﬃces to apply induction.

Remark 4.6. It is possible to obtain a more precise vanishing result following the approach for W = Sn in [9].

5 Comparison with Serre’s negligible classes Stable cohomology was deﬁned by the ﬁrst author in [2] and [3]. J.-P. Serre deﬁned a related but somewhat diﬀerent notion [14, p. 170]. In his terminology, negligible elements α ∈ H ∗ (G, M ) are those which are ˜ is killed under every surjective homomorphism GK → G, where K = k(X) ˜ = G\X ◦ . Negligible elements form an ideal the function ﬁeld of a quotient X in the total ring H ∗ (G, M ). We are considering a smaller set of homomorphisms GK → G, namely, from Galois groups of ﬁelds of type K = k(V )G , and the ideal of negligible classes deﬁned by Serre is smaller. The resulting groups are diﬀerent (for example, for Z/2-coeﬃcients). The quotient ring obtained by Serre’s construction surjects ∗ (G, M ), for any algebraically closed k. onto the ring Hk,s

6 Unramiﬁed cohomology Here we study unramiﬁed cohomology, which was introduced in [4] and [6] (see also [7]). Let K = k(X) be a function ﬁeld over an algebraically closed ﬁeld k, and M an étale sheaf on X. For every divisorial valuation ν ∈ ValK of K we have a split exact sequence 1 → Iν → GKν → GK ν → 1 where Kν is the completion of K with respect to ν, K ν the residue ﬁeld and Iν is the inertia group. This gives an exact sequence in Galois cohomology ν H i−1 (GK ν , Iν M ) H i (GKν , M Iν ) → H i (GKν , M ) −→

δ

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where M Iν , resp. Iν M , are the sheaves of invariants, resp. coinvariants. Unramiﬁed cohomology is deﬁned by i Hk,un (GK , M ) := ∩ν∈ValK Ker(δν ) ⊂ H i (GK , M ).

Lemma 6.1. Let π : X → Y be a surjective morphism of algebraic varieties over k, and M an étale sheaf on Y . Then there is a natural homomorphism: i i π ∗ : Hk,un (Y, M ) → Hk,un (X, π ∗ (M )).

Moreover, if π is ﬁnite, then there is a natural homomorphism i i π∗ : Hk,un (X, π ∗ (M )) → Hk,un (Y, M )

and the composition π∗ ◦ π ∗ is multiplication by the degree of π. Proof. We have an embedding π ∗ : k(X) "→ k(Y ) of function ﬁelds and the corresponding map π∗ : Gk(Y ) → Gk(X) of Galois groups. A divisorial valuation ν of k(Y ) is either trivial on π ∗ (k(X)) or deﬁnes a divisorial valuation ν ∗ on k(X). If ν is trivial on π ∗ (k(X)), then π∗ (Iν ) for the inertia subgroup Iν ⊂ Gk(Y )ν and hence δν is zero on π ∗ H ∗ (Gk(X) , M ). If ν on π ∗ (k(X)) coincides with ν ∗ then under the induced map π∗ : Gk(Y )ν → Gk(X)ν we have π∗ (Iν ) ⊂ Iν ∗ . Thus δν ∗ (α) = 0 implies that δν π ∗ (α) = 0 for all ∗ α ∈ Hk,un (Gk(X) , M ).

Let G be a ﬁnite group and V a faithful G-representation as above. Let K = k(V )G be the function ﬁeld of the quotient. We can consider its stable i cohomology groups Hk,s (G, M ) as subgroups of H i (GK , M ). Deﬁne unramiﬁed cohomology groups i i (G, M ) ∩ Hk,un (GK , M ). Hk,un (G, M ) := Hk,s

Proposition 6.2. Assume that char(k) |M |. Then the unramiﬁed cohomoli (G, M ) ogy groups Hk,un • •

do not depend on the representation V ; are functorial in G.

Proof. Let V, V be two faithful representations of G and consider the diagram G\(V × V ) → G\V ↓ G\V. The case of constant coeﬃcients follows from [8] and the observation that both arrows in the above diagram are natural vector bundles on the quotients. A small modiﬁcation of the argument proves the claim for a general G-module.

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7 General vanishing results ¯ p . Here we collect general arguments In this section, we work over k = F proving triviality of stable and unramiﬁed cohomology groups. Theorem 7.1. Let G be a ﬁnite group and M a ﬁnite p-torsion G-module. i (G, M ) = 0, for all i > 1. Then, Hk,s Proof. See [14, Chapter 2, Proposition 3].

The main reason for introducing unramiﬁed cohomology groups is: Theorem 7.2. Let V be a faithful representation of G. If K = k(V )G is a purely transcendental extension of k, then, for all i > 0, we have i Hk,un (G, Z/) = 0.

Proof. Immediate from [6, Corollary 1.2.1].

Theorem 7.3 (Lang). Let G be an algebraic group over k. Let F be an automorphism of G(k) which is a composition of an element in Aut(G)(k) and ¯ p . Let G = GF ⊂ G(k) be the ﬁnite subgroup ﬁxed by F . a Frobenius of k = F Then G\G G, hence is a rational variety. Proof. Consider the map τ :G→ G x → F (x)−1 x. The action of τ on the Lie algebra of G is surjective with kernel a ﬁnite subgroup G = GF . Note that τ coincides with the composition τ : G → G\G → G. Indeed, if τ (x) = τ (y), then F (x)−1 x = F (y)−1 y, or F (xy −1 )−1 xy −1 = 1, or xy −1 ∈ G. It follows that x = gy, g ∈ G. The converse is clear. Thus G\G is rational.

Lemma 7.4. Let be a prime and π : X → Y a separable morphism ki varieties of ﬁnite degree prime to . Assume that Hk,un (X, Z/) = 0. Then i Hk,un (Y, Z/) = 0. Proof. Immediate from Lemma 6.1: the degree deg(π) is prime to and muli (Y, Z/).

tiplication by deg(π) is invertible on Hk,un This lemma will be applied to Y = G\V . The goal will be to construct X with vanishing unramiﬁed cohomology.

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Corollary 7.5. Let G = GF ⊂ G(k) be as above. Let X be a G-linear variety over k. Assume that there exist a variety Y over k, with trivial G-action, and a G-equivariant ﬁnite morphism π : G × Y → X. Let S be the set of all primes dividing the degree of π. Then i Hk,un (G, Z/) = 0, for all i > 0 and ∈ / S ∪ {p}.

¯ p is rational. By Lang’s Proof. Any aﬃne connected algebraic group over k = F theorem, the quotient G\G is isomorphic to G, and thus rational. For primes ∈ / S ∪ {p} not dividing the degree of π, the induced map on cohomology is injective. This concludes the proof.

Theorem 7.6. Let G be a Lie group over k. Let G = GF ⊂ G(k) be a ﬁnite subgroup. Put ⎧ for G of type C or Dn , n ≥ 5; ⎨ {p, 2} s(G) := {p, 2, 3} for G of type D4 , F4 , E6 , E7 ; ⎩ {p} otherwise. Then i (G, Z/) = 0 Hk,un

for all ∈ / s(G). We have a natural homomorphism H i (G, Z/) → H i (G\G, Z/). By Lang’s Theorem 7.3, G\G G, as algebraic varieties. Thus we get a homomorphism i (G, Z/). ρ : H i (G, Z/) → Het Assume that G is semi-simple. Then Pic(G) π1 (G) is a ﬁnite group and ˜ ˜ is the universal cover of G. We obtain a natural G = G/Pic(G), where G homomorphism i i (Pic(G), Z/) → Het (G, Z/). η : Het Theorem 7.7. Let G be a semi-simple Lie group over k. Let G = GF ⊂ G(k) be a ﬁnite subgroup. Consider the diagram i Het (Pic(G), Z/) σ∗i ◦ η

H i (G, Z/)

σ∗i ◦ ρ

i / Hk,s (G, Z/).

Then the image of ρ is contained in the image of η.

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Proof. Standard computation using restriction of the ﬁbration G → G/T to G◦ = T × AN and the transgression homomorphism. Geometrically, the map σ ∗ ◦ ν in the diagram corresponds to the embedding of the maximal Bruhat cell U+ TU− into the G\G = G, where T is the maximal torus. We have ∗ (T, Z/) = H ∗ (T, Z/), Hk,s

for = p, and these coincide with H ∗ (Pic(T) ⊗ Z/, Z/).

Corollary 7.8. Assume that G is simply-connected and that the natural translation action of G on itself is aﬃne. Then i (G, Z/) = 0, Hk,s

for all i > 0 and q. Proof. Standard computation using restriction of the ﬁbration G → G/T to G◦ = T × AN and the transgression homomorphism. Corollary 7.9. Assume that G is simply-connected and that the natural translation action of G on itself is aﬃne. Then i Hk,s (G, Z/) = 0,

for all i > 0 and q.

8 Reduction to Sylow subgroups Let G be a ﬁnite group. For H ⊂ G let NG (H) denote the normalizer of H. Let Syl (G) be an -Sylow subgroup of G. Recall the following classical result (see, e.g., [1, Section III.5]): H i (G, Z/) = H i (Syl (G), Z/)NG (Syl (G)) . Theorem 8.1. Let G be a ﬁnite group. Let be a prime distinct from the characteristic of k. Then there is an isomorphism ∼

i i Hk,s (G, Z/) −→ Hk,s (Syl (G), Z/)NG (Syl (G)) .

Similarly, ∼

i i (G, Z/) −→ Hk,un (Syl (G), Z/)NG (Syl (G)) . Hk,un

Proof. Let V be a faithful representation of G over k. Then the map π : Syl (G)\V → G\V is a ﬁnite, separable, and surjective map of degree prime to . Hence π∗ ◦ π ∗ is invertible in cohomology. This implies the ﬁrst claim. The fact that local ramiﬁcation indices of π are coprime to implies the second claim.

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Lemma 8.2. Let G, H be ﬁnite groups. Let ρ : H → W be a faithful krepresentation of H. Assume that H\W is stably rational. Let U be a faithful ˜ := H S G, where S is a ﬁnite G-set. Then G\U ˜ representation of G is stably birationally equivalent to G\V , where V is a faithful representation of G. Proof. Put ρS = ⊕s∈S ρs : HS :=

Hs → Aut(WS ),

WS = ⊕s∈S Ws ,

s∈S

where Hs = H, Ws = W , for all s ∈ S, and ρs = ρ on the factor Hs and trivial on Hs , for s = s. We construct U := V ⊕ WS , where V is a faithful representation of G and extend the action of G to VS via the G-action on ˜ = H S G in Aut(U ). The quotient space S. This gives a representation of G is a ﬁbration over G\V with ﬁbers (H\W )|S| . We can assume that H\W is rational. The action of G on (H\W )|S| permutes the coordinates. It follows ˜

that G\U is birationally equivalent to a vector bundle over G\V . Corollary 8.3. Let G = Syl (Sn ) and let V be a faithful representation of G. Then G\V is stably rational. Proof. The -Sylow subgroups of Sn are products of wreath products of groups Z/ · · · Z/ (see [1, VI.1]). The quotient H\W is rational, for a faithful representation W of H = Z/. We apply induction to conclude that the quotient G\V is stably rational.

Corollary 8.4. Let G = Syl (GLn (Fq )), with q, and let V be a faithful representation of G. Then G\V is stably rational. Proof. The structure of -Sylow subgroups of GLn (Fq ) is known (cf. [1, VII.4]): it is also a product of iterated wreath products of cyclic -groups. Thus we can apply Lemma 8.2.

Corollary 8.5. Let k be an algebraically closed ﬁeld of characteristic zero. Then i (GLn (Fq ), Z/) = 0 for all i > 0, and q. Hk,un Remark 8.6. Similar computations can be performed for some other ﬁnite groups of Lie type, e.g., for O± 2m (Fq ) and Spn (Fq ).

References 1. A. Adem and R. J. Milgram – Cohomology of ﬁnite groups, second ed., Grundlehren der Mathematischen Wissenschaften, vol. 309, Springer-Verlag, Berlin, 2004. 2. F. Bogomolov – “Bloch’s conjecture for torsion cohomology of algebraic varieties”, 1985, Amsterdam University preprint 85-20.

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3. — , “Stable cohomology”, 1986, Arbeitstagung talk, MPI preprint. 4. — , “The Brauer group of quotient spaces of linear representations”, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 3, p. 485–516, 688. 5. F. Bogomolov, J. Maciel, and T. Petrov – “Unramiﬁed Brauer groups of ﬁnite simple groups of Lie type Al ”, Amer. J. Math. 126 (2004), no. 4, p. 935– 949. 6. J.-L. Colliot-Thélène and M. Ojanguren – “Variétés unirationnelles non rationnelles: au-delà de l’exemple d’Artin et Mumford”, Invent. Math. 97 (1989), no. 1, p. 141–158. 7. J.-L. Colliot-Thélène – “Birational invariants, purity and the Gersten conjecture, K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), Proc. Sympos. Pure Math. 58, Part 1, American Mathematical Society, Providence, RI, 1995, p. 1–64. 8. J.-L. Colliot-Thélène and J.-J. Sansuc – “The rationality problem for ﬁelds of invariants under linear algebraic groups (with special regards to the Brauer group)”, Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai, 2007, p. 113–186. 9. S. Garibaldi, A. Merkurjev, and J.-P. Serre – Cohomological invariants in Galois cohomology, University Lecture Series, vol. 28, American Mathematical Society, Providence, RI, 2003. 10. B. Kunyavski – “The Bogomolov multiplier of ﬁnite simple groups”, 2009, this volume. 11. E. Peyre – “Unramiﬁed cohomology and rationality problems”, Math. Ann. 296 (1993), no. 2, p. 247–268. 12. — , “Unramiﬁed cohomology of degree 3 and Noether’s problem”, Invent. Math. 171 (2008), no. 1, p. 191–225. 13. D. J. Saltman – “Noether’s problem over an algebraically closed ﬁeld”, Invent. Math. 77 (1984), no. 1, p. 71–84. 14. J.-P. Serre – Cohomologie galoisienne, ﬁfth ed., Lecture Notes in Mathematics, vol. 5, Springer-Verlag, Berlin, 1994. 15. J. Tits – “Sur les degrés des extensions de corps déployant les groupes algébriques simples”, C. R. Acad. Sci. Paris Ser. I Math. 315 (1992), no. 11, p. 1131–1138.

Sextic Double Solids Ivan Cheltsov1 and Jihun Park2 1

2

School of Mathematics The University of Edinburgh Mayﬁeld Road, Edinburgh EH9 3JZ, UK [email protected] Department of Mathematics POSTECH, Pohang, Kyungbuk 790-784, Korea [email protected]

Summary. We study properties of double covers of P3 ramiﬁed along nodal sextic surfaces such as nonrationality, Q-factoriality, potential density, and elliptic ﬁbration structures. We also consider some relevant problems over ﬁelds of positive characteristic.

Key words: Double solids, ﬁbrations, potential density, rational points 2000 Mathematics Subject Classiﬁcation codes: 14E05, 14E08, 14G05, 14G15 All varieties are assumed to be projective, normal, and deﬁned over the ﬁeld C unless otherwise stated.

1 Introduction For a given variety, one of the substantial questions is whether it is rational or not. Global holomorphic diﬀerential forms are natural birational invariants of smooth algebraic varieties which solve the rationality problem for algebraic curves and surfaces (see [147]). However, these birational invariants are not sensitive enough to tell whether a given higher dimensional algebraic variety is nonrational. There are only four known methods to prove the nonrationality of a higher dimensional algebraic variety (see [79]). The nonrationality of a smooth quartic 3-fold was proved in [80] using the group of birational automorphisms as a birational invariant. The nonrationality of a smooth cubic 3-fold was proved in [39] through the study of its intermediate Jacobian. Birational invariance of the torsion subgroups of the 3rd integral cohomology groups was used in [4] to prove the nonrationality of some unirational varieties. The nonrationality of a wide class of rationally

F. Bogomolov, Y. Tschinkel (eds.), Cohomological and Geometric Approaches to Rationality Problems, Progress in Mathematics 282, DOI 10.1007/978-0-8176-4934-0_4, © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

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connected varieties was proved in [88] via reductions into ﬁelds of positive characteristic (see [34], [89], and [90]). Meanwhile, the method of intermediate Jacobians works only in 3-folds. In most of the interesting cases, the 3rd integral cohomology groups have no torsion. The method of paper [88] works in every dimension, but its direct application gives the nonrationality just for a very general element of a given family. Even though the method of [80] works in every dimension, the area of its application is not so broad. For this paper we mainly use the method that has evolved out of [80]. The most signiﬁcant concept in the method is the birational super-rigidity that was implicitly introduced in [80]. Definition 1.1. A terminal Q-factorial Fano variety V with Pic(V ) ∼ = Z is birationally super-rigid if the following three conditions hold: 1. the variety V cannot be birationally transformed into a ﬁbration3 whose general enough ﬁber is a smooth variety of Kodaira dimension −∞; 2. the variety V cannot be birationally transformed into another terminal Q-factorial Fano variety with Picard group Z that is not biregular to V ; 3. Bir(V ) = Aut(V ). Implicitly the paper [80] proved that all the smooth quartic 3-folds in P4 are birationally super-rigid. Moreover, some Fano 3-folds with nontrivial group of birational automorphisms were also handled by the technique of [80], which gave the following weakened version of the birational super-rigidity: Definition 1.2. A terminal Q-factorial Fano variety V with Pic(V ) ∼ = Z is called birationally rigid if the ﬁrst two conditions of Deﬁnition 1.1 are satisﬁed. It is clear that the birational rigidity implies the nonrationality. Initially the technique of [80] was applied only to smooth varieties such as quartic 3folds, quintic 4-folds, certain complete intersections, double spaces, and so on, but later, to singular varieties in [44], [65], [66], [67], [103], [111], [113], and [119]. Moreover, similar results were proved for many higher-dimensional conic bundles (see [125] and [126]) and del Pezzo ﬁbrations (see [115]). Recently, Shokurov’s connectedness principle in [130] shed a new light on the birational rigidity, which simpliﬁed the proofs of old results and helped to obtain new results (see [25], [29], [42], [49], [114], and [118]). A quartic 3-fold with a single simple double point is not birationally superrigid, but it is proved in [111] to be birationally rigid (for a simple proof, see [42]). However, a quartic 3-fold with one nonsimple double point may not necessarily be birationally rigid as shown in [44]. On the other hand, Qfactorial quartic 3-folds with only simple double points are birationally rigid (see [103]). Double covers of P3 with at most simple double points, so-called double solids, were studied in [37] with a special regard to quartic double solids, i.e., 3

For every ﬁbration τ : Y → Z, we assume that dim(Y ) > dim(Z) = 0 and τ∗ (OY ) = OZ .

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double covers of P3 ramiﬁed along quartic nodal surfaces. It is natural to ask whether a double solid is rational or not. We can immediately see that all double solids are nonrational when their ramiﬁcation surfaces are of degree greater than six. However, if the ramiﬁcation surfaces have lower degree, then the problem is not simple. Smooth quartic double solids are known to be nonrational (see [38], [94], [133], [134], [135], [136], and [144]), but singular ones can be birationally transformed into conic bundles. Quartic double solids cannot have more than 16 simple double points (see [12], [53], [92], [106], and [123]) and in the case of one simple double point they are nonrational as well (see [13] and [138]). There are non-Q-factorial quartic double solids with six simple double points that can be birationally transformed into smooth cubic 3-folds (see [91]) and therefore are not rational due to [39]. On the other hand, some quartic double solids with seven simple double points are rational (see [91]). In general, the rationality question of singular quartic double solids can be very subtle and must be handled through the technique of intermediate Jacobians (see [13], [128], and [129]). In the present paper we will consider the remaining case—the nonrationality question of sextic double solids, i.e., double covers of P3 ramiﬁed along sextic nodal surfaces. To generate various examples of sextic double solids, we note that a double cover π : X → P3 ramiﬁed along a sextic surface S ⊂ P3 can be considered as a hypersurface u2 = f6 (x, y, z, w) of degree 6 in the weighted projective space P(1, 1, 1, 1, 3), where x, y, z, and w are homogeneous coordinates of weight 1, u is a homogeneous coordinate of weight 3, and f6 is a homogeneous polynomial of degree 6. A smooth sextic double solid is proved to be birationally super-rigid in [77]. Moreover, a smooth double space of dimension n ≥ 3 was considered in [110]. The birational super-rigidity of a double cover of P3 ramiﬁed along a sextic with one simple double point was proved in [113]. To complete the study in this direction, one needs to prove the following: Theorem A. Let π : X → P3 be a Q-factorial double cover ramiﬁed along a sextic nodal surface S ⊂ P3 . Then X is birationally super-rigid. As an immediate consequence, we obtain: Corollary A. Every Q-factorial double cover of P3 ramiﬁed along a sextic nodal surface is nonrational and not birationally isomorphic to a conic bundle. Remark 1.3. Our proof of Theorem A does not require the base ﬁeld to be algebraically closed. Therefore, the statement of Theorem A is valid over an arbitrary ﬁeld of characteristic zero.

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One can try to prove the nonrationality of a sextic double solid using the technique of intermediate Jacobians (see [13], [128], and [129]), but it seems to be very hard and still undone even in the smooth case (see [22]) except for the nonrationality of a suﬃciently general smooth sextic double solid via a degeneration technique (see [13], [36], and [138]). It is worthwhile to put emphasis on the Q-factoriality condition of Theorem A. Indeed, rational sextic double solids do exist if we drop the Qfactoriality condition. Example 1.4. Let X be the double cover of P3 ramiﬁed in the Barth sextic (see [6]) given by the equation 4(τ 2 x2 − y 2 )(τ 2 y 2 − z 2 )(τ 2 z 2 − x2 ) − w2 (1 + 2τ )(x2 + y 2 + z 2 − w2 )2 = 0 √

in Proj(C[x, y, z, w]), where τ = 1+2 5 . Then X has only simple double points and the number of singular points is 65. Moreover, there is a determinantal quartic 3-fold V ⊂ P4 with 42 simple double points such that the diagram V ρ X

f

π

/ P4 γ / P3

commutes (see [53] and [108]), where ρ is a birational map and γ is the projection from one simple double point of the quartic V . Therefore, the double cover X is rational because determinantal quartics are rational (see [103] and [108]). In particular, X is not Q-factorial by Theorem A. Indeed, one can show that Pic(X) ∼ = Z and Cl(X) ∼ = Z14 (see Example 3.7 in [53]). A point p on a double cover π : X → P3 ramiﬁed along a sextic surface S is a simple double point on X if and only if the point π(p) is a simple double point on S. Sextic surfaces cannot have more than 65 simple double points (see [7], [82], and [143]). Furthermore, for each positive integer m not exceeding 65 there is a sextic surface with m simple double points (see [6], [21], and [132]), but in many cases it is not clear whether the corresponding double cover is Q-factorial or not (see [37], [46], and [53]). Example 1.4 shows that the Q-factoriality condition is crucial for Theorem A. Accordingly, it is worthwhile to study the Q-factoriality of sextic double solids. A variety X is called Q-factorial if a multiple of each Weil divisor on the variety X is a Cartier divisor. The Q-factoriality depends on both local types of singularities and their global position (see [35], [37], and [103]). Moreover, the Q-factoriality of the variety X depends on the ﬁeld of deﬁnition of the variety X as well. When X is a Fano 3-fold with mild singularities and deﬁned over C, the global topological condition

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rank(H 2 (X, Z)) = rank(H4 (X, Z)) is equivalent to the Q-factoriality. The following three examples are inspired by [5], [91], and [103]. Example 1.5. Let π : X → P3 be the double cover ramiﬁed along a sextic S and given by u2 + g32 (x, y, z, w) = h1 (x, y, z, w)f5 (x, y, z, w) ⊂ P(1, 1, 1, 1, 3), where g3 , h1 , and f5 are suﬃciently general polynomials deﬁned over R of degree 3, 1, and 5, respectively; x, y, z, w are homogeneous coordinates of weight 1; u is a homogeneous coordinate of weight 3. Then the double cover X is not Q-factorial over C because the divisor h1 = 0 splits into two non-QCartier divisors conjugated by Gal(C/R) and given by the equation √ √ (u + −1g3 (x, y, z, w))(u − −1g3 (x, y, z, w)) = 0. The sextic surface S ⊂ Proj(C[x, y, z, w]) has 15 simple double points at the intersection points of the three surfaces {h1 (x, y, z, w) = 0} ∩ {g3 (x, y, z, w) = 0} ∩ {f5 (x, y, z, w) = 0}, which gives 15 simple double points of X. Introducing two new variables s and t of weight 2 deﬁned by ⎧ √ u + −1g3 (x, y, z, w) f5 (x, y, z, w) ⎪ ⎪ √ = ⎪ ⎨s = h1 (x, y, z, w) u − −1g3 (x, y, z, w) √ ⎪ f5 (x, y, z, w) −1g (x, y, z, w) u − ⎪ 3 ⎪ √ = ⎩t = h1 (x, y, z, w) u + −1g3 (x, y, z, w) we can unproject X ⊂ P(1, 1, 1, 1, 3) in the sense of [121] into two complete intersections ! " √ ⎧ sh1 (x, y, z, w) = u + −1g3 (x, y, z, w) ⎪ ⎪ ⎪ V = ⊂ P(1, 1, 1, 1, 3, 2) √ ⎪ ⎨ s s(u − −1g3 (x, y, z, w)) = f5 (x, y, z, w) " ! √ ⎪ th1 (x, y, z, w) = u − −1g3 (x, y, z, w) ⎪ ⎪ ⎪ ⊂ P(1, 1, 1, 1, 3, 2), √ ⎩ Vt = t(u + −1g3 (x, y, z, w)) = f5 (x, y, z, w) respectively, which are not deﬁned over R. Eliminating variable u, we get ! √ Vs = {s2 h1 − 2 −1sg3 − f5 = 0} ⊂ P(1, 1, 1, 1, 2) √ Vt = {t2 h1 + 2 −1tg3 − f5 = 0} ⊂ P(1, 1, 1, 1, 2) and for the unprojections ρs : X Vs and ρt : X Vt we obtain a commutative diagram

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Ys Yt ~ @@@ φ ~ @@@ ψ ~ ~ φ s t @ ~ @@ t ~ @@ ~ @@ ~~ @@ ~~~ ~ ~ ~ ~ ρs ρt Vs o_ _ _ _ _ _ _ X _ _ _ _ _ _ _/ Vt ψs

with birational morphisms φs , ψs , φt , and ψt such that ψs and ψt are extremal contractions in the sense of [41], while φs and φt are ﬂopping contractions. Both the weighted hypersurfaces Vs and Vt are quasi-smooth (see [75]) and Q-factorial with Picard groups Z (Lemma 3.5 in [43], Lemma 3.2.2 in [50], Théoréme 3.13 of Exp. XI in [68], see also [20]). Moreover, Vs and Vt are projectively isomorphic in P(1, 1, 1, 1, 2) by the action of Gal(C/R) ∼ = Z2 . In particular, Pic(Ys ) ∼ = Pic(Yt ) ∼ = Z ⊕ Z; Ys and Yt are Q-factorial; Cl(X) = Z ⊕ Z. However, the Gal(C/R)-invariant part of the group Cl(X) is Z. Thus the 3-fold X is Q-factorial over R. It is therefore birationally super-rigid and nonrational over R by Theorem A. It is also not rational over C because Vs ∼ = Vt is birationally rigid (see [43]). Moreover, the involution of X interchanging ﬁbers of π induces a nonbiregular involution τ ∈ Bir(Vs ) which is regularized by ρs , i.e., the self-map ρ−1 s ◦τ ◦ρs : X → X is biregular (see [32]). Example 1.6. Let V ⊂ P4 be a quartic 3-fold with one simple double point o. Then the quartic V is Q-factorial and Pic(V ) ∼ = Z. In fact, V can be given by the equation t2 f2 (x, y, z, w)+ tf3 (x, y, z, w)+ f4 (x, y, z, w) = 0 ⊂ P4 = Proj(C[x, y, z, w, t]). Here, the point o is located at [0 : 0 : 0 : 0 : 1]. It is well known that the quartic 3-fold V is birationally rigid and hence nonrational (see [42], [103], and [111]). However, the quartic V is not birationally super-rigid because Bir(V ) = Aut(V ). Indeed, the projection φ : V P3 from the point o has degree 2 at a generic point of V and induces a nonbiregular involution τ ∈ Bir(V ). Let f : Y → V be the blowup at the point o. The linear system | − nKY | is free for some natural number n 0 and gives a birational morphism g = φ|−nKY | : Y → X contracting every curve Ci ⊂ Y such that f (Ci ) is a line on V passing through the point o. We then obtain the double cover π : X → P3 ramiﬁed along the sextic surface S ⊂ P3 given by the equation f32 (x, y, z, w) − 4f2 (x, y, z, w)f4 (x, y, z, w) = 0. The variety X, a priori, has canonical Gorenstein singularities. We suppose that V is general enough. Each line f (Ci ) then corresponds to an intersection point of three surfaces {f2 (x, y, z, w) = 0} ∩ {f3 (x, y, z, w) = 0} ∩ {f4 (x, y, z, w) = 0}

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in P3 = Proj(C[x, y, z, w]) which gives 24 diﬀerent smooth rational curves C1 , C2 , . . . , C24 on Y . For each curve Ci we have NY /Ci ∼ = OCi (−1) ⊕ OCi (−1) and hence the morphism g is a standard ﬂopping contraction which maps every curve Ci to a simple double point of the 3-fold X. In particular, the sextic S ⊂ P3 has exactly 24 simple double points. However, the 3-fold X is not Q-factorial and Cl(X) = Z ⊕ Z. Put ρ := g ◦ f −1 . Then the involution γ = ρ ◦ τ ◦ ρ−1 is biregular on X and interchanges the ﬁbers of the double cover π. Thus the map ρ is a regularization of the birational nonbiregular involution τ in the sense of [32], while the commutative diagram Y @ @@ g ~~ @@ ~ ~ @@ ~ ~~~ ρ _ _ _ _ _ _ _/ X V f

Y @ @@ f ~~ @@ ~ ~ @@ ~ ~~~ / X o_ _ _ _ρ _ _ _ V g

γ

is a decomposition of the birational involution τ ∈ Bir(V ) in a sequence of elementary links (or Sarkisov links) with a midpoint X (see [41], [43], and [78]). Suppose that f2 (x, y, z, w) and f4 (x, y, z, w) are deﬁned over Q and √ f3 (x, y, z, w) = 2g3 (x, y, z, w), where g3 (x, y, z,√ t) is deﬁned over Q as well. Then the quartic 3-fold √ V is deﬁned over Q( 2) and not invariant under the action of Gal(Q( 2)/Q). However, the sextic surface S ⊂ P3 is given by the equation 2g32 (x, y, z, w) − 4f2 (x, y, z, w)f4 (x, y, z, w) = 0 ⊂ P3 = Proj(Q[x, y, z, w]), which implies that the 3-fold X is deﬁned over Q as well. Moreover, the √ Gal(Q( 2)/Q)-invariant part of the group Cl(X) is Z. Therefore, the 3-fold X is Q-factorial and birationally super-rigid over Q by Theorem A and Remark 1.3. Example 1.7. Let V be a smooth divisor of bidegree (2, 3) in P1 × P3 . The 3-fold V is then deﬁned by the bihomogeneous equation f3 (x, y, z, w)s2 + g3 (x, y, z, w)st + h3 (x, y, z, w)t2 = 0, where f3 , g3 , and h3 are homogeneous polynomials of degree 3. In addition, we denote the natural projection of V to P3 by π : V −→ P3 . Suppose that the polynomials f3 , g3 , and h3 are general enough. The 3-fold V then has exactly 27 lines C1 , C2 , · · · , C27 such that −KV · Ci = 0 because the intersection {f3 (x, y, z, w) = 0} ∩ {g3 (x, y, z, w) = 0} ∩ {h3 (x, y, z, w) = 0}

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in P3 consists of exactly 27 points. The projection π has degree 2 in the outside of the 27 points π(Ci ). The anticanonical model ⎞ ⎛ H 0 (V, OV (−nKV ))⎠ Proj ⎝ n≥0

of V is the double cover X of P3 ramiﬁed along the nodal sextic S deﬁned by g32 (x, y, z, w) − 4f3 (x, y, z, w)h3 (x, y, z, w) = 0. It has exactly 27 simple double points each of which comes from each line Ci . The morphism φ|−KV | : V −→ X given by the anticanonical system of V contracts these 27 lines to the simple double points. Therefore, it is a small contraction and hence the double cover X cannot be Q-factorial. A generic divisor of bidegree (2, 3) in P1 × P3 over C is known to be nonrational (see [5], [33], and [131]), and hence the double cover X is also nonrational. As shown in Examples 1.5, 1.6, and 1.7, there are non-Q-factorial sextic double solids with 15, 24, and 27 simple double points. However, we will prove the following: Theorem B. Let π : X → P3 be a double cover ramiﬁed along a nodal sextic surface S ⊂ P3 . Then the 3-fold X is Q-factorial when #| Sing(S)| ≤ 14 and it is not Q-factorial when #| Sing(S)| ≥ 57. Using Theorem A with the theorem above, we immediately obtain: Corollary B. Let π : X → P3 be a double cover ramiﬁed along a sextic S ⊂ P3 with at most 14 simple double points. Then X is birationally superrigid. In particular, X is not rational and not birationally isomorphic to a conic bundle. In [21], there are explicit constructions of sextic surfaces in P3 with each number of simple double points not exceeding 64, which give us many examples of nonrational singular sextic double solids with at most 14 simple double points. Besides the birational super-rigidity, a Q-factorial double cover of P3 ramiﬁed in a sextic nodal surface has other interesting properties. Implicitly the method of [80] to prove the birational (super-)rigidity also gives us information on birational transformations to elliptic ﬁbrations and Fano varieties with canonical singularities. Construction A. Consider a double cover π : X → P3 ramiﬁed along a sextic S ⊂ P3 with a simple double point o. Let f : W → X be the blowup at the point o. Then the anticanonical linear system | − KW | is free and the morphism φ|−KW | : W → P2 is an elliptic ﬁbration such that the diagram

Sextic Double Solids

W φ|−KW |

f

83

/X π

γ P2 o_ _ _ _ _ _ P3

is commutative, where γ : P3 P2 is the projection from the point π(o). It is a surprise that some double covers of P3 ramiﬁed in nodal sextics can be birationally transformed into elliptic ﬁbrations in a way very diﬀerent from the one described in Construction A. Construction B. Let π : X → P3 be a double cover ramiﬁed along a sextic S ⊂ P3 such that the surface S contains a line L ⊂ P3 and the line L passes through exactly four simple double points of S. For a general enough point p ∈ X, there is a unique hyperplane Hp ⊂ P3 containing π(p) and L. The set L ∩ (C \ Sing(S)) consists of a single point qp , where C ⊂ Hp is the quintic curve given by S ∩ Hp = L ∪ C. The two points π(p) and qp determine a line Lp in P3 . Deﬁne a rational map ΞL : X Grass(2, 4) by ΞL (p) = Lp . The image of the map ΞL is isomorphic to P2 , hence we may assume that the map ΞL is a rational map of X onto P2 . Obviously the map ΞL is not deﬁned over L, the normalization of its general ﬁber is an elliptic curve, and a resolution of indeterminacy of the map ΞL birationally transforms the 3-fold X into an elliptic ﬁbration. In this paper we will prove that these two constructions are essentially the only ways to transform X birationally into an elliptic ﬁbration when X is Q-factorial. Theorem C. Let π : X → P3 be a Q-factorial double cover ramiﬁed along a nodal sextic S. Suppose that we have a birational map ρ : X Y , where τ : Y → Z is an elliptic ﬁbration. Then one of the following holds: 1. There are a simple double point o on X and a birational map β : P2 Z such that the projection γ from the point π(o) makes the diagram ρ X _ _ _ _ _ _ _/ Y π

τ

γ β P3 _ _ _ / P2 _ _ _ / Z

commute. 2. The sextic S contains a line L ⊂ P3 with #| Sing(S) ∩ L| = 4 and there is a birational map β : P2 Z such that the diagram ρ X _ _ _ _ _ _/ Y τ ΞL β P2 _ _ _ _ _ _ / Z

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is commutative, where ΞL is the rational map deﬁned in Construction B. In the case of one simple double point, Theorem C was proved in [27]. Corollary C1. All birational transformations of a Q-factorial double cover of P3 ramiﬁed along a sextic nodal surface into elliptic ﬁbrations4 are described by Constructions A and B. The following result was also proved in [24]. Corollary C2. A smooth double cover X of P3 ramiﬁed along a sextic surface S ⊂ P3 cannot be birationally transformed into any elliptic ﬁbration. Remark 1.8. Let X be a double cover of P3 ramiﬁed in a sextic surface S ⊂ P3 such that the surface S has a double line (see [67]). Then the set of birational transformations of X into elliptic ﬁbrations is inﬁnite and cannot be eﬀectively described (see [31]). The statement of Theorem C is valid over an arbitrary ﬁeld F of characteristic zero, but in Construction A the singular point must be deﬁned over F as we see in the example below. Similarly the same has to be satisﬁed for Theorem D, but the total number of singular points on a line must be counted in geometric sense (over the algebraic closure of F). Example 1.9. Let X be the double cover of P3 ramiﬁed in a sextic S ⊂ P3 and deﬁned by the equation u2 = x6 + xy 5 + y 6 + (x + y)(z 5 − 2zw4 ) + y(z 4 − 2w4 )(z − 3w) in P(1, 1, 1, 1, 3). Then X is smooth in the outside of four simple double points given by x = y = z 4 − 2w4 = 0. Hence, X is Q-factorial, birationally superrigid, and nonrational over C by Theorems A and B. Moreover, x = y = 0 cuts a curve C ⊂ X such that −KX · C = 1 and π(C) ⊂ S is a line. Therefore, X can be birationally transformed over C into exactly ﬁve elliptic ﬁbrations given by Constructions A and B. However, the 3-fold X deﬁned over Q is birationally isomorphic to only one elliptic ﬁbration given by Construction B. Birational transformations of other higher-dimensional algebraic varieties into elliptic ﬁbrations were studied in [24], [25], [26], [28], [29], [30], [31], and [124]. It turns out that classiﬁcation of birational transformations into elliptic ﬁbrations implicitly gives classiﬁcation of birational transformations into canonical Fano 3-folds. In the present paper we will prove the following result. 4

Fibrations τ1 : U1 → Z1 and τ2 : U2 → Z2 can be identiﬁed if there are birational maps α : U1 U2 and β : Z1 Z2 such that τ2 ◦ α = β ◦ τ1 and the map α induces an isomorphism between generic ﬁbers of τ1 and τ2 .

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Theorem D. Let π : X → P3 be a Q-factorial double cover ramiﬁed in a nodal sextic S ⊂ P3 . Then X is birationally isomorphic to a Fano 3-fold with canonical singularities that is not biregular to X if and only if the sextic S contains a line L passing through ﬁve simple double points of the surface S ⊂ P3 . During the proof of Theorem D, we will explicitly describe the constructions of all possible birational transformations of sextic double solids into Fano 3-folds with canonical singularities. Example 1.10. Let X be the double cover of P3 ramiﬁed in a sextic S ⊂ P3 and deﬁned by the equation u2 = x6 + xy 5 + y 6 + (x + y)(z 5 − zw4 ) in P(1, 1, 1, 1, 3). Then X is smooth in the outside of ﬁve simple double points given by x = y = z(z 4 − w5 ) = 0. For the same reason as in Example 1.9, the double cover X is Q-factorial, birationally super-rigid, and nonrational. As for elliptic ﬁbrations, it can be birationally transformed into ﬁve elliptic ﬁbrations given by Construction A. Also, the 3-fold X is birationally isomorphic to a unique Fano 3-fold with canonical singularities that is not biregular to X. The statements of Theorems A, C, and D are valid over all ﬁelds of characteristic zero, but over ﬁelds of positive characteristic some diﬃculties may occur. Indeed, the vanishing theorem of Y. Kawamata and E. Viehweg (see [84], [142]) is no longer true in positive characteristic. Even though there are some vanishing theorems over ﬁelds of positive characteristic (see [55], [127]), they are not applicable to our case. A smooth resolution of indeterminacy of a birational map may fail as well because it implicitly uses resolution of singularities (see [74]) which is completely proved only in characteristic zero. However, resolution of singularities for 3-folds is proved in [1] for the case of characteristic > 5 (see also [45]). Consider the following very special example. Example 1.11. Suppose that the base ﬁeld is F5 = Z/5Z. Let X be the double cover of P3 = Proj(F5 [x, y, z, w]) ramiﬁed along the sextic S given by the equation x5 y + x4 y 2 + x2 y 3 z − y 5 z − 2x4 z 2 + xz 5 + yz 5 + x3 y 2 w + 2x2 y 3 w −xyz 3 w − xyz 2 w2 − x2 yw3 + xy 2 w3 + x2 zw3 + xyw4 + xw5 + 2yw5 = 0. Then X is smooth (see [52] and [63]) and Pic(X) ∼ = Z by Lemma 3.2.2 in [50] or Lemma 3.5 in [43] (see [20] and [68]). Moreover, X contains a curve C given by the equations x = y = 0 whose image in P3 is a line L contained in the sextic S ⊂ P3 . For a general enough point p ∈ X, there is a unique hyperplane Hp ⊂ P3 containing π(p) and L. The residual quintic curve Q ⊂ Hp given by

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S ∩ Hp = L ∪ Q intersects L at a single point qp with multqp (Q|L ) = 5. The two points π(p) and qp determine a line Lp in P3 . As in Construction B we can deﬁne a rational map Ψ : X P2 by the lines Lp . As we see, the situation is almost the same as that of Construction B. We, at once, see that a resolution of indeterminacy of the map Ψ birationally transforms the 3-fold X into an elliptic ﬁbration. Therefore, Theorem C and even Corollary C2 are not valid over some ﬁelds of positive characteristic. We will, however, prove the following result: Theorem E. Let π : X → P3 be a double cover deﬁned over a perfect ﬁeld F and ramiﬁed along a sextic nodal surface S ⊂ P3 . Suppose that X is Qfactorial and Pic(X) ∼ = Z. Then X is birationally super-rigid and birational maps of X into elliptic ﬁbrations are described by Constructions A and B if char(F) > 5. Nonrationality and related questions like nonruledness or birational rigidity over ﬁelds of positive characteristic may be interesting in the following cases: 1. arithmetics of algebraic varieties over ﬁnite ﬁelds (see [54], [93], and [107]); 2. classiﬁcation of varieties over ﬁelds of positive characteristic (see [102] and [127]); 3. algebro-geometric coding theory (see [18], [61], [62], [73], [137], and [140]); 4. proofs of the nonrationality of certain higher-dimensional varieties by means of reduction into ﬁelds of positive characteristic (see [34], [88], [89], and [90]), where even nonperfect ﬁelds may appear in some very subtle questions as in [90]. In arithmetic geometry, it is an important and diﬃcult problem to measure the size of the set of rational points on a given variety deﬁned over a number ﬁeld F. One of the most profound works in this area is, for example, Faltings’ theorem that a smooth curve of genus at least two deﬁned over a number ﬁeld F has ﬁnitely many F-rational points (see [56]). One of the higher-dimensional generalizations of the theorem is the Weak Lang Conjecture that the set of rational points of a smooth variety of general type deﬁned over a number ﬁeld is not Zariski dense, which is still far away from proofs. A counterpart of the Weak Lang Conjecture is the conjecture that for a smooth variety X with ample −KX deﬁned over a number ﬁeld F there is a ﬁnite ﬁeld extension of the ﬁeld F over which the set of rational points of X is Zariski dense. We can easily check that this conjecture is true for curves and surfaces, where the condition implies that X is rational over some ﬁnite ﬁeld extension. Therefore, smooth Fano 3-folds are the ﬁrst nontrivial cases testing the conjecture. Definition 1.12. The set of rational points of a variety X deﬁned over a number ﬁeld F is said to be potentially dense if for some ﬁnite ﬁeld extension K of the ﬁeld F the set of K-rational points of X is Zariski dense in X.

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Using elliptic ﬁbrations, [15] and [70] have proved: Theorem 1.13. The set of rational points is potentially dense on all smooth Fano 3-folds deﬁned over a number ﬁeld F possibly except double covers of P3 ramiﬁed along smooth sextics. Arithmetic properties of algebraic varieties are closely related to their biregular and birational geometry (see [8], [9], [10], [11], [58], [95], [96], [97], [98], [99], [100], and [101]). For example, the possible exception appears in Theorem 1.13 because smooth double covers of P3 ramiﬁed in sextics are the only smooth Fano 3-folds that are not birationally isomorphic to elliptic ﬁbrations (see [81]). Besides Fano varieties, on several other classes of algebraic varieties the potential density of rational points has been proved (see [15], [16], and [17]). In Section 8 we prove the following result: Theorem F. Let π : X → P3 be a double cover deﬁned over a number ﬁeld F and ramiﬁed along a sextic nodal surface S ⊂ P3 . If Sing(X) = ∅, then the set of rational points on X is potentially dense. As shown in Theorem C, the sextic double solid can be birationally transformed into an elliptic ﬁbration if it has a simple double point. Therefore, we can adopt the methods of [15] and [70] in this case. Acknowledgment. We would like to thank V. Alexeev, F. Bogomolov, A. Corti, M. Grinenko, V. Iskovskikh, M. Mella, A. Pukhlikov, V. Shokurov, Yu. Tschinkel, and L. Wotzlaw for helpful conversations. This work has been done during the ﬁrst author’s stay at KIAS and POSTECH in Korea. We would also like to thank them for their hospitality. The second author has been supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2007-412-J02302).

2 Movable log pairs and Nöther–Fano inequalities To study sextic double solids we frequently use movable log pairs introduced in [2]. In this section we overview their properties and Nöther–Fano inequalities that are the most important tools for birational (super-)rigidity. n Definition 2.1. On a variety X a movable boundary MX = i=1 ai Mi is a formal ﬁnite Q-linear combination of linear systems Mi on X such that the base locus of each Mi has codimension at least two and each coeﬃcient ai is nonnegative. A movable log pair (X, MX ) is a variety X with a movable boundary MX on X. Every movable log pair can be considered as a usual log pair by replacing each linear system by its general element. In particular, for a given movable

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log pair (X, MX ) we may handle the movable boundary MX as an eﬀective divisor. We can also consider the self-intersection M2X of MX as a well-deﬁned eﬀective codimension-two cycle when X is Q-factorial. We call KX + MX the log canonical divisor of the movable log pair (X, MX ). Throughout the rest of this section, we will assume that log canonical divisors are Q-Cartier divisors. Definition 2.2. Movable log pairs (X, MX ) and (Y, MY ) are birationally equivalent if there is a birational map ρ : X Y such that MY = ρ(MX ). The notions such as discrepancies, (log) terminality, and (log) canonicity can be deﬁned for movable log pairs as for usual log pairs (see [86]). Definition 2.3. A movable log pair (X, MX ) has canonical (terminal, resp.) singularities if for every birational morphism f : W → X each discrepancy a(X, MX , E) in KW + f −1 (MX ) ∼Q f ∗ (KX + MX ) + a(X, MX , E)E E: f -exceptional divisor

is nonnegative (positive, resp.). Example 2.4. Let M be a linear system on a 3-fold X with no ﬁxed components. Then the log pair (X, M) has terminal singularities if and only if the linear system M has only isolated simple base points which are smooth points on the 3-fold X. The Log Minimal Model Program holds good for three-dimensional movable log pairs with canonical (terminal) singularities (see [2] and [86]). In particular, it preserves their canonicity (terminality). Every movable log pair is birationally equivalent to a movable log pair with canonical or terminal singularities. Away from the base loci of the components of its boundary, the singularities of a movable log pair coincide with those of its variety. Definition 2.5. A proper irreducible subvariety Y ⊂ X is called a center of the canonical singularities of a movable log pair (X, MX ) if there are a birational morphism f : W → X and an f -exceptional divisor E ⊂ W such that the discrepancy a(X, MX , E) ≤ 0 and f (E) = Y . The set of all the centers of the canonical singularities of the movable log pair (X, MX ) will be denoted by CS(X, MX ). Note that a log pair (X, MX ) is terminal if and only if CS(X, MX ) = ∅. Let (X, MX ) be a movable log pair and Z ⊂ X be a proper irreducible subvariety such that X is smooth along the subvariety Z. Then elementary properties of blowups along smooth subvarieties of smooth varieties imply that Z ∈ CS(X, MX ) ⇒ multZ (MX ) ≥ 1

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and in the case when codim(Z ⊂ X) = 2 we have Z ∈ CS(X, MX ) ⇐⇒ multZ (MX ) ≥ 1. For a movable log pair (X, MX ) we consider a birational morphism f : W → X such that the log pair (W, MW := f −1 (MX )) has canonical singularities. Definition 2.6. The number κ(X, MX ) = dim(φ|nm(KW +MW )| (W )) for n 0 is called the Kodaira dimension of the movable log pair (X, MX ), where m is a natural number such that m(KW + MW ) is a Cartier divisor. When |nm(KW + MW )| = ∅ for all n ∈ N, the Kodaira dimension κ(X, MX ) is deﬁned to be −∞. Proposition 2.7. The Kodaira dimension of a movable log pair is welldeﬁned. In particular, it does not depend on the choice of the birationally equivalent movable log pair with canonical singularities. Proof. Let (X, MX ) and (Y, MY ) be movable log pairs with canonical singularities such that there is a birational map ρ : Y X with MX = ρ(MY ). Choose positive integers mX and mY such that both mX (KX + MX ) and mY (KY + MY ) are Cartier divisors. We must show that either |nmX (KX + MX )| = |nmY (KY + MY )| = ∅ for all n ∈ N or dim(φ|nmX (KX +MX )| (X)) = dim(φ|nmY (KY +MY )| (Y )) for n

0.

We consider a Hironaka hut of ρ : Y X, i.e., a smooth variety W with birational morphisms g : W → X and f : W → Y such that the diagram W } BBB g BB }} } BB }} B ~}} ρ Y _ _ _ _ _ _ _/ X f

commutes. We then obtain KW + MW ∼Q g ∗ (KX + MX ) + ΣX ∼Q f ∗ (KY + MY ) + ΣY , where MW = g −1 (MX ), ΣX and ΣY are the exceptional divisors of g and f , respectively. Because the movable log pairs (X, MX ) and (Y, MY ) have canonical singularities, the exceptional divisors ΣX and ΣY are eﬀective and hence the linear systems |n(KW +MW )|, |g ∗ (n(KX +MX ))|, and |f ∗ (n(KY + MY ))| have the same dimension for a big and divisible enough natural number n. Moreover, if these linear systems are not empty, then we have φ|n(KW +MW )| = φ|g∗ (n(KX +MX ))| = φ|f ∗ (n(KY +MY ))| , which implies the claim.

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By deﬁnition, the Kodaira dimension of a movable log pair is a birational invariant and a nondecreasing function of the coeﬃcients of the movable boundary. Definition 2.8. A movable log pair (V, MV ) is called a canonical model of a movable log pair (X, MX ) if there is a birational map ψ : X V such that MV = ψ(MX ), the movable log pair (V, MV ) has canonical singularities, and the divisor KV + MV is ample. Proposition 2.9. A canonical model of a movable log pair is unique if it exists. Proof. Let (X, MX ) and (Y, MY ) be canonical models such that there is a birational map ρ : Y X with MX = ρ(MY ). Take a smooth variety W with birational morphisms g : W → X and f : W → Y such that the diagram W } BBB g BB }} } BB }} B ~}} ρ _ _ _ _ _ _ _ /X Y f

commutes. We have KW + MW ∼Q g ∗ (KX + MX ) + ΣX ∼Q f ∗ (KY + MY ) + ΣY , where MW = g −1 (MX ) = f −1 (MY ), ΣX and ΣY are the exceptional divisors of birational morphisms g and f , respectively. Let n ∈ N be a big and divisible enough number such that n(KW + MW ), n(KX + MX ), and n(KY + MY ) are Cartier divisors. For the same reason as in the proof of Proposition 2.7 we obtain φ|n(KW +MW )| = φ|g∗ (n(KX +MX ))| = φ|f ∗ (n(KY +MY ))| . Therefore, the birational map ρ is an isomorphism because KX + MX and

KY + MY are ample. The existence of the canonical model of a movable log pair implies that its Kodaira dimension is equal to the dimension of the variety. Nöther–Fano inequalities can be immediately reinterpreted in terms of canonical singularities of movable log pairs. For reader’s understanding, we give the theorems and their proofs on the relation between singularities of movable log pairs and birational (super-)rigidity. In addition, with del Pezzo surfaces of Picard number 1 deﬁned over nonclosed ﬁelds, we demonstrate how to apply the theorems, which is so simple that one can easily understand. The following result is known as a classical Nöther-Fano inequality. Theorem 2.10. Let X be a terminal Q-factorial Fano variety with Pic(X) ∼ = Z. If every movable log pair (X, MX ) with KX + MX ∼Q 0 has canonical singularities, then X is birationally super-rigid.

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Proof. Suppose that there is a birational map ρ : X V such that V is a Fano variety with Q-factorial terminal singularities and Pic(V ) ∼ = Z. We are to show that ρ is an isomorphism. Let MV = r| − nKV | and MX = 0 and a rational number r > 0 such ρ−1 (MV ) for a natural number n that KX + MX ∼Q 0. Because | − nKV | is free for n 0 and V has at worst terminal singularities, the log pair (V, MV ) has terminal singularities. In addition, the equality κ(X, MX ) = κ(V, MV ) = 0 implies that the divisor KV + MV is nef; otherwise the Kodaira dimension κ(V, MV ) would be −∞. Let f : W → X be a birational morphism of a smooth variety W such that g = ρ ◦ f is a morphism. Then ∗

KW + MW = f (KX + MX ) +

l1

a(X, MX , Fi )Fi +

i=1

= g ∗ (KV + MV ) +

l2

m

a(X, MX , Ek )Ek

k=1

a(V, MV , Gj )Gj +

j=1

m

a(V, MV , Ek )Ek ,

k=1

where MW = f −1 (MX ), each divisor Fi is f -exceptional but not g-exceptional, each divisor Gj is g-exceptional but not f -exceptional, and each Ek is both f -exceptional and g-exceptional. Applying Lemma 2.19 in [87], we obtain a(X, MX , Ek ) = a(V, MV , Ek ) for each k and we see that there is no g-exceptional but not f -exceptional divisor, i.e., l2 = 0 because the log pair (V, MV ) has terminal singularities. Furthermore, there exits no f -exceptional but not g-exceptional divisor, i.e., l1 = 0 because the Picard numbers of V and X are the same. Therefore, the log pair (X, MX ) has at worst terminal singularities. For some d ∈ Q>1 , both the movable log pairs (X, dMX ) and (V, dMV ) are canonical models. Hence, ρ is an isomorphism by Proposition 2.9. We now suppose that we have a birational map χ : X Y of X into a ﬁbration τ : Y → Z, where Y is smooth and a general ﬁber of τ is a smooth variety of Kodaira dimension −∞. Let MY = c|τ ∗ (H)| and MX = χ−1 (MY ), where H is a very ample divisor on Z and c is a positive rational number such that KX + MX ∼Q 0. Then the Kodaira dimension κ(X, MX ) is zero because the log pair (X, MX ) has at worst canonical singularities and KX + MX ∼Q 0. However, the Kodaira dimension κ(Y, MY ) = −∞. This contradiction completes the proof.

The proof of Theorem 2.10 shows a condition for the Fano variety X to be birationally rigid as follows:

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Corollary 2.11. Let X be a terminal Q-factorial Fano variety with Pic(X) ∼ = Z. Suppose that for every movable log pair (X, MX ) with KX + MX ∼Q 0 either the singularities of the log pair (X, MX ) are canonical or the divisor −(KX + ρ(MX )) is ample for some birational automorphism ρ ∈ Bir(X). Then X is birationally rigid. The Log Minimal Model Program tells us that the condition in Theorem 2.10 is a necessary and suﬃcient one for X to be birationally super-rigid. Proposition 2.12. Let X be a terminal Q-factorial Fano 3-fold with Pic(X) ∼ = Z. Then X is birationally super-rigid if and only if every movable log pair (X, MX ) with KX + MX ∼Q 0 has at worst canonical singularities. Proof. Suppose that X is birationally super-rigid. In addition, we suppose that there is a movable log pair (X, MX ) with noncanonical singularities such that KX + MX ∼Q 0. Let f : W → X be a birational morphism such that the log pair (W, MW := f −1 (MX )) has canonical singularities. Then KW + MW = f ∗ (KX + MX ) +

k i=1

a(X, MX , Ei )Ei ∼Q

k

a(X, MX , Ei )Ei ,

i=1

where Ei is an f -exceptional divisor and a(X, MX , Ej ) < 0 for some j. Applying the relative Log Minimal Model Program to the log pair (W, MW ) over X we may assume KW + MW is f -nef. Then, Lemma 2.19 in [87] immediately implies that a(X, MX , Ei ) ≤ 0 for all i. The Log Minimal Model Program for (W, MW ) gives a birational map ρ of W into a Mori ﬁbration space Y , i.e., a ﬁbration π : Y → Z such that −KY is π-ample, the variety Y has Q-factorial terminal singularities, and Pic(Y /Z) ∼ = Z. However, the bira tional map ρ ◦ f −1 is not an isomorphism. Despite its formal appearance, Theorem 2.10 can be eﬀectively applied in many diﬀerent cases. For example, the following result in [95] and [96] is an application of Theorem 2.10. Theorem 2.13. Let X be a smooth del Pezzo surface deﬁned over a perfect 2 ≤ 3. Then X is birationally rigid and ﬁeld F with Pic(X) ∼ = Z and KX nonrational over F. Proof. Suppose that X is not birationally rigid. Then there is a movable log pair (X, MX ) deﬁned over F such that KX + MX ∼Q 0 and that is not canonical at some smooth point o ∈ X. Therefore, multo (MX ) > 1 and 2 2 ¯ > deg(o ⊗ F), ¯ 3 ≥ KX = MX ≥ mult2o (MX ) deg(o ⊗ F)

¯ is the algebraic closure of the ﬁeld F. In the case K 2 = 1, the strict where F X 2 inequality is a contradiction. Moreover, if KX = 2, then the point o is deﬁned

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3 over F, and if KX = 3, then the point o splits into no more than two points ¯ over the ﬁeld F. 2 Suppose that KX is either 2 or 3. Let f : V → X be the blowup at the point o. Then 2 ¯ − deg(o ⊗ F) KV2 = KX

and V is a smooth del Pezzo surface because Pic(X) = Z, the inequality multo (MX ) > 1 holds, and the boundary MX is movable. The double cover φ|−KV | induces an involution τ ∈ Bir(X) that is classically known as Bertini or Geizer involution. Simple calculations show the ampleness of divisor −(KX + τ (MX )), which contradicts Corollary 2.11.

The proofs of Theorems 2.10 and 2.13 and Lemma 5.3.1 in [90] imply that a result similar to Theorem 2.13 holds over a nonperfect ﬁeld as well. Indeed, one can prove that a nonsingular del Pezzo surface X deﬁned over nonperfect ﬁeld F is nonrational over F and is not birationally isomorphic over F to any nonsingular del Pezzo surface Y with Pic(Y ) = Z, which is smooth in 2 codimension one, if Pic(X) ∼ ≤ 3. = Z and KX Most applications of Theorem 2.10 have the pattern of the proof of Theorem 2.13 implicitly. The following result can be considered as a weak Nöther–Fano inequality. Theorem 2.14. Let X be a terminal Q-factorial Fano variety with Pic(X) ∼ = Z, ρ : X Y a birational map, and π : Y → Z a ﬁbration. Suppose that a general enough ﬁber of π is a smooth variety of Kodaira dimension zero. Then the singularities of the movable log pair (X, MX ) are not terminal, where MX = rρ−1 (|π ∗ (H)|) for a very ample divisor H on Z and r ∈ Q>0 such that KX + MX ∼Q 0. Proof. Suppose CS(X, MX ) = ∅. Let MY = r|π ∗ (H)|. Then we see κ(X, cMX ) = κ(Y, cMY ) ≤ dim(Z) < dim(X). However, CS(X, cMX ) = ∅ for small c > 1 and hence κ(X, cMX ) = dim(X), which is a contradiction.

The easy result below shows how to apply Theorem 2.14. Proposition 2.15. Let X be a smooth del Pezzo surface of degree one with Pic(X) ∼ = Z deﬁned over a perfect ﬁeld F and o the unique base point of the anticanonical linear system of the surface X. Let ρ : X Y be a birational map, where Y is a smooth surface. Suppose that π : Y → Z is a relatively minimal elliptic ﬁbration with connected ﬁbers such that a general enough ﬁber of π is smooth. Then the birational map ρ is the blowup at some F-rational point p on the del Pezzo surface X and the morphism π is induced by |− nKY | for some n ∈ N. Furthermore, p ∈ Cˆ and the equality pn = idCˆ holds, where Cˆ is the smooth part of the unique curve C of arithmetic genus one in | − KX | passing through the point p and considered as a group scheme with the identity idCˆ = o.

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Proof. Let MX = cρ−1 (|π ∗ (H)|), where H is a very ample on curve Z and c ∈ Q>0 , such that the equivalence KX + MX ∼Q 0 holds. Then the set CS(X, MX ) contains a point p on the surface X by Theorem 2.14. In particular, multp (MX ) ≥ 1, but 2 2 ¯ ≥ deg(p ⊗ F) ¯ ≥ 1, 1 = KX = MX ≥ mult2p (MX ) deg(p ⊗ F)

¯ is the algebraic closure of the ﬁeld F. Hence, multp (MX ) = 1 and where F the point p is deﬁned over the ﬁeld F. Let f : V → X be the blow up at the point p. Then KV2 = 0 and −KV ∼Q MV = f −1 (MX ), which implies that the linear system | − rKV | is free for a natural number r 0. The morphism φ|−rKV | is a relatively minimal elliptic ﬁbration and MV · E = 0 for a general enough ﬁber E of the elliptic ﬁbration φ|−rKV | . Therefore the linear system (ρ ◦ f )−1 (|π ∗ (H)|) is contained in the ﬁbers of the ﬁbration φ|−rKV | . Relative minimality of the ﬁbrations π and φ|−rKV | implies ρ ◦ f is an isomorphism. Suppose p = o. Let C ∈ | − KX | be a curve passing through p. Because 2 1 = KX = C · MX ≥ multp (MX ) multp (C) = multp (C) ≥ 1,

the curve C is smooth at the point p. Let C˜ = f −1 (C) ∼ −KV . Then ¯ ˜ = 1 and the curve C˜ is Gal(F/F)-invariant. In particular, the h0 (V, OV (C)) ˜ and we have Z ∼ curve Z has an F-point φ|−rKV | (C) = P1 . Take the smallest ˜ > 1. The exact sequence natural n such that h0 (V, OV (nC)) ˜ → OV (nC) ˜ → O ˜ (nC| ˜ ˜) → 0 0 → OV ((n − 1)C) C C ˜ O ˜ (nC| ˜ ˜ )) = 0, which implies the claim. implies h0 (C, OC (n(p− o))) = h0 (C, C C

Corollary 2.16. Let X be a terminal Q-factorial Fano variety with Pic(X) ∼ = Z such that every movable log pair (X, MX ) with KX +MX ∼Q 0 has terminal singularities. Then X is not birationally isomorphic to a ﬁbration of varieties of Kodaira dimension zero. Unfortunately, Corollary 2.16 is almost impossible to use. As far as we know, there are no known examples of Fano varieties that are not birationally isomorphic to ﬁbrations of varieties of Kodaira dimension zero. The only known example of a rationally connected variety that cannot be birationally transformed into a ﬁbration of varieties of Kodaira dimension zero is a conic bundle with a big enough discriminant locus in [30]. Theorem 2.17. Let X be a terminal Q-factorial Fano variety with Pic(X) ∼ = Z and ρ : X Y be a nonbiregular birational map onto a Fano variety Y with canonical singularities. Then KX + MX ∼Q 0 and

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CS(X, MX ) = ∅, where MX =

1 −1 (| nρ

Proof. Let MY =

1 n|

− nKY |) for a natural number n

0.

− nKY |. We then see κ(X, MX ) = κ(Y, MY ) = 0,

which implies KX + MX ∼Q 0. Suppose CS(X, MX ) = ∅. Both the log pair (X, rMX ) and (Y, rMY ) are canonical models for a rational number r > 1 suﬃciently close to 1. It is a contradiction that ρ is an isomorphism by Proposition 2.9.

The following easy result shows how to apply Theorem 2.17. Proposition 2.18. Let X be a smooth del Pezzo surface of degree one with Pic(X) ∼ = Z deﬁned over an arbitrary perfect ﬁeld F. Then the surface X is not birationally isomorphic to a del Pezzo surface with du Val singularities which is not isomorphic to the surface X. Proof. Let ρ : X Y be a birational map over the ﬁeld F and MX = 1 −1 (| − nKY |) for a natural number n 0, where Y is a del Pezzo surface nρ with du Val singularities and ρ is not an isomorphism. Then KX + MX ∼Q 0 and CS(X, MX ) contains some smooth point o on the del Pezzo surface X by Theorem 2.17. In particular, multo (MX ) ≥ 1, but 2 2 ¯ ≥ deg(o ⊗ F) ¯ ≥ 1, = MX ≥ mult2o (MX ) deg(o ⊗ F) 1 = KX

¯ is the algebraic closure of the ﬁeld F. Hence, multo (MX ) = 1 and where F the point o is deﬁned over the ﬁeld F. Let f : V → X be the blow up at the point o. Then KV2 = 0 and −KV ∼Q MV = f −1 (MX ), which implies freeness of the linear system |−rKV | for a natural number r 0. The morphism φ|−rKV | is an elliptic ﬁbration and MV · E = 0 for a general enough ﬁber E of φ|−rKV | . Therefore, the linear system (ρ ◦ f )−1 (| − nKY |) is compounded from a pencil, which is impossible.

The paper [80] by V. Iskovskikh and Yu. Manin was based on the idea of G. Fano that can be summarized by Nöther–Fano inequalities. Since 1971 the method of Iskovskikh and Manin has evolved to show birational rigidity of various Fano varieties. Recently, Shokurov’s connectedness principle improved the method so that one can extremely simplify the proof of the result of Iskovskikh and Manin (see [42]). Furthermore, it also made it possible to prove the birational super-rigidity of smooth hypersurfaces of degree n in Pn , n ≥ 4 (see [118]). In what follows we will explain Shokurov’s connectedness principle and how it can be applied to birational rigidity.

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Movable boundaries always can be considered as eﬀective divisors and movable log pairs as usual log pairs. Therefore, we may use compound log pairs that contain both movable and ﬁxed components. From now on, we will not assume any restrictions on the coeﬃcients of boundaries. In particular, boundaries may not be eﬀective unless otherwise stated. Definition 2.19. A log pair (V, B V ) is called the log pullback of a log pair (X, BX ) with respect to a birational morphism f : V → X if B

V

=f

−1

(BX ) −

n

a(X, BX , Ei )Ei ,

i=1

where a(X, BX , Ei ) is the discrepancy of an f -exceptional divisor Ei over (X, BX ). In particular, it satisﬁes KV + B V ∼Q f ∗ (KX + BX ). Definition 2.20. A proper irreducible subvariety Y ⊂ X is called a center of the log canonical singularities of (X, BX ) if there are a birational morphism f : W → X and a divisor E ⊂ W such that E is contained in the support of the eﬀective part of the divisor !B W " and f (E) = Y . The set of all the centers of the log canonical singularities of a log pair (X, BX ) will be denoted by LCS(X, BX ). In addition, the union of all the centers of log canonical singularities of (X, MX ) will be denoted by LCS(X, BX ). Consider a log pair (X, BX ), where BX = ki=1 ai Bi is eﬀective and Bi ’s are prime divisors on X. Choose a birational morphism f : Y → X such that Y is smooth and the union of all the proper transforms of the divisors Bi and all f -exceptional divisors forms a divisor with simple normal crossing. The morphism f is called a log resolution of the log pair (X, BX ). By deﬁnition, the equality KY + B Y ∼Q f ∗ (KX + BX ) holds, where (Y, B Y ) is the log pullback of the log pair (X, BX ) with respect to the birational morphism f . Definition 2.21. The subscheme L(X, BX ) associated with the ideal sheaf I(X, BX ) = f∗ (OY (#−B Y $)) is called the log canonical singularity subscheme of the log pair (X, BX ). The support of the subscheme L(X, BX ) is exactly the locus of LCS(X, BX ). The following result is called Shokurov vanishing (see [130]). Theorem 2.22. Let (X, BX ) be a log pair with an eﬀective divisor BX . Suppose that there is a nef and big Q-divisor H on X such that D = KX +BX +H is Cartier. Then H i (X, I(X, BX ) ⊗ OX (D)) = 0 for i > 0. Proof. Let f : W −→ X be a log resolution of (X, BX ). Because f ∗ H is nef and big on W and f ∗ D = KW + B W + f ∗ H, we obtain

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Ri f∗ (f ∗ OX (D) ⊗ OW (#−B W $)) = 0 for i > 0 from relative Kawamata–Viehweg vanishing (see [84] and [142]). The degeneration of local-to-global spectral sequence and R0 f∗ (f ∗ OX (D) ⊗ OW (#−B W $)) = I(X, BX ) ⊗ OX (D) imply that for all i H i (X, I(X, BX ) ⊗ OX (D)) = H i (W, f ∗ OX (D) ⊗ OW (#−B W $)), while H i (W, f ∗ OX (D) ⊗ OW (#−B W $)) = 0 for i > 0 by Kawamata–Viehweg vanishing.

Consider the following application of Theorem 2.22. Lemma 2.23. Let V be a variety isomorphic to P1 ×P1 . Let BV be an eﬀective Q-divisor on V of type (a, b), where a and b ∈ Q ∩ [0, 1). Then LCS(V, BV ) = ∅. Proof. Intersecting the boundary BV with the rulings of V , we see that the set LCS(V, BV ) does not contain a curve on V . Suppose that the set LCS(V, BV ) contains a point o. There is a Q-divisor H on V of type (1 − a, 1 − b) such that the divisor D = KV + BV + H is Cartier. Since the divisor H is ample, Theorem 2.22 implies the sequence H 0 (V, OV (D)) → H 0 (L(V, BV ), OL(V,BV ) (D)) → 0 is exact. However, H 0 (V, OV (D)) = 0, which is a contradiction.

For every Cartier divisor D on X, the sequence 0 → I(X, BX ) ⊗ D → OX (D) → OL(X,BX ) (D) → 0 is exact and Theorem 2.22 implies the following two connectedness theorems of Shokurov. Theorem 2.24. Let (X, BX ) be a log pair with an eﬀective boundary BX . If the divisor −(KX + BX ) is nef and big, then the locus LCS(X, BX ) is connected. Theorem 2.25. Let (X, BX ) be a log pair with an eﬀective boundary. Let g : X −→ Z be a contraction. If the divisor −(KX + BX ) is g-nef and gbig, then LCS(X, BX ) is connected in a neighborhood of each ﬁber of the contraction g. The following result is Theorem 17.4 of [87].

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Theorem 2.26. Let g : X Z be a contraction, where the varieties X and → m Z are normal. Let DX = i=1 di Di be a Q-divisor on X such that the divisor −(KX +DX ) is g-nef and g-big. Suppose that codim(g(Di ) ⊂ Z) ≥ 2 whenever di < 0. Then, for a log resolution h : V → X of the log pair (X, DX ), the locus ∪aE ≤−1 E is connected in a neighborhood of every ﬁber of the morphism g ◦ h, where E is a divisor on V and the rational number aE is the discrepancy of E with respect to (X, DX ). Proof. Let f = g ◦ h, A = aE >−1 aE E, and B = aE ≤−1 −aE E. Then #A$ − !B" = KV − h∗ (KX + DX ) + {−A} + {B} and R1 f∗ (OV (#A$ − !B")) = 0 by Kawamata–Viehweg vanishing. Hence, the map f∗ (OV (#A$)) → f∗ (OB (#A$)) is surjective. Every irreducible component of #A$ is either h-exceptional or the proper transform of some Dj with dj < 0. Thus h∗ (#A$) is g-exceptional and f∗ (OV (#A$)) = OZ . Consequently, the map OZ → f∗ (OB (#A$)) is surjective, which implies the connectedness of !B" in a neighborhood of every ﬁber of the morphism f because the divisor #A$ is eﬀective and has no common component with !B".

We deﬁned the notions of centers of canonical singularities and the set of centers of canonical singularities for movable log pairs. However, the movability of boundaries has nothing to do with all these notions. So we are free to use them for usual log pairs as well. The following theorem, frequently referred to as adjunction, leads us to the bridge between Shokurov’s connectedness principle and Nöther–Fano inequalities. Theorem 2.27. Let (X, BX ) be a log pair with an eﬀective divisor BX , Z an element in CS(X, BX ), and H an eﬀective irreducible Cartier divisor on X. Suppose that both the varieties X and H are smooth at a generic point of Z and Z ⊂ H ⊂ Supp(BX ). Then, the set LCS(H, BX |H ) is not empty. ˆ = f −1 (H). Proof. Let f : W → X be a log resolution of (X, BX + H). Put H Then ˆ = f ∗ (KX + BX + H) + KW + H a(X, BX + H, E)E ˆ E=H

and by our assumption the subvarieties Z and H are centers of the log canonical singularities of the pair (X, BX + H). Therefore, applying Theorem 2.26

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ˆ on W to the log pullback of (X, BX + H) on W , we obtain a divisor E = H ˆ such that f (E) = Z, a(X, BX , E) ≤ −1, and H ∩ E = ∅. Now the equalities ˆ ˆ = f |∗ˆ (KH + BX |H ) + KHˆ = (KW + H)| a(X, BX + H, E)E|Hˆ H H ˆ E=H

imply the claim.

By taking Nöther–Fano inequalities into consideration, it is signiﬁcant for us to study the singularities of certain movable log pairs on Fano varieties. It requires us to investigate the multiplicities of certain movable boundaries or their self-intersections. The following result is Theorem 3.1 of [42]. Theorem 2.28. Let S be a smooth surface and MS an eﬀective movable boundary on the surface S. Suppose that there is a point o in LCS(S, (1 − a1 )B1 + (1 − a2 )B2 + MS ), where ai ’s are nonnegative rational numbers and Bi ’s are irreducible and reduced curves on S intersecting normally at the point o. Then, we have ! 4a1 a2 if a1 ≤ 1 or a2 ≤ 1 2 multo (MS ) ≥ 4(a1 + a2 − 1) if a1 > 1 and a2 > 1. Furthermore, the inequality is strict if the singularities of the log pair (S, (1 − a1 )B1 + (1 − a2 )B2 + MS ) are not log canonical in a neighborhood of the point o. Proof. Let D = (1 − a1 )B1 + (1 − a2 )B2 + MS and f : S → S be a birational morphism such that the surface S is smooth. We consider KS + f −1 (D) = Ei )Ei , where Ei is an f -exceptional curve. We suppose that a(S, D, E1 ) ≤ −1 and the curve E1 is contracted to the point o. Then the birational morphism f is a composition of k blowups at smooth points. Claim 1. The statement is true when a1 > 1 and a2 > 1 if the statement holds when a1 ≤ 1 or a2 ≤ 1. Deﬁne the numbers a(S, Ei ), m(S, MS , Ei ), and m(S, Bj , Ei ) as follows: k

a(S, Ei )Ei = KS − f ∗ (KS ),

i=1 k

m(S, MS , Ei )Ei = f −1 (MS ) − f ∗ (MS ),

i=1 k i=1

m(S, Bj , Ei )Ei = f −1 (Bj ) − f ∗ (Bj ).

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We then observe that the equality a(S, D, Ei ) = a(S, Ei )−m(S, MS , Ei )+m(S, B1 , Ei )(a1 −1)+m(S, B2 , Ei )(a2 −1) holds. We may assume that m(S, B1 , E1 ) ≥ m(S, B2 , E1 ). Then, −1 ≥ a(S, D, E1 ) ≥ a(S, E1 ) − m(S, MS , E1 ) + m(S, B2 , E1 )(a1 + a2 − 2) and hence o ∈ LCS(S, (2−a1 −a2 )B2 +MS ). Because the log pair (S, (2−a1 − a2 )B2 + MS ) satisﬁes our assumption, we obtain multo (M2S ) ≥ 4(a1 + a2 − 1). Claim 2. The statement holds when a1 ≤ 1 or a2 ≤ 1. We may assume that a1 ≤ 1. Let h : T → S be the blowup at the point o and E be an exceptional curve of h. Then f factors through h such that f = g ◦ h for some birational morphism g : S → T which is a composition of k − 1 blowups at smooth points. Then ¯1 + (1 − a2 )B ¯2 + (1 − a1 − a2 + m)E + MT = h∗ (KS + D), KT + (1 − a1 )B ¯j = h−1 (Bj ), m = multo (MS ), and MT = h−1 (MS ). where B We are to use the induction on k. In the case k = 1, we have S = T , E1 = E, and a(S, D, E1 ) = a1 + a2 − m − 1 ≤ −1. Thus multo (M2S ) ≥ m2 ≥ (a1 + a2 )2 ≥ 4a1 a2 and we are done. We therefore suppose that k > 1 and g(E1 ) is a point p ∈ E. We see ¯1 + (1 − a2 )B ¯2 + (1 − a1 − a2 + m)E + MT ). p ∈ LCS(T, (1 − a1 )B ¯2 , and p ∈ B ¯1 ∪ B ¯2 . By ¯1 , p ∈ E ∩ B There are three possible cases: p ∈ E ∩ B the induction hypothesis, the statement holds for the log pair ¯1 + (1 − a1 − a2 + m)E + MT ) (T, (1 − a1 )B ¯1 , for the log pair in the case p ∈ E ∩ B ¯2 + (1 − a1 − a2 + m)E + MT ) (T, (1 − a2 )B ¯2 , and for the log pair in the case p ∈ E ∩ B (T, (1 − a1 − a2 + m)E + MT ) ¯2 because all conditions of the theorem are satisﬁed in ¯1 ∪ B in the case p ∈ B these cases and the morphism g consists of k − 1 blowups at smooth points. Also we have multo (M2S ) ≥ m2 + multp (M2T ). ¯ 1 , we obtain In the case p ∈ E ∩ B

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multo (M2S ) ≥ m2 + 4a1 (a1 + a2 − m) = (2a1 − m)2 + 4a1 a2 ≥ 4a1 a2 . ¯2 . If either a2 ≤ 1 or a1 + a2 − m ≤ 1, then Consider the case p ∈ E ∩ B we can proceed as in the previous case. If not, then we have multo (M2S ) ≥ m2 + 4(a1 + 2a2 − m − 1) > 4a2 ≥ 4a1 a2 . ¯1 ∪ B ¯2 , then we obtain If p ∈ B multo (M2S ) ≥ m2 + 4(a1 + a2 − m) > m2 + 4a1 (a1 + a2 − m) ≥ 4a1 a2 , which completes the proof.

Instead of Theorem 2.28, the following simpliﬁed version, which is a special case of Theorem 2.1 in [49], is more often applied. Theorem 2.29. Let S be a smooth surface, o a point on S, and MS an eﬀective movable boundary on S such that o ∈ LCS(S, MS ). Then multo (M2S ) ≥ 4. Moreover, if the equality holds, then multo (MS ) = 2. Even though Theorems 2.28 and 2.29 are results on surfaces, they can be applied to 3-folds via Theorem 2.27. The following result is Corollary 7.3 of [116], which holds even over ﬁelds of positive characteristic and implicitly goes back to the classical paper [80]. Theorem 2.30. Let X be a smooth 3-fold and MX an eﬀective movable boundary on X. Suppose that a point o belongs to CS(X, MX ). Then the inequality multo (M2X ) ≥ 4 holds, with equality only when multo (MX ) = 2. Proof. Let H be a general very ample divisor on X containing o. Then the point o is a center of log canonical singularities of the log pair (H, MX |H ) by Theorem 2.27. On the other hand, multo (M2X ) = multo ((MX |H )2 ) and multo (MX ) = multo (MX |H ). Hence, the claim follows from Theorem 2.29.

As a matter of fact, Theorem 2.30 can be proved in a more geometric way. Lemma 2.31. Let X be a smooth 3-fold and MX an eﬀective movable boundary on X. Suppose that the log pair (X, MX ) has canonical singularities and CS(X, MX ) contains a point o. Then there is a birational morphism f : V → X such that V has Q-factorial terminal singularities, f contracts exactly one exceptional divisor E to the point o, and KV + MV = f ∗ (KX + MX ), where MV = f −1 (MX ).

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Proof. Because the log pair (X, MX ) has at worst canonical singularities, there are ﬁnitely many divisorial discrete valuations ν of the ﬁeld of rational functions of X whose centers on X are the point o and whose discrepancies a(X, MX , ν) are nonpositive. Therefore, we may consider a birational morphism g : W → X such that the 3-fold W is smooth, g contracts k divisors, KW + MW = g ∗ (KX + MX ) +

k

ai Ei ,

i=1

the movable log pair (W, MW ) has canonical singularities, and the set CS(W, MW ) does not contain subvarieties of ∪ki=1 Ei , where MW = g −1 (MX ), g(Ei ) = o, and ai ∈ Q. Applying the relative version of the Log Minimal Model Program (see [86]) to the movable log pair (W, MW ) over X, we may assume that W has Q-factorial terminal singularities and KW + MW = g ∗ (KX + MX ) because of the canonicity of (X, MX ). Applying the relative Minimal Model Program to W over the variety X, we get the necessary 3-fold and the birational morphism.

The following result was conjectured in [41] and proved in [83]. Theorem 2.32. Let X be a smooth 3-fold and f : V → X be a birational morphism of a 3-fold V with Q-factorial terminal singularities. Suppose that the morphism f contracts exactly one exceptional divisor E and contracts it to a point o. Then the morphism f is the weighted blowup at the point o with weights (1, n1 , n2 ) in suitable local coordinates on X, where the natural numbers n1 and n2 are coprime. With Theorem 2.32, Theorem 2.30 was proved in [41] in the following way, which explains the geometrical nature of the inequality in Theorem 2.30. Proposition 2.33. Let X be a smooth 3-fold with an eﬀective movable boundary MX on X. Suppose that CS(X, MX ) contains a point o. Let f : V → X be the weighted blowup at the point o with weights (1, n1 , n2 ) in suitable local coordinates on X such that KV + MV = f ∗ (KX + MX ), where natural numbers n1 and n2 are coprime and MV = f −1 (MX ). Then multo (M2X ) ≥

(n1 + n2 )2 (n1 − n2 )2 =4+ ≥ 4. n1 n2 n1 n2

Moreover, if n1 = n2 , then f is the regular blowup at the point o and multo (MX ) = 2.

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Proof. Let E ⊂ V be the f -exceptional divisor. Then KV = f ∗ (KX ) + (n1 + n2 )E and MV = f ∗ (MX ) − mE for some m ∈ Q>0 . Thus, m = n1 + n2 and multo (M2X ) ≥ m2 E 3 =

(n1 + n2 )2 . n1 n2

The following application of Theorem 2.27 is Theorem 3.10 in [42]. Theorem 2.34. Let X be a 3-fold with a simple double point o and BX an eﬀective boundary on X such that o ∈ CS(X, BX ). Then the inequality multo (BX ) ≥ 1 holds. Proof. Let f : W → X be the blowup at the point o and E be the f exceptional divisor. Then KW + BW = f ∗ (KX + BX ) + (1 − multo (BX ))E, where BW = f −1 (BX ). Suppose that multo (BX ) < 1. Then, there is a center Z ∈ CS(W, BW ) that is contained in E, and hence LCS(E, BW |E ) = ∅ by Theorem 2.27. But it is impossible because of Lemma 2.23.

3 Birational super-rigidity The goal of this section is to prove Theorem A. Let π : X → P3 be a Q-factorial double cover ramiﬁed along a nodal sextic S ⊂ P3 . We then see that Pic(X) ∼ = ZKX , −KX ∼ π ∗ (OP3 (1)), and 3 −KX = 2. Consider an arbitrary movable boundary MX on the 3-fold X such that the divisor −(KX + MX ) is ample. To prove Theorem A we must show that CS(X, MX ) = ∅ and then apply Theorem 2.10. We suppose that CS(X, MX ) = ∅. In what follows, we will derive a contradiction. Lemma 3.1. Smooth points of the 3-fold X are not contained in CS(X, MX ). Proof. Suppose that CS(X, MX ) has a smooth point o on X. Let H be a general enough divisor in the linear system | − KX | passing through the point o. We then obtain 2 > H · M2X ≥ multo (M2X ) ≥ 4 2 = H · KX

from Theorem 2.30, which is absurd.

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Lemma 3.2. Singular points of the 3-fold X are not contained in CS(X, MX ). Proof. If CS(X, MX ) contains a singular point o on X, then Theorem 2.34 gives us 2 > H · M2X ≥ 2 mult2o (MX ) ≥ 2, 2 = H · KX where H is a general enough divisor in | − KX | passing through the point o. It is absurd.

Lemmas 3.1 and 3.2 together show that any element of the set CS(X, MX ) cannot be a point of X. Therefore, it must contain a curve C ⊂ X. To complete the proof of Theorem A it is enough to show that the set CS(X, MX ) cannot contain a curve. Lemma 3.3. The intersection number −KX · C is 1. Proof. Let H be a general enough divisor in the anticanonical linear system | − KX |. Then 2 2 = H · KX > H · M2X ≥ multC (M2X )H · C ≥ −KX · C,

which implies −KX · C = 1.

Corollary 3.4. The curve π(C) ⊂ P3 is a line and C ∼ = P1 . Lemma 3.5. The curve C is not contained in the smooth locus of the 3-fold X. Proof. Suppose that the curve C lies on the smooth locus of the 3-fold X. Let f : W → X be the blowup along the curve C and E be the f -exceptional divisor. We then get multC (MX ) ≥ 1 and MW = f −1 (MX ) = f ∗ (MX ) − multC (MX )E. The linear system | − KW | = |f ∗ (−KX ) − E| has just one base curve C˜ such that ˜ = π(C) ⊂ P3 . π ◦ f (C) We see that C˜ ⊂ E if and only if π(C) ⊂ S. Let H = f ∗ (−KX ). Then the divisor 3H − E has nonnegative intersection ˜ We are to show that the divisor with all the curves on W possibly except C. ˜ 3H − E is nef. We obtain (3H − E) · C = 0 unless C˜ is contained in E. Therefore, we suppose that the curve C˜ is contained in E. The normal bundle NX/C of the curve C ∼ = P1 on the 3-fold X splits into NX/C ∼ = OC (a) ⊕ OC (b) for some integers a ≥ b. The exact sequence 0 → TC → TX |C → NX/C → 0

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shows deg(NX/C ) = a + b = −KX · C + 2g(C) − 2 = −1. On the other hand, the curve C is contained in the smooth locus of the proper transform Sˆ ∼ = S of the sextic S ⊂ P3 . The exact sequence → NX/C → NX/Sˆ |C → 0 0 → NS/C ˆ ∼ and NS/C = OC (−4) imply b ≥ −4. In particular, a − b ≤ 7. ˆ Let s∞ be the exceptional section of the ruled surface f |E : E → C. Because E 3 = − deg(NX/C ) = 1 and −KX · C = 1, we obtain (3H − E) · s∞ =

7+b−a ≥ 0, 2

which implies that the divisor 3H − E is nef. Because 3H − E is nef, we get (3H − E) · M2W ≥ 0, but (3H − E) · M2W = 6r2 − 4 mult2C (MX ) − 2r multC (MX ) < 0, where r ∈ Q ∩ (0, 1) such that MX ∼Q −rKX .

Corollary 3.6. The curve C contains a simple double point of the 3-fold X. Lemma 3.7. The line π(C) is contained in the sextic surface S. Proof. Suppose π(C) ⊂ S. Let H be the linear subsystem in |−KX | of surfaces containing the curve C. The base locus of H consists of the curve C and the # such that π(C) = π(C). # Choose a general enough surface D in the curve C pencil H. The restriction MX |D is not movable, but MX |D = multC (MX )C + multCe (MX )C˜ + RD , where RD is a movable boundary. The surface D is smooth outside of the singular points pi of the 3-fold X which are contained in the curve C. Moreover, each point pi is a simple double point on the surface D. Thus, on the surface D, we have k C 2 = C˜ 2 = −2 + , 2 #2 < 0 where k is the number of the points pi on C. Hence, we obtain C 2 = C on the surface D because k ≤ 3. Immediately, the inequality #2 ≥ (multC (MX ) − 1)C · C # + RD · C #≥0 (1 − multC˜ (MX ))C follows, which implies multC˜ (MX ) ≥ 1. Therefore, for a general member H ∈ | − KX | we have a contradiction 2 2 = H · KX > H · M2X ≥ mult2C (MX )H · C + mult2C˜ (MX )H · C˜ ≥ 2.

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Lemma 3.8. The line π(C) is not contained in the sextic surface S. Proof. Suppose π(C) ⊂ S. Let p be a general enough point on the curve C and L ⊂ P3 be a general line tangent to S at the point π(p). Then the ˜ ⊂ X of L is an irreducible curve which is singular at the proper transform L point p. By construction, the curve L is not contained in the base locus of the components of the movable boundary MX . Thus, we obtain contradictory inequalities ˜ multp (MX ) ≥ 2 multC (MX ) ≥ 2. ˜ · MX ≥ multp L 2>L

We have shown that the set CS(X, MX ) is empty. Now, we can immediately obtain Theorem A from Theorem 2.10.

4 Q-factoriality In this section we study the Q-factoriality on double covers of P3 ramiﬁed along nodal sextics and prove Theorem B. The Q-factoriality depends both on local types of singularities and on their global position. In the case of Fano 3-folds, the Q-factoriality is equivalent to the global topological condition rank(H 2 (X, Z)) = rank(H4 (X, Z)). In the case of the double solids, the condition means the 4th integral homology group of X generated by the class of the pullback of a hyperplane in P3 via the covering morphism. Using the method in [37], we study the Q-factoriality on a double cover X of P3 ramiﬁed along a sextic S. As before, we assume that X has only simple double points. Note that Pic(X) ∼ = H 2 (X, Z) when X has at worst rational singularities. For us in order to see whether a double solid X is Q-factorial, the main job is to compute the rank of the group H4 (X, Z). Indeed, the double solid X is Q-factorial if and only if rank(H4 (X, Z)) = 1 because rank H 2 (X, Z) = 1. The paper [37] gives us a method to compute it by studying the number of singularities of S, their position in P3 , and the sheaf I ⊗ OP3 (5), where I is the ideal sheaf of the set Σ of singular points of S in P3 . The following result was proved in [37] (see also [48] and [47]). Theorem 4.1. Under the same notation, we have rank(H4 (X, Z)) = #(Σ) − I + 1, where I is the number of independent conditions which vanishing on Σ imposes on homogeneous forms of degree 5 on P3 .

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We deﬁne the defect of X to be the nonnegative integer #(Σ) − I. Then we can restate the Q-factoriality as follows: Corollary 4.2. The double cover X is Q-factorial if and only if the defect of X is 0. On the other hand, from the exact sequence 0 → I ⊗ OP3 (5) → OP3 (5) →

C→0

p∈Σ

we obtain a long exact sequence H 0 (P3 , I⊗OP3 (5)) "→ H 0 (P3 , OP3 (5)) → H 0 (P3 ,

C) → H 1 (P3 , I⊗OP3 (5)) → 0,

p∈Σ

which tells us defect of X = dim(H 1 (P3 , I ⊗ OP3 (5))). An immediate application of this method is the second part of Theorem B. Since dim(H 0 (P3 , OP3 (5))) = 56, the defect of X is positive if #(Σ) ≥ 57. We can easily observe that if #(Σ) ≤ 6, then the sequence C) → 0 0 → H 0 (P3 , I ⊗ OP3 (5)) → H 0 (P3 , OP3 (5)) → H 0 (P3 , p∈Σ

is exact regardless of their position. Therefore, when #(Σ) ≤ 6 the defect of X is trivially 0, i.e., the sextic double solid X is Q-factorial. As a matter of fact, we can go farther. As Theorem B states, if the surface S has at most 14 nodes, then the 3-fold X is Q-factorial regardless of their position. In what follows, we prove the ﬁrst part of Theorem B. Definition 4.3. We say that a set of points Γ on P3 is on sextic-node position if no 5k + 1 points of Γ can lie on a curve of degree k in P3 for every positive integer k. Lemma 4.4. Let Σ be the set of singular points of the sextic S. Then the set Σ is on sextic-node position. Proof. Suppose that the surface S is deﬁned by a homogeneous polynomial equation F (x0 , x1 , x2 , x3 ) = 0 of degree six. We consider the linear system $ 3 $ $ ∂F $ $ $ L := $ λi = 0$ . $ $ ∂xi i=0

The base locus of the linear system L is exactly the singular locus of the surface S. A curve of degree k in P3 intersects a generic member of the linear system L exactly 5k times since L ⊂ |OP3 (5)|. Therefore, the set Σ is on sextic-node position.

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For convenience, we state an elementary lemma. Lemma 4.5. Let Γ = {q1 , . . . , qs } be a set of s ≥ 4 points in P3 . For a given point q ∈ Γ , there is a hyperplane H which contains at least three points of Γ but not the point q unless all the points q, q1 , . . . , qs lie on a single hyperplane. Proof. Because not all the points q, q1 , · · · , qs lie on a single hyperplane, we may assume there are two distinct hyperplanes H1 and H2 such that H1 ∪ H2 contains the point q and four points, say q1 , q2 , q3 , and q4 , of Γ , q1 ∈ H1 \ H2 , and q2 ∈ H2 \ H1 . Then one of the hyperplanes generated by {q1 , q2 , q3 } and {q1 , q2 , q4 } must not pass through the point q; otherwise all of the ﬁve points

q, q1 , . . . , q4 would be on a single hyperplane. Also, the following result of [14] is useful. 2 Theorem 4.6. Let π : Y → P2 be sthe blowup at points p1 , . . . , ps ∈ P . Then ∗ the linear system |π (OP2 (d)) − i=1 Ei | is base-point-free for all

s≤

1 0 2 (h (P , OP2 (d + 3)) − 5), 3

where d ≥ 3 and Ei = π −1 (pi ), if at most k(d + 3 − k) − 2 of the points pi lie on a curve of degree 1 ≤ k ≤ 12 (d + 3). Theorem 4.1 tells us that the ﬁrst part of Theorem B immediately follows from the lemma below. Lemma 4.7. Let γ : V → P3 be the blowup at k diﬀerent points Γ = {p1 , . . . , pk } and p be a point in V \ ∪ki=1 Ei such that the set Γ ∪ {γ(p)} is on sextic-node position, where Ei = γ −1 (pi ). If k ≤ 13, then the linear k system |γ ∗ (OP3 (5)) − i=1 Ei | is base-point-free at the point p. Proof. It is enough to ﬁnd a quintic hypersurface in P3 that passes through all the points of Γ but not the point q := γ(p). We may assume that k = 13. Let r be the maximal number of points of Γ that belong to a single hyperplane of P3 together with the point q. Note that 2 ≤ r ≤ 13. Without loss of generality, we may also assume that the ﬁrst r points of Γ , i.e., p1 , . . . , pr , are contained in a hyperplane H together with the point q. We prove the statement case by case. Case 1. (r = 2) We divide the set Γ into ﬁve subsets of Γ such that each subset contains exactly three points of Γ and the union of all the ﬁve subsets is Γ . Because r = 2, the hyperplane generated by each subset cannot contain the point q. The product of these ﬁve hyperplanes is what we want. Before we proceed, we note that the points q and p1 , . . . , pr do not lie on a single line. If they do, then the hyperplane H must contain more than r points of Γ .

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Case 2. (r = 3) By Lemma 4.5, we can ﬁnd three points of Γ outside of H such that generate a hyperplane not passing through the point q. Since r = 3, we can repeat this procedure two more times with the remaining points of Γ in the outside of H. Only one point, say p13 , then remains in the outside of H. Because the four points, q, p1 , p2 , p3 , cannot lie on a line, there is a quadric hypersurface passing through the points p1 , p2 , p3 , p13 , but not the point q. Case 3. (r = 4) As in the previous case, we can ﬁnd two hyperplanes which together contains six points of Γ in the outside of H but not q. We then have three remaining points of Γ in the outside of H. There is a line passing though two points, say p1 , p2 , of p1 , . . . , p4 , but not the point q. Then these two points together with one of the remaining points in the outside of H generate a hyperplane not containing the point q. Now, we have four points, two of them are on H and the others not on H. Obviously, these four points belong to a quadric hypersurface not passing through the point q. Therefore, the product of the quadric hypersurface and the hyperplanes gives us a quintic hypersurface that we are looking for. Case 4. (r = 5) First of all, by Lemma 4.5, we ﬁnd a hyperplane which contains three points, say p6 , p7 , p8 , of Γ in the outside of H but not the point q. We split the case into two subcases. Subcase 4.1. When four points of Γ on H together with the point q lie on a line. Assume that the points q and p1 , . . . , p4 lie on a single line. The hyperplane generated by the points p4 , p5 , and p9 cannot contain q. The hyperplane generated by {p3 , p10 , p11 } cannot pass through the point q; otherwise the number r would be bigger than ﬁve. By the same reason, we can ﬁnd a hyperplane which contains {p2 , p12 , p13 } but not the point q. Choose a hyperplane which passes through the point p1 but not the point q. Then we are done. Subcase 4.2. When no four points of Γ on H lie on a line together with the point q. In this case, two pairs of points of Γ on H give two lines which do not contain the point q. Therefore, we can ﬁnd a quadric hypersurface containing six points of Γ , four from H and two from Γ \ (H ∪ {p6 , p7 , p8 }), but not the point q. Furthermore, because the number r is ﬁve we may choose the two points from Γ \ (H ∪ {p6 , p7 , p8 }) so that the other three points in the outside of H cannot belong to a single line together with the point q. We then have four points which we have not covered yet, three points, say p11 , p12 , p13 , from the outside of H, and one point, say p1 , on H. Because the points p11 , p12 , p13 and q do not lie on a line, we can easily ﬁnd a quadric hypersurface passing through all the four points but not the point q.

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Case 5. (r = 6) Again, by Lemma 4.5, we ﬁnd a hyperplane which contains three points, say p7 , p8 , p9 , of Γ in the outside of H but not the point q. By the sextic-node position condition, we can ﬁnd two lines on H which together contain four points of Γ on H but not the point q. They give us a quadric hypersurface in P3 which pass though six points of Γ \ {p7 , p8 , p9 }. Among these six points, two points are from the outside of H and the others from H. Therefore, we have four points that have not been yet covered. Because two of them are in the outside of H, we can easily ﬁnd a quadric hypersurface which contains these four points but not the point q. Case 6. (r = 7) In this case, we can ﬁnd three pairs of points of Γ on H such that each pair gives us a line not passing through the point q. It implies that we can construct a cubic hypersurface which passes through six points of Γ on H and three points of Γ in the outside of H but not the point q. Moreover, we may assume that the remaining three points in the outside of H do not lie on a single line together with the point q due to the sextic-node position condition. It is easy to ﬁnd a quadric hypersurface containing the remaining points of Γ but not q. So we are done. Case 7. (r = 8 or 9) We can ﬁnd four pairs of points of Γ on H such that each pair gives us a line not passing through the point q. From this fact, we easily obtain a quartic hypersurface passing eight points of Γ on H and four points of Γ outside of H but not the point q. We then have only one point of Γ that is not covered. Just take a hyperplane passing through this point but not the point q, and we are done. Case 8. (r = 10) Because of the sextic-node position condition, we can ﬁnd three pairs, say {p1 , p2 }, {p3 , p4 }, {p5 , p6 }, of points of Γ on H such that each pair gives us a line not passing through the point q and, in addition, no three of the points p7 , p8 , p9 , p10 cannot lie on a line passing through point q. This shows there is a quintic hypersurface which passes through Γ but not the point q. Case 9. (r = 11) We have eleven points of Γ on H and two points, p12 , p13 , of Γ in the outside of H. We can ﬁnd a quintic curve C on H which passes through the eleven points on H but not the point q by Theorem 4.6. Note that the support of the curve C is not a line because of the sextic-node position condition. A generic hyperplane passing through p12 , p13 meets C at more than two points. Choose two points p and p among these intersection points. Let v be the

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point at which two lines p12 , p and p13 , p intersect. Then the cone over the curve C with vertex v has all the points of Γ but not the point q. Case 10. (r = 12) All the points except one point, p13 , lie on the hyperplane H. It immediately follows from Theorem 4.6 that we can ﬁnd a plane quintic curve which passing {p1 , . . . , p12 } but not the point q. Taking the cone over the plane quintic curve with vertex p13 , we obtain a quintic hypersurface that we want. Case 11. (r = 13) In this case, all the points lie on the hyperplane H. It immediately follows from Theorem 4.6 that we can ﬁnd a plane quintic passing all the points except the point q, which gives us a quintic hypersurface in P3 that we want. Consequently, we complete the proof.

Therefore, the ﬁrst part of Theorem B has been proved. The three-dimensional conjecture of Fujita (see [51], [85], and [122]) implies Lemma 4.7 in the case when the points in Γ are in very general position. Moreover, in the case when points in Σ are in very general position, the Qfactoriality of X follows from Lefschetz theory (see Theorem 1.34 in [37]). However, neither the three-dimensional nor the two-dimensional conjecture of Fujita can be applied, in general, to an appropriate adjoint linear system in our case. The crucial point here is that the proof of Theorem 4.6 is based on the vanishing theorem of Ramanujam (see [19] and [120]) for 2-connected eﬀective divisors on an algebraic surface (see Proposition 2 in [141]) which is slightly stronger in some cases than the vanishing theorem of Kawamata and Viehweg (see [84] and [142]). The method of [37] also explains the non-Q-factoriality of Examples 1.5, 1.6, and 1.7 over C. Let X −→ P3 be a double cover ramiﬁed along a sextic S. Suppose that the sextic S ⊂ P3 is given by the equation g32 (x, y, z, w) + hr (x, y, z, w)f6−r (x, y, z, w) = 0, where g3 , hr , and f6−r , 1 ≤ r ≤ 3, are generic homogeneous polynomials over C of degree 3, r, and 6 − r, respectively. Then the number of singular points is 18r − 3r2 . All of them are simple double points. The defect of V is h1 (P3 , I ⊗ OP3 (5)) = h0 (P3 , I ⊗ OP3 (5)) − h0 (P3 , OP3 (5)) + h0 (P3 , C) p∈Σ

= h (P , I ⊗ OP3 (5)) − 56 + 18r − 3r . 0

3

2

Let H be the hypersurface of degree r deﬁned by hr = 0. Then it is easy to check h0 (P3 , I ⊗ OP3 (5)) is bigger than or equal to h0 (P3 , OP3 (4)) + h0 (H, OH (2)) + h0 (H, OH ) = 42 when r = 1, h0 (P3 , OP3 (3)) + h0 (H, OH (2)) + h0 (H, OH (1)) = 33 when r = 2, h0 (P3 , OP3 (2)) + h0 (H, OH (2)) + h0 (H, OH (2)) = 30 when r = 3.

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In all the cases, the defect of V is positive. Therefore, the double cover X is not Q-factorial.

5 Elliptic ﬁbrations This section is devoted to Theorem C. Let π : X → P3 be a Q-factorial double cover ramiﬁed along a nodal sextic S ⊂ P3 . Consider a ﬁbration τ : Y → Z whose general enough ﬁber is a smooth elliptic curve. Suppose that we have a birational map ρ of X onto Y . We then put MX = n1 M with M = ρ−1 (|τ ∗ (HZ )|), where HZ is a very ample divisor on Z and n is the natural number such that M ⊂ | − nKX |. ρ X _ _ _ _ _ _/ Y π

P3

τ

Z

It immediately follows from Theorem 2.14 that the set CS(X, MX ) is nonempty. Remark 5.1. The linear system M is not composed from a pencil and cannot be contained in the ﬁbers of any dominant rational map χ : X P1 . Using the proof of Lemma 3.1, we can easily show that the set CS(X, MX ) does not contain any smooth point of X. Lemma 5.2. Let o be a simple double point on X that belongs to CS(X, MX ). Then there is a birational map β : P2 Z such that the diagram ρ X _ _ _ _ _ _ _/ Y π

τ

γ β P3 _ _ _/ P2 _ _ _/ Z

commutes, where γ is the projection from the point π(o). Proof. Let f : W → X be the blowup at the point o and C be a general enough ﬁber of the elliptic ﬁbration φ|−KW | : W → P2 . Then for a general surface D in f −1 (M), 2(n − multo (M)) = C · D ≥ 0, while multo (M) ≥ n by Theorem 2.34. We can therefore conclude that multo (M) = n and f −1 (M) lies in the ﬁbers of the elliptic ﬁbration φ|−KW | : W → P2 , which implies the claim.

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Corollary 5.3. The set CS(X, MX ) cannot contain two singular points of the 3-fold X. We assume that CS(X, MX ) does not contain any point and that it contains a curve C ⊂ X. Lemma 5.4. The intersection number −KX · C is 1. Proof. Let H be a general enough divisor in the linear system | − KX |. Then we have 2 = H · M2X ≥ multC (M2X )H · C ≥ −KX · C, 2 = H · KX

which implies −KX · C ≤ 2. Suppose −KX · C = 2. Then Supp(M2X ) = C and multC (M2X ) = mult2C (MX ) = 1, which means that for two diﬀerent divisors M1 and M2 in the linear system M we have multC (M1 · M2 ) = n2 ,

multC (M1 ) = multC (M2 ) = n,

and set-theoretically M1 ∩ M2 = C. However, the linear system M is not composed from a pencil. Therefore, for a general enough point p ∈ C the linear subsystem D of M passing through the point p has no base components. Let D1 and D2 be general enough divisors in D. Then in set-theoretic sense p ∈ D1 ∩ D1 = M1 ∩ M2 = C,

which is a contradiction. Corollary 5.5. The curve π(C) ⊂ P3 is a line and C ∼ = P1 .

Remark 5.6. In the second part of the proof of Lemma 5.4, we have never used the irreducibility of the curve C. Hence, we may assume CS(X, MX ) = {C}. Moreover, the same arguments imply multC (M2 ) < 2n2 . Lemma 5.7. The line π(C) is contained in the sextic surface S. Proof. It follows from the proof of Lemma 3.7 and Remark 5.6.

Before we proceed, we observe ! #| Sing(X) ∩ C| ≤

3, π(C) ⊂ S 5, π(C) ⊂ S,

by intersecting S with either the line π(C) or a hyperplane in P3 passing through π(C). Furthermore, when π(C) ⊂ S, the equality #| Sing(X)∩C| = 5 holds if and only if all the hyperplanes tangent to the sextic surface S at points of π(C\ Sing(X)) coincide.

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Lemma 5.8. The curve C passes through at least four singular points of X. Proof. Let H be a general hyperplane in P3 containing the line π(C). Then the curve D = H ∩ S = π(C) ∪ Q is reduced, where Q is a quintic curve. The curve D is singular at each singular point pi of S such that pi ∈ π(C) for i ∈ {1, . . . , k}. The set π(C) ∩ Q consists of at most ﬁve points and Sing(D) ∩ π(C) ⊂ π(C) ∩ Q. Thus k = #| Sing(X) ∩ C| ≤ 5. Suppose k ≤ 3. Then the intersection π(C) ∩ Q contains two points o1 and o2 diﬀerent from pi due to the generality in the choice of H. The hyperplane H is therefore tangent to the sextic S at o1 and o2 . Hyperplanes passing through the line π(C) form a pencil whose proper transforms on the 3-fold X are K3 surfaces in | − KX | passing through C. Hence, the lines tangent to the sextic surface S at a general point of the line π(C) span whole P3 . Note that this is no longer true in the case k = 5 as we mentioned right before the lemma. Let L1 and L2 be general enough lines in H passing through the points o1 and o2 , respectively. Then Lj is tangent to the sextic surface S at the point ˜ j ⊂ X of the curve Lj is an irreducible oj . Therefore, the proper transform L ˜ curve such that −KX · Lj = 2. Also, it is singular at the point o˜j = π −1 (oj ). ˜ of the surface H on X and a general surface Consider the proper transform H M in the linear system M. Then M |H˜ = multC (M)C + R, ˜ such that C ⊂ Supp(R). Moreover, where R is an eﬀective divisor on H ˜ j ) multC (M )+ ˜ j ) ≥ 2n, ˜ j ≥ multo˜j (L multp (M )·multp (L 2n = M · L ˜j p∈(M\C)∩L

˜ j ⊂ C set-theoretically. However, on H ˜ the curves L ˜1 which implies M ∩ L ˜ and L2 span two pencils with the base loci consisting of the points o˜1 and o˜2 , respectively. Therefore, we see R = ∅ due to the generality in the choice of two curves L1 and L2 . Note that if k = 4, then this is not true. ˜ = C for a general divisor H ˜ ∈ | − KX | Hence, set-theoretically M ∩ H ˜ passing through the curve C and a divisor M ∈ M with H ⊂ Supp(M ). Let p˜ ˜ and Mp˜ be the linear system of surfaces be a general point on the surface H in M containing p˜. Then Mp˜ has no base components due to Remark 5.1. ˜ in Mp˜ Therefore, for a general divisor M ˜ ∩H ˜ =C p˜ ∈ M ˜ ⊂ Supp(M ˜ ), which contradicts the generality of the point p˜ ∈ H. ˜ because H

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Lemma 5.9. Suppose that the curve C contains exactly four singular points of the 3-fold X. Then there is a birational map β : P2 Z such that the diagram ρ X _ _ _ _ _ _/ Y Ξπ(C) τ β P2 _ _ _ _ _ _ / Z is commutative, where Ξπ(C) is a rational map deﬁned as in Construction B. Proof. By our assumption, the curve C passes through four singular points p1 , ˜ → X at the points p1 , . . . , p4 p2 , p3 , p4 of X. We consider the blowup g1 : X ˜ and the blowup g2 : W → X along the proper transform of the curve C on ˜ Put g := g2 ◦ g1 : W → X. We then get X. −KW = g ∗ (−KX ) −

4

Ei − F,

i=1

where Ei and F are the g-exceptional divisors such that g(Ei ) = pi and g(F ) = C. Let L be a curve on W such that π ◦ g(L) is a line tangent to S at some general point of π(C). Then MW · L ≤ 2 − 2 multC (MX ) ≤ 0, where MW = g −1 (MX ). Because such curves as L span a Zariski dense subset in W , we obtain multC (MX ) = 1. Each elliptic curve L is a ﬁber of the elliptic ﬁbration Ξπ(C) ◦ g : W → P2 . Thus MW lies in the ﬁbers of Ξπ(C) ◦ g, which implies the claim.

Lemma 5.10. The curve C passes through at most four singular points of X. Proof. Suppose that the curve C passes through ﬁve singular points p1 , . . . , p5 ˜ → X at the points p1 , . . . , p5 and of X. Again, we consider the blowup g1 : X ˜ ˜ Put the blowup g2 : W → X along the proper transform of the curve C on X. g := g2 ◦ g1 : W → X. Then we obtain −KW = g ∗ (−KX ) −

5

Ei − F,

i=1

where Ei and F are the g-exceptional divisors such that g(Ei ) = pi and g(F ) = C. Let f : U → W be a birational morphism such that h = ρ ◦ g ◦ f is a morphism. Then we obtain ∗

KU + MU = (g ◦ f ) (KX + MX ) +

r i=0

ai Gi ,

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where MU = (g ◦ f )−1 (MX ), Gi are the (g ◦ f )-exceptional divisors, and ai ∈ Q. Whenever ai ≤ 0, we have g ◦ f (Gi ) = C. But multC (MX ) < 2 by Remark 5.6 and hence there is exactly one i, say i = 0, such that a0 ≤ 0. It implies f (G0 ) = F and a0 = 0. ˆ of the morphism τ ◦ h : U → Z. Then Consider a general enough ﬁber L ˆ ˆ ˆ = 0 by construction. KU ·L = 0 because the curve L is elliptic. However, MU ·L ˆ ˆ So we see Gi · L = 0 for i = 0, which means that f is an isomorphism near L. −1 ˜ ˜ ˆ Thus MW · L = 0, where MW = f (MX ) and L = f (L). There is a surface D ⊂ W such that π ◦ g(D) ⊂ P3 is the plane tangent to the sextic surface S along the whole line π(C). The surface D is the closure of the set spanned by curves whose images via π ◦ g are lines tangent to the surface S at some point of π(C). By the same argument as in the proof of Lemma 5.9, we obtain that multC (MX ) = 1, and hence D ∼ MW − F +

5

bi Ei

i=1

ˆ · Gi = 0 for i = 0, we get for some bi ∈ Z. On the other hand, because L ˜ = f ∗ (Ej ) · L ˆ= Ej · L

r

ˆ=0 cij Gi · L

i=1

˜ · D < 0, which means L ˜ ⊂ D. This is impossible where cij ∈ N. Therefore, L ˜ because the curves L span a Zariski dense subset in W .

Therefore, Theorem C is proven.

6 Canonical Fano 3-folds To prove Theorem D, we let π : X → P3 be a Q-factorial double cover ramiﬁed in a nodal sextic S ⊂ P3 . We then suppose that there is a nonbiregular birational map ρ : X Y of X onto a Fano 3-fold Y with canonical singularities. We are to show that there is a curve C ⊂ X such that π(C) is a line on the surface S passing through ﬁve nodes of the sextic S. 0. We put M = ρ−1 (| − nKY |) and MX = n1 M for a natural number n We then see that KX + MX ∼Q 0 and the singularities of the movable log pair (X, MX ) are not terminal by Theorem 2.17. By our construction, the linear system M cannot be contained in the ﬁbers of any dominant rational map χ : X Z with 0 < dim(Z) ≤ 2. Proposition 6.1. The set CS(X, MX ) consists of a single curve C ⊂ X which satisﬁes 1. −KX · C = 1,

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2. π(C) ⊂ S, 3. #| Sing(X) ∩ C| = 5. Proof. For the proof, we literally repeat the proofs in Section 5 except those of Lemmas 5.2 and 5.9.

Let p1 , p2 , p3 , p4 , p5 ∈ C be singular points of X. We consider the blowup ˜ → X at all the points pi and the blowup f2 : W → X ˜ along the proper f1 : X ˜ Put f = f2 ◦ f1 : W → X. We then note that transform of the curve C on X. W is smooth and 5 −KW ∼ f ∗ (−KX ) − Ei − G, i=1

where Ei and G are the f -exceptional divisors with f (Ei ) = pi and f (G) = C. Each surface Ei is isomorphic to the blowup of P1 × P1 at one point. We have the proper transforms F1i and F2i of two rulings of the quadric P1 × P1 with self-intersection −1 on each surface Ei . The normal bundle NW/Fji of the curve Fji ∼ = P1 in the 3-fold W splits into NW/Fji ∼ = OFji (a) ⊕ OFji (b) for some integers a ≥ b. The exact sequence 0 → TFji → TW |Fji → NW/Fji → 0 implies deg(NW/Fji ) = a + b = −KW · Fji + 2g(Fji ) − 2 = −2. On the other hand, the exact sequence 0 → NEi /Fji → NW/Fji → NW/Ei |Fji → 0 together with NEi /Fji ∼ = OFji (−1) implies b ≥ −1. Therefore, a = b = −1 and % →W we can make a standard ﬂop for each curve F i . Indeed, we let h : W j

be the blowup along all the curves Fji and Rji be the h-exceptional divisor dominating the curve Fji . Then Rji ∼ = P1 × P1 and there is a birational morˆ and %→W ˆ which contracts each surface Ri to a curve Fˆ i ⊂ W phism ˆ h:W j j −1 ˆ ◦ h is not an isomorphism in a neighborhood of each curve F i . for which h j ˆ ◦ h−1 (Ei ) ⊂ W ˆi = h ˆ . Then E ˆi ∼ Let E = P2 and ˆi | ˆ ∼ E Ei = OP2 (−2), ˆi can be contracted to a terminal cyclic which implies that each divisor E ˆ → V be the contraction of quotient singularity of type 12 (1, 1, 1). Let fˆ : W ˆ all the Ei . Then V has exactly ﬁve singular points oi of type 12 (1, 1, 1), it is Q-factorial, and Pic(V ) ∼ = Z ⊗ Z.

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ˆ ◦ h−1 (G). Then there is a birational morphism g : V → X Let F = fˆ ◦ h contracting the surface F to the curve C. % W @@ ~ ~ @@hˆ h ~ ~ @@ ~ ~ @ ~~ h◦h ˆ −1 _ _ _ _ _ _ _ /W ˆ W f

fˆ

g V X o ρ φ|−rKV | Y _ _ _ _ _ _ _/ U

At a generic point of C the morphism g is a blowup. In fact, the morphism g is the blowup of the ideal sheaf of the curve C ⊂ X by Proposition 1.2 in [139]. Moreover, the proof of Lemma 3.8 implies multC (MX ) = 1. Hence, −KV ∼Q MV ∼Q g ∗ (−KX ) − F, where MV = g −1 (MX ). The morphism g|F : F → C has ﬁve reducible ﬁbers consisting of two copies of P1 intersecting transversally at the corresponding singular point oi that is a simple double point on the surface F . Let C˜ ⊂ F be the unique base curve of the pencil | − KV |. Then the numerical equivalence C˜ ≡ KV2 holds. Therefore, we have −KV is nef ⇐⇒ −KV · C˜ ≥ 0 ⇐⇒ −KV3 ≥ 0. Because elementary calculations imply −KV3 = 12 , the anticanonical divisor −KV is nef and big. Hence, | − rKV | is base-point-free for a natural number r 0 by the Base Point Freeness theorem (see [86]). The morphism φ|−rKV | : V → U is birational and U is a canonical Fano 3-fold with −KU3 = 12 . The image of every element in the set CS(V, MV ) on the 3-fold X is an element in CS(X, MX ) because KV + MV = g ∗ (KX + MX ). Hence, every element in CS(V, MV ) must be a curve dominating the curve C due to the assumption made in Remark 5.6, which implies multC (M) ≥ 2n2 . However, it is impossible because of Remark 5.6. Therefore, the set CS(V, MV ) = ∅. For a rational number c slightly bigger than 1, the singularities of the log pair (V, cMV ) are still terminal and the equivalence KV + cMV = φ∗|−rKV | (KU + cMU ) holds, where MU = φ|−rKV | (MV ). Hence, the movable log pair (U, cMU ) is a canonical model. On the other hand, the movable log pair (Y, nc | − nKY |) is a canonical model as well. Consequently, the map φ|−rKV | ◦ (ρ ◦ g)−1 : Y U

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is an isomorphism by Proposition 2.9. All the statements above do not depend on the existence of a birational map ρ of X onto Y . They depend only on the condition that X has a curve C such that π(C) ⊂ S is a line passing through ﬁve nodes of the sextic surface S. Once such a curve C ⊂ X exists, we can construct a birational transformation of X into a canonical Fano 3-fold by means of blowing up the ideal sheaf of the curve C ⊂ X and the birational morphism given by a plurianticanonical linear system. We have proved Theorem D. In addition, we have obtained explicit classiﬁcation of all birational transformations of a double cover X into Fano 3-folds with canonical singularities. As we mentioned before, ﬁve singular points of the surface S lying on the line π(C) ⊂ S force every hyperplane in P3 tangent to S at some point of π(C) smooth on S to be tangent to the surface S along the whole line π(C). Such a tangent hyperplane is unique and its proper transform on V is the only divisor in the linear system | − KV − F | which is contracted by the birational morphism φ|−rKV | to a nonterminal point of the canonical Fano 3-fold U .

7 Sextic double solids over ﬁnite ﬁelds We consider a double cover π : X → P3 deﬁned over a perfect ﬁeld F of characteristic char(F) > 5. Suppose that the 3-fold X is Q-factorial and that it is ramiﬁed along a nodal sextic surface S ⊂ P3 . Actually, we may assume that the ﬁeld F is algebraically closed because F is perfect. We are to adjust the proofs of both Theorems A and C to the case char(F) > 5. We ﬁrst list valid statements in Sections 3 and 5 in the case char(F) > 5. The following remain valid: 1. 2. 3. 4. 5.

Propositions 2.7, 2.9, and Theorem 2.30; negativity of exceptional loci (see [3] and Lemma 2.19 in [87]); resolution of singularities of 3-folds (see [1] and [45]); numerical intersection theory on smooth 3-folds (see [59]); elementary properties of blowups (see [71]).

Lemma 7.1. Theorems 2.10 and 2.14 are valid in the case char(F) > 5. Proof. The proofs for the case char(F) = 0 depend only on the facts listed above.

The following may not remain valid in the case char(F) = 0: 1. Theorem 2.34; 2. special cases of Bertini’s theorem (see [64]). For the birational super-rigidity, we need Theorem 2.34 and Bertini’s theorem.

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The characteristic-free method for the proof of Theorem 2.30 in [116] can be used to prove Theorem 2.34. However, we used Theorem 2.34 just to prove Lemma 3.2. So instead of proving Theorem 2.34 in the case char(F) > 5, we prove Lemma 3.2 only with Theorem 2.30, which is enough for the birational super-rigidity. Lemma 7.2. Let (X, MX ) be a movable log pair such that −(KX + MX ) is ample and let o ∈ X be a simple double point. Then the point o does not belong to CS(X, MX ). Proof. Suppose that the point o belongs to the set CS(X, MX ). Let f : W → X be the blowup at the point o and C be a general enough ﬁber of the elliptic ﬁbration φ|−KW | : W → P2 . Then 2(1 − multo (MX )) > C · MW ≥ 0, where MW = f −1 (MX ). This implies multo (MX ) < 1. We consider KW + MW = f ∗ (KX + MX ) + (1 − multo (MX ))G, where G is the f -exceptional divisor. We then see that there is a center B ∈ CS(W, MW ) with B ⊂ G. The intersection number of MW with each ruling of G ∼ = P1 × P1 is multo MX < 1. On the other hand, we have multB (MW ) ≥ 1. Therefore, the center B must be a point and multB (M2W ) ≥ 4 by Theorem 2.30. Let H1 and H2 be two general surfaces in | − KW | passing through the point B. Then H1 ∩ H2 consists of the ﬁber E of the elliptic ﬁbration φ|−KW | with B ∈ E. Consider general enough divisors D ∈ | − 2KW | and F1 , F2 ∈ |f ∗ (−KX )|. Then the divisors D, F1 , and F2 do not pass through the point B at all. The divisors H1 + F1 , H2 + F2 , and D + G are elements of the linear subsystem H ⊂ |f ∗ (−2KX ) − G| of surfaces passing B. The intersection Supp(H1 + F1 ) ∩ Supp(H2 + F2 ) ∩ Supp(D + G) contains B and consists of a ﬁnite number of points. In particular, the linear system H has no base curves but B is a base point of H. Let H be a general surface in H. Then we obtain 4 > H · M2W ≥ multB (H) multB (M2W ) ≥ 4, which is absurd.

During excluding a one-dimensional member of CS(X, MX ), we implicitly used Bertini’s theorem only one time just for the following special case.

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Lemma 7.3. Let C ⊂ X be a curve with −KX · C = 1 and π(C) ⊂ S. Then a general enough surface H ∈ | − KX | passing through C is smooth along C \ Sing(X). Proof. The simple double points of the 3-fold X correspond to the simple double points of the sextic surface S because char(F) = 2. Meanwhile, the curve L := π(C) on P3 is a line. The line L cannot pass through more than three singular points of S; otherwise it would be contained in S. The surface D = π(H) ⊂ P3 is a plane containing L. The singularities of surface H correspond to the singularities of the curve D ∩ S which is the ramiﬁcation divisor of the double cover π : H → D. For a general enough surface H ∈ |− KX |, the plane D is not tangent to the sextic S at the points of L \ Sing(S), which implies the claim.

Therefore, the birational super-rigidity remains true over the ﬁeld F. Now, we consider the statements in Section 5 over the ﬁeld F. They also require both Theorem 2.34 and Bertini’s theorem. The reason why Theorem 2.34 is required again is the lemma below. However, it can be proved only with Theorem 2.30. Lemma 7.4. Let ρ : X Y be a birational map and τ : Y → Z be a ﬁbration whose general ﬁber is a smooth elliptic curve. Let (X, MX ) be the movable log pair such that M := ρ−1 (|τ ∗ (H)|) and MX = n1 M, where H is a very ample divisor on surface Z and n is the natural number such that M ⊂ | − nKX |. Suppose that the set CS(X, MX ) contains a singular point o ∈ X. Then there is a birational map β : P2 Z such that the diagram ρ X _ _ _ _ _ _ _/ Y π

τ

γ β P3 _ _ _/ P2 _ _ _/ Z

commutes, where γ is the projection from the point π(o). Proof. Consider the blowup f : W → X at the point o. Let C be a general ﬁber of φ|−KW | . Then 2n − 2 multo (M) = C · f −1 (M) ≥ 0, which implies multo (MX ) ≤ 1. Furthermore, the multiplicity multo (MX ) cannot be less than 1. Indeed, if multo (MX ) < 1, then the proof of Lemma 7.2 shows contradictory inequalities 4 ≥ H · M2W ≥ multB (H) multB (M2W ) > 4, where MW = f −1 (MX ), B is a center of CS(W, MW ), and H is a general surface in |f ∗ (−2KX ) − E| passing through B. In the case multo (MX ) = 1, the linear system f −1 (M) lies in the ﬁbers

of the elliptic ﬁbration φ|−KW | : W → P2 , which implies the claim.

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Bertini’s Theorem is required again only for the following statement that can be proved without using Bertini’s theorem. Lemma 7.5. Let C be a curve on X such that −KX · C = 1 and π(C) ⊂ S. Suppose that #| Sing(X)∩C| ≤ 3. Then a general surface H ∈ |−KX | passing through curve C has at least two diﬀerent simple double points on the curve C ⊂ X at which the 3-fold X is smooth. Proof. The surface π(H) ⊂ P3 is a plane passing through the line L := π(C) ⊂ S. Therefore, π(H) ∩ S = L ∪ Q, where Q is a plane quintic. Whenever H moves in the pencil of surfaces in | − KX | passing through C, the quintic Q moves in a pencil on S whose base locus is Sing(S) ∩ L. It gives a ﬁnite morphism γ : L → P1 of degree 5−#| Sing(S)∩L| such that in the outside of the set Sing(S)∩L the morphism γ is ramiﬁed at the points where L ∪ Q is not a normal crossing divisor on the plane π(H). These points correspond to nonsimple double points of the surface H contained in the curve C and diﬀerent from Sing(X) ∩ C. However, this morphism cannot be ramiﬁed everywhere because we assumed char(F) > 5.

Corollary 7.6. Lemma 5.8 remains true in the case char(F) > 5. Proof. Apply Lemma 7.5 to the proof of Lemma 5.8.

Because the proofs of Lemmas 5.9 and 5.10 are characteristic-free, Theorem E is true.

8 Potential density Now, we prove Theorem F. Consider a double cover π : X → P3 deﬁned over a number ﬁeld F and ramiﬁed along a nodal sextic surface S ⊂ P3 . We suppose that Sing(X) = ∅. We will show that the set of rational points of the 3-fold X is potentially dense, which means that there exists a ﬁnite extension K of the ﬁeld F such that the set of all K-rational points of the 3-fold X is Zariski dense. The rationality and the unirationality of the 3-fold X over the ﬁeld Q would automatically imply potential density of rational points on X. However, the 3-fold X is nonrational in general due to Theorem A and the unirationality of the 3-fold X is unknown. Moreover, X is expected to be nonunirational in general. Actually, the degree of a rational dominant map from P3 to a double cover of P3 ramiﬁed in a very generic smooth sextic surface must be divisible by 2 and 3 due to [89] and [90]. The following result was proved in [16]:

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Theorem 8.1. Let τ : D → P2 be a double cover deﬁned over a number ﬁeld F and ramiﬁed along a reduced sextic curve R ⊂ P2 . Suppose Sing(D) = ∅. Then the set of rational points on the surface D is potentially dense if and only if the curve R ⊂ P2 is not a union of six lines intersecting at a single point. Actually, Theorem 8.1 is a special case of the following result in [17]. Theorem 8.2. Let D be a K3 surface deﬁned over a number ﬁeld F such that D has either a structure of an elliptic ﬁbration or an inﬁnite group of automorphisms. Then the set of rational points on D is potentially dense. Hence, taking Theorem C into consideration, we see that Theorem F is a three-dimensional analogue of Theorem 8.1. When singularities of the sextic surface S are worse than simple double points but are not too bad, the double cover X tends to be more rational (see [23]). Thus Theorem F must be true for sextic surfaces with any singularities possibly except cones over sextic curves. If the sextic surface S ⊂ P3 is a reduced union of six hyperplanes passing through one line, the set of rational points on X is not potentially dense due to Faltings’ theorem ([56] and [57]) because the 3-fold X is birationally isomorphic to a product P2 × C, where C is a smooth curve of genus 2. As a matter of fact, the sets of rational points are potentially dense on double covers of Pn ramiﬁed along general enough sextic hypersurfaces for n 0 due to the following result ([40]): Theorem 8.3. Let V be a double cover of Pn ramiﬁed in a suﬃciently general hypersurface of degree 2d > 4. Then V is unirational if n ≥ c(d), where c(d) ∈ N depends only on d. We will prove the potential density of the set of rational points on X using the technique of [15], [16], and [70] which relies on the following result proved in [104]. Theorem 8.4. Let F be a number ﬁeld. Then there is an integer nF such that no elliptic curve deﬁned over F has an F-rational torsion point of order n > nF . Let o be a simple double point on X. The point π(o) is a node of the sextic surface S. Replacing the ﬁeld F by a ﬁnite extension of F, we may assume that the point o and some other ﬁnitely many points that we will need in the sequel are deﬁned over F. Let f : V → X be the blowup at the point o with f -exceptional divisor E. Then the linear system | − KV | is free and the morphism φ|−KV | : V → P2 is an elliptic ﬁbration. The surface E is a multisection of φ|−KV | of degree 2. Let H be a general surface in | − f ∗(KX )|. Then H is a multisection of φ|−KV | of degree 2 as well. The following lemma is a corollary of Proposition 2.4 in [15].

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Lemma 8.5. Suppose that there is a multisection M of φ|−KV | of degree d ≥ 2 such that the morphism φ|−KV | |M is branched at a point p ∈ M which is contained in a smooth ﬁber of the elliptic ﬁbration φ|−KV | . Then the divisor p1 − p2 ∈ Pic(Cb ) is not a torsion for some distinct points p1 and p2 of the intersection M ∩ Cb , where Cb = φ−1 |−KV | (b) and b is a C-rational point in the complement to a countable union of proper Zariski closed subsets in P2 .

Proof. See [15].

Lemma 8.6. Let M ∈ |H| be an irreducible multisection of φ|−KV | of degree 2 deﬁned over F such that the set of rational points on M is potentially dense in M and φ|−KV | |M is branched at a point contained in a smooth ﬁber of φ|−KV | . Then the set of rational points on X is potentially dense. Proof. For each n ∈ N, we let Φn be the set of points p of M satisfying the following two conditions: 1. the point p is contained in a smooth ﬁber Cp of the elliptic ﬁbration φ|−KV | ; 2. 2np = nH|Cp in Pic(Cp ). Let Φn be the Zariski closure of the set Φn in M . Suppose Φn = M for some n. Take a very general ﬁber C of φ|−KV | and let C ∩ M = {p1 , p2 }, where p1 = p2 . Then either 2np1 ∼ nH|C or 2np2 ∼ nH|C because Φn = M . However, p1 + p2 ∼ H|C . Thus 2np1 ∼ 2np2 ∼ nH|C and the element p1 − p2 is a torsion in Pic(C). Therefore, the C-rational point φ|−KV | (C) is contained in the countable union of proper Zariski closed subsets in P2 of Lemma 8.5, which contradicts the very general choice of the ﬁber C. Accordingly, the set Φn is not Zariski dense in M for any n ∈ N. Moreover, it follows from Theorem 8.4 that each set Φn for n > nF , where nF is the number deﬁned in Theorem 8.4, is disjoint from the set of F-rational points on M . Because of the assumption on the multisection M , we may assume that the set of F-rational points on the surface M is Zariski dense. Take an F-rational point F Φi ), q ∈ M := M \(Z ∪ni=1 where the set Z ⊂ M consists of points contained in singular ﬁbers of φ|−KV | . Let Cq be the ﬁber of φ|−KV | passing through q. Then both the curve Cq and the point φ|−KV | (q) are deﬁned over the ﬁeld F. The divisor 2q − H|Cq ∈ Pic(Cq ) is deﬁned over F as well. Moreover, 2q − H|Cq is not a torsion divisor. By Riemann–Roch theorem, for each n ∈ N there is a unique F-rational point qn ∈ Cq such that

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qn + (2n − 1)q = nH|Cq in Pic(Cq ). Because 2q − H|Cq is not a torsion divisor, we see that qi = qj if and only if i = j. We obtain an inﬁnite collection of F-rational points on Cq . Consequently, for each F-rational point q in M , the curve Cq is contained in the Zariski closure of the set of F-rational points of V . Because the set M is a Zariski dense subset of M , the set of rational points on the 3-fold X is potentially dense.

In order to prove Theorem F, it is enough to ﬁnd an element in |H| satisfying the conditions of Lemma 8.6. To ﬁnd such an element, we let T be the set of points (p, q) ∈ S × S satisfying the following conditions: 1. 2. 3. 4. 5.

p = q; the points p and q are smooth points on the sextic surface S; the point q is contained in the hyperplane D ⊂ P3 tangent to S at p; the point q is a smooth point of the intersection S ∩ D; the intersection S ∩ D is reduced.

We also let ψ : T → S be the projection on the second factor. Lemma 8.7. The image ψ(T ) contains a Zariski open subset of the sextic S ⊂ P3 . Proof. Let p be a general point on the sextic S ⊂ P3 and D be the hyperplane tangent to the surface S at the point p in P3 . To prove the claim we just need to show that D ∩S is reduced, which is nothing but the ﬁniteness of the Gauss map at a generic point of S. When the surface S is smooth, the intersection D ∩ S is known to be reduced (see [60], [76], or [112]). Even though S can have double points in our case, the intersection D ∩ S is reduced because S is not ruled (see [105]). Here, we prove it only with simple calculation. Suppose that D ∩ S is not reduced and D ∩ S = mC + F ⊂ D ∼ = P2 , where m ≥ 2. Then C is a line, a conic, or a plane cubic curve. Let γ : S˜ → S be the blowup at the double points of S and C˜ = γ −1 (C). Then S is a surface of general type, KS˜ = γ ∗ (OP3 (2)|S ), and C˜ is either a rational curve or an elliptic curve. Moreover, the selfintersection number C˜ 2 of C˜ is negative by adjunction formula, but C˜ moves in a family on the surface S˜ when we move the point p in S, which is a contradiction.

Therefore, by Lemma 8.7 we can ﬁnd a hyperplane D ⊂ P3 such that D∩S is reduced and singular at some smooth point of S. Moreover, we may assume

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that D does not contain the point π(o) and there is a line L ⊂ P3 passing through the point π(o) such that L ∩ D ∩ S = ∅ and L intersects the sextic S transversally at four diﬀerent smooth points of ˜ be the surface in the linear system |H| such that π ◦ f (D) ˜ = D. S. Let D ˜ Then D is an irreducible multisection of the elliptic ﬁbration φ|−KV | of degree ˜ contained in the ﬁber C 2 such that φ|−KV | |D˜ is branched at a point q ∈ D of φ|−KV | such that π ◦ f (C) = L. By construction, the ﬁber C is a smooth ˜ Moreover, elliptic curve, π ◦ f (q) = L ∩ D ∩ S, and q is a smooth point on D. ˜ is deﬁned over F. Hence, the set extending the ﬁeld F we can assume that D ˜ by Theorem 8.1. Theorem F is of rational points is potentially dense on D proven. It would be natural to prove Theorem F in the case when the sextic S is singular and reduced (see Theorem 8.1). Most of the arguments in this section work for any reduced singular sextic surface. Actually, in the case when the sextic S has nonisolated singularities (for example, when it is reducible) we do not need to use Lemma 8.7 at all, but in the case when the sextic S is irreducible and has isolated singularities we can prove Lemma 8.7 using the ﬁniteness of the Gauss map for curves (see [69]) in the assumption S is not a scroll (see [105], [145], and [146]), which is satisﬁed automatically if S is not a cone. Moreover, in general the proof of Theorem F must be simpler for bad singularities. For instance, in the case when the sextic S has a singular point of multiplicity 4, the double cover X is unirational and nonrational in general due to [138], but it is rational when S has a singular point of multiplicity 5. However, when S is a cone over a smooth sextic curve R ⊂ P2 , the double cover X is birationally equivalent to P1 ×D, where D is a double cover of P2 ramiﬁed along R. The potential density of rational points on X is therefore equivalent to the potential density of rational points on D, which is still unknown in general (see [17]).

References 1. S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, Princeton University Press (1998). 2. V. Alexeev, General elephants of Q-Fano 3-folds, Compos. Math. 91 (1994), 91–116. 3. M. Artin, Some numerical criteria of contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485–496. 4. M. Artin, D. Mumford, Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. 25 (1972), 75–95. 5. F. Bardelli, Polarized mixed Hodge structure: on irrationality of threefolds via degeneration, Ann. Mat. Pura Appl. 137 (1984), 287–369. 6. W. Barth, Two projective surfaces with many nodes, admitting the symmetries of the icosahedron, J. Algebraic Geom. 5 (1996), 173–186.

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Moduli Stacks of Vector Bundles on Curves and the King–Schofield Rationality Proof Norbert Hoﬀmann Mathematisches Institut der Georg-August-Universität 37073 Göttingen, Germany [email protected] Summary. The King–Schoﬁeld proof of rationality certain moduli spaces of vector bundles on curves is explained in the language of algebraic stacks.

Key words: Rationality, moduli spaces, vector bundles on curves 2000 Mathematics Subject Classiﬁcation codes: 14E08, 14H60, 14D20

1 Introduction Let C be a connected smooth projective curve of genus g ≥ 2 over an algebraically closed ﬁeld k. Consider the coarse moduli scheme Bunr,d (resp. Bunr,L ) of stable vector bundles on C with rank r and degree d ∈ Z (resp. determinant isomorphic to the line bundle L on C). Motivated by work of A. Tyurin [10, 11] and P. Newstead [7, 8], it has been believed for a long time that Bunr,L is rational if r and the degree of L are coprime. Finally, this conjecture was proved in 1999 by A. King and A. Schoﬁeld [4]; they deduce it from their following main result: Theorem 1.1 (King–Schofield). Bunr,d is birational to the product of an aﬃne space An and Bunh,0 where h be the highest common factor of r and d. The present text contains the complete proof of King and Schoﬁeld translated into the language of algebraic stacks. Following their strategy, the moduli stack Bunr,d of rank r, degree d vector bundles is shown to be birational to a Grassmannian bundle over Bunr1 ,d1 for some r1 < r; then induction is used. However, this Grassmannian bundle is in some sense twisted. Mainly for that reason, King and Schoﬁeld need a stronger induction hypothesis than 1.1: They add the condition that their birational map preserves a certain Brauer class ψr,d on Bunr,d . One main advantage of the stack language here

F. Bogomolov, Y. Tschinkel (eds.), Cohomological and Geometric Approaches to Rationality Problems, Progress in Mathematics 282, DOI 10.1007/978-0-8176-4934-0_5, © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

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is that this extra condition is not needed: The stack analogue of Theorem 1.1 is proved by a direct induction. (In more abstract terms, this can be understood roughly as follows: A Brauer class corresponds to a gerbe with band Gm . But the gerbe on Bunr,d corresponding to ψr,d is just the moduli stack Bunr,d. Thus a rational map of coarse moduli schemes preserving this Brauer class corresponds to a rational map of the moduli stacks.) This paper consists of four parts. Section 2 contains the precise formulation of the stack analogue 2.2 to Theorem 1.1; then the original results of King and Schoﬁeld are deduced. Section 3 deals with Grassmannian bundles over stacks because they are the main tool for the proof of Theorem 2.2 in Section 4. Finally, Appendix A summarizes the basic properties of the moduli stack Bunr,d that we use. In particular, a proof of Hirschowitz’ theorem about the tensor product of general vector bundles on C is given here, following Russo and Teixidor [9]. The present article has grown out of a talk in the joint seminar of U. Stuhler and Y. Tschinkel in Göttingen. I thank them for encouraging me to write this text. I am grateful to J. Heinloth for valuable suggestions and for many useful discussions about these stacks.

2 The King–Schoﬁeld theorem in stack form We denote by Bunr,d the moduli stack of vector bundles of rank r and degree ¯ This stack d on our smooth projective curve C of genus g ≥ 2 over k = k. is algebraic in the sense of Artin, smooth of dimension (g − 1)r2 over k and irreducible; these properties are discussed in more detail in the appendix. Our main subject here is the birational type of Bunr,d. We will frequently use the notion of a rational map between algebraic stacks; it is deﬁned in the usual way as an equivalence class of morphisms deﬁned on dense open substacks. A birational map is a rational map that admits a two-sided inverse. Definition 2.1. A rational map of algebraic stacks M _ _ _/ M is birationally linear if it admits a factorization ∼ M _ _ _/ M × An

pr1

/ M

into a birational map followed by the projection onto the ﬁrst factor. Now we can formulate the stack analogue of the King–Schoﬁeld Theorem 1.1; its proof will be given in Section 4. Theorem 2.2. Let h be the highest common factor of the rank r ≥ 1 and the degree d ∈ Z. There is a birationally linear map of stacks μ : Bunr,d _ _ _/ Bunh,0

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and an isomorphism between the Picard schemes Picd (C) and Pic0 (C) such that the following diagram commutes: μ Bunr,d _ _ _/ Bunh,0 det

Picd (C)

∼

(1)

det

/ Pic0 (C)

Remark 2.3. One cannot expect an isomorphism of Picard stacks here: If (1) were a commutative diagram of stacks, then choosing a general vector bundle E of rank r and degree d would yield a commutative diagram of automorphism groups ∼ / Gm Gm (_)r

Gm

(_)h

∼

/ Gm

which is impossible for r = h. Remark 2.4. In the theorem, we can furthermore achieve that μ preserves scalar automorphisms in the following sense: Let E and E = μ(E) be vector bundles over C that correspond to a general point in Bunr,d and its image in Bunh,0 . Then E and E are stable (because we have assumed g ≥ 2) and hence simple. The rational map μ induces a morphism of algebraic groups μE : Gm = Aut(E) −→ Aut(E ) = Gm which is an isomorphism because μ is birationally linear. Thus μE is either the identity or λ → λ−1 ; it is independent of E because Bunr,d is irreducible. Modifying μ by the automorphism E → E dual of Bunh,0 if necessary, we can achieve that μE is the identity for every general E. Clearly, the map μ in the theorem restricts to a birationally linear map between the dense open substacks of stable vector bundles. But any rational (resp. birational, resp. birationally linear) map between these induces a rational (resp. birational, resp. birationally linear) map between the corresponding coarse moduli schemes; cf. Proposition A.6 in the appendix for details. Hence the original theorem of King and Schoﬁeld follows: Corollary 2.5 (King–Schofield). Let Bunr,d be the coarse moduli scheme of stable vector bundles of rank r and degree d on C. Then there is a birationally linear map of schemes μ : Bunr,d _ _ _/ Bunh,0 .

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Of course, this is just a reformulation of Theorem 1.1. Remark 2.6. As mentioned before, King and Schoﬁeld also prove that the rational map μ : Bunr,d _ _ _/ Bunh,0 preserves their Brauer class ψr,d . This is equivalent to the condition that μ induces a rational map between the corresponding Gm -gerbes, i.e., a rational map Bunr,d _ _ _/ Bunh,0 that preserves scalar automorphisms in the sense of Remark 2.4. We recall the consequences concerning the rationality of Bunr,L . Because Diagram (1) commutes, μ restricts to a rational map between ﬁxed determinant moduli schemes; thus one obtains: Corollary 2.7 (King–Schofield). Let L be a line bundle on C, and let Bunr,L be the coarse moduli scheme of stable vector bundles of rank r and determinant L on C. Then there is a birationally linear map of schemes μ : Bunr,L _ _ _/ Bunh,O where h is the highest common factor of r and deg(L). In particular, Bunr,L is rational if the rank r and the degree deg(L) are coprime; this proves the conjecture mentioned in the introduction. More generally, it follows that Bunr,L is rational if Bunh,O is. For h ≥ 2, it seems to be still an open question whether Bunh,O is rational or not.

3 Grassmannian bundles Let V be a vector bundle over a dense open substack U ⊆ Bunr,d. Recall that a part of this datum is a functor from the groupoid U(k) to the groupoid of vector spaces over k. So for each appropriate vector bundle E over C, we do not only get a vector space VE over k, but also a group homomorphism AutOC (E) → Autk (VE ). Note that both groups contain the scalars k ∗ . Definition 3.1. A vector bundle V over a dense open substack U ⊆ Bunr,d has weight w ∈ Z if the diagram k∗

(_)w

k∗

/ AutOC (E) / Autk (VE )

commutes for all vector bundles E over C that are objects of the groupoid U(k). Example 3.2. The trivial vector bundle On over Bunr,d has weight 0.

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We denote by E univ the universal vector bundle over C × Bunr,d, and by its restriction to {p} × Bunr,d for some point p ∈ C(k).

Epuniv

Example 3.3. Epuniv is a vector bundle of weight 1 on Bunr,d, and its dual (Epuniv )dual is a vector bundle of weight −1. For another example, we ﬁx a vector bundle F over C. By semicontinuity, there is an open substack U ⊆ Bunr,d that parametrizes vector bundles E of rank r and degree d over C with Ext1 (F, E) = 0; assume U = ∅. The spaces Hom(F, E) are ﬁbers of a vector bundle Hom(F, E univ ) over U according to Grothendieck’s theory of cohomology and base change in EGA III. Similarly, there is a vector bundle Hom(E univ , F ) deﬁned over an open substack of Bunr,d whose ﬁber over any point [E] with Ext1 (E, F ) = 0 is the vector space Hom(E, F ). Example 3.4. Hom(F, E univ ) is a vector bundle of weight 1 and Hom(E univ, F ) is a vector bundle of weight −1. Note that any vector bundle of weight 0 over an open substack U ⊆ Bunr,d contained in the stable locus descends to a vector bundle over the corresponding open subscheme U ⊆ Bunr,d of the coarse moduli scheme, cf. Proposition A.6. Vector bundles of nonzero weight do not descend to the coarse moduli scheme. Proposition 3.5. Consider all vector bundles V of ﬁxed weight w over dense open substacks of a ﬁxed stack Bunr,d. Assume that V0 has minimal rank among them. Then every such V is generically isomorphic to V0n for some n. Proof. The homomorphism bundles End(V0 ) and Hom(V0 , V) are vector bundles of weight 0 over dense open substacks of Bunr,d. Hence they descend to vector bundles A and M over dense open subschemes of Bunr,d , cf. Proposition A.6. The algebra structure on End(V0 ) and its right(!) action on Hom(V0 , V) also descend; they turn A into an Azumaya algebra and M into a right Amodule. In particular, the generic ﬁber MK is a right module under the central simple algebra AK over the function ﬁeld K := k(Bunr,d ). By our choice of V0 , there are no nontrivial idempotent elements in AK ; hence AK is a skew ﬁeld. We have just constructed a functor V → MK from the category in question to the category of ﬁnite-dimensional right vector spaces over AK . This functor is a Morita equivalence; its inverse is deﬁned as follows: Given such a right vector space MK over AK , we can extend it to a right A-module M over a dense open subscheme of Bunr,d , i.e., to a right End(V0 )module of weight 0 over a dense open substack of Bunr,d; we send MK to the vector bundle of weight w V := M ⊗End(V0 ) V0 . Using this Morita equivalence, the proposition follows from the correspond ing statement for right vector spaces over AK .

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Corollary 3.6. There is a vector bundle of weight w = 1 (resp. w = −1) and rank h = hcf(r, d) over a dense open substack of Bunr,d. Proof. Because the case of weight w = −1 follows by dualizing the vector bundles, we only consider vector bundles of weight w = 1. Here Epuniv is a vector bundle of rank r over Bunr,d, and Hom(Ldual , E univ ) is a vector bundle of rank r(1 − g + deg(L)) + d over a dense open substack if L is a suﬃciently ample line bundle on C. Consequently, the rank of V0 divides r and r(1 − g + deg(L)) + d; hence it also divides their highest common factor h.

To each vector bundle V over a dense open substack U ⊆ Bunr,d, we can associate a Grassmannian bundle Grj (V) −→ U ⊆ Bunr,d. By deﬁnition, Grj (V) is the moduli stack of those vector bundles E over C which are parametrized by U, endowed with a j-dimensional vector subspace of VE . Grj (V) is again a smooth Artin stack locally of ﬁnite type over k, and its canonical morphism to U is representable by Grassmannian bundles of schemes. If V is a vector bundle of some weight, then all scalar automorphisms of E preserve all vector subspaces of VE . This means that the automorphism groups of the groupoid Grj (V)(k) also contain the scalars k ∗ . In particular, it makes sense to say that a vector bundle over Grj (V) has weight w ∈ Z: There is an obvious way to generalize Deﬁnition 3.1 to this situation. m be the moduli To give some examples, we ﬁx a point p ∈ C(k). Let Parr,d stack of rank r, degree d vector bundles E over C endowed with a quasiparabolic structure of multiplicity m over p. Recall that such a quasiparabolic structure is just a coherent subsheaf E ⊆ E with the property that E/E is isomorphic to the skyscraper sheaf Opm . m Example 3.7. Parr,d is canonically isomorphic to the Grassmannian bundle univ dual Grm ((Ep ) ) over Bunr,d.

Here we have regarded a quasiparabolic vector bundle E • = (E ⊆ E) as the vector bundle E together with a dimension m quotient of the ﬁber Ep . But we can also regard it as the vector bundle E together with a dimension m vector subspace in the ﬁber at p of the twisted vector bundle E (p). Choosing a trivialization of the line bundle OC (p) over p, we can identify the ﬁbers of E (p) and E at p; hence we also obtain: m is isomorphic to the Grassmannian bundle Grm (Epuniv ) Example 3.8. Parr,d over Bunr,d−m where E univ is the universal vector bundle over C × Bunr,d−m.

These two Grassmannian bundles θ

θ

1 2 m Bunr,d ←− Parr,d −→ Bunr,d−m

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form a correspondence between Bunr,d and Bunr,d−m, the Hecke correspondence. Its eﬀect on the determinant line bundles is given by det θ1 (E • ) = det(E) ∼ = OC (mp) ⊗ det(E ) = OC (mp) ⊗ det θ2 (E • )

(2)

for each parabolic vector bundle E • = (E ⊆ E) with multiplicity m at p. Proposition 3.9. Let V and W be two vector bundles of the same weight w over dense open substacks of Bunr,d. If j ≤ rk(W) ≤ rk(V), then there is a birationally linear map ρ : Grj (V) _ _ _/ Grj (W) over Bunr,d. Proof. According to Proposition 3.5, there is a vector bundle W of weight w such that V ∼ = W ⊕ W over some dense open substack U ⊆ Bunr,d. We may assume without loss of generality that U is contained in the stable locus and denote by U ⊆ Bunr,d the corresponding open subscheme, cf. Proposition A.6. We use the following simple fact from linear algebra: If W and W are vector spaces over k with dim(W ) ≥ j, then every j-dimensional vector subspace of W ⊕ W whose image S in W also has dimension j is the graph of a unique linear map S → W . This means that Grj (W ⊕ W ) contains as a dense open subscheme the total space of the vector bundle Hom(S univ , W ) over Grj (W ) where S univ is the universal subbundle of the constant vector bundle W over Grj (W ). In our stack situation, these considerations imply that Grj (V) is birational to the total space of the vector bundle Hom(S univ , W ) over Grj (W) where S univ is the universal subbundle of the pullback of W over Grj (W). This deﬁnes the rational map ρ. The vector bundle Hom(S univ , W ) has weight 0 because S univ and W both have weight w. Since the scalars act trivially, we can descend Grj (W) and this vector bundle over it to a Grassmannian bundle over U and a vector bundle over it, cf. Proposition A.6. In particular, our homomorphism bundle is trivial over a dense open substack of Grj (W). This proves that ρ is birationally linear.

Corollary 3.10. Let V be a vector bundle of weight w = ±1 over a dense open substack of Bunr,d. If j is divisible by hcf(r, d), then the Grassmannian bundle Grj (V) −→ Bunr,d is birationally linear. Proof. By Corollary 3.6, there is a vector bundle W of weight w and rank j. Due to the proposition, Grj (V) is birationally linear over Grj (W) Bunr,d.

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4 Proof of Theorem 2.2 The aim of this section is to prove Theorem 2.2, i.e., to construct the birationally linear map μ : Bunr,d _ _ _/ Bunh,0 where h is the highest common factor of the rank r and the degree d. We proceed by induction on r/h. For r = h, the theorem is trivial: Tensoring with an appropriate line bundle ∼ / Bunh,0 with the required deﬁnes even an isomorphism of stacks Bunr,d properties. Thus we may assume r > h. Lemma 4.1. There are unique integers rF and dF that satisfy (1 − g)rF r + rF d − rdF = h

(3)

r < hrF < 2r.

(4)

and Proof. (3) has an integer solution rF , dF because h is also the highest common factor of r and (1 − g)r + d; here rF is unique modulo r/h. Furthermore, rF is nonzero modulo r/h since −rdF = h has no solution. Hence precisely one of the solutions rF , dF of (3) also satisﬁes (4).

We ﬁx rF , dF and deﬁne r1 := hrF − r,

d1 := hdF − d,

h1 := hcf(r1 , d1 ).

Then r1 < r, and h1 is a multiple of h. In particular, r1 /h1 < r/h. Lemma 4.2. There is an exact sequence 0 −→ E1 −→ F ⊗k V −→ E −→ 0

(5)

where E1 , F , E are vector bundles over C and V is a vector space over k with rk(E1 ) = r1 , deg(E1 ) = d1 ,

rk(F ) = rF , deg(F ) = dF ,

rk(E) = r, deg(E) = d

dim(V ) = h

such that the following two conditions are satisﬁed: i) Ext1 (F, E) = 0, and the induced map V → Hom(F, E) is bijective. ii) Ext1 (E1 , F ) = 0, and the induced map V dual → Hom(E1 , F ) is injective. Proof. We may assume h = 1 without loss of generality: If there is such a sequence for r/h and d/h instead of r and d, then the direct sum of h copies is the required sequence for r and d. By our choice of rF and dF and Riemann–Roch, all vector bundles F and E of these ranks and degrees satisfy χ(F, E) := dimk Hom(F, E) − dimk Ext1 (F, E) = h = 1.

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If F and E are general, then Hom(F, E) ∼ =k

and

Ext1 (F, E) = 0

according to a theorem of Hirschowitz [2, Section 4.6], and there is a surjective map F → E by an argument of Russo and Teixidor [9]. Thus we obtain an exact sequence 0 −→ E1 −→ F −→ E −→ 0 (6) that satisﬁes condition i (with V = k). (For the convenience of the reader, a proof of the cited results is given in the appendix, cf. Theorem A.7.) Furthermore, all vector bundles of the given ranks and degrees satisfy χ(E1 , F ) = χ(F, E) − χ(E, E) + χ(E1 , E1 ) > χ(F, E) = h = 1 because r1 < r. Now we can argue as above: For general E1 and F , we have Ext1 (E1 , F ) = 0 by Hirschowitz, and there is an injective map E1 → F with torsion-free cokernel by Russo–Teixidor; cf. also Theorem A.7 in the appendix. Thus we obtain an exact sequence (6) that satisﬁes condition ii (with V = k). Finally, we consider the moduli stack of all exact sequences (6) of vector bundles with the given ranks and degrees. As explained in the appendix (cf. Corollary A.5), it is an irreducible algebraic stack locally of ﬁnite type over k. But i and ii are open conditions, so there is a sequence that satisﬁes both.

From now on, let F be a ﬁxed vector bundle of rank rF and degree dF that occurs in such an exact sequence (5). Definition 4.3. The rational map of stacks λF : Bunr,d _ _ _/ Bunr1 ,d1 is deﬁned by sending a general rank r, degree d vector bundle E over C to the kernel of the natural evaluation map

E : Hom(F, E) ⊗k F −→ E. We check that this does deﬁne a rational map. Let UF ⊆ Bunr,d be the open substack that parametrizes all E for which Ext1 (F, E) = 0 and E is surjective. Then the E are the restrictions of a surjective morphism univ of vector bundles over C × UF . So the kernel of univ is also a vector bundle; it deﬁnes a morphism λF : UF → Bunr1 ,d1 . This gives the required rational map because UF is nonempty by our choice of F . For later use, we record the eﬀect of λF on determinant line bundles: det λF (E) ∼ = det(F )⊗h ⊗ det(E)dual .

(7)

Following [4], the next step is to understand the ﬁbers of λF . We denote by Hom(E1univ , F ) the vector bundle over an open substack of Bunr1 ,d1 whose ﬁber over any point [E1 ] with Ext1 (E1 , F ) = 0 is the vector space Hom(E1 , F ).

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Proposition 4.4. Bunr,d is over Bunr1,d1 naturally birational to the Grassmannian bundle Grh (Hom(E1univ , F )). Proof. If E is a rank r, degree d vector bundle over C for which Ext1 (F, E) = 0 and the above map := E is surjective, then the exact sequence

0 −→ ker( ) −→ Hom(F, E) ⊗k F −→ E −→ 0 satisﬁes the condition i of the previous lemma. This identiﬁes the above open substack UF ⊆ Bunr,d with the moduli stack of all exact sequences (5) that satisfy i. Similarly, let UF ⊆ Grh (Hom(E1univ , F )) be the open substack that parametrizes all pairs (E1 , S ⊆ Hom(E1 , F )) for which Ext1 (E1 , F ) = 0 and the natural map α : E1 → S dual ⊗k F is injective with torsion-free cokernel. For such a pair (E1 , S), the exact sequence α

0 −→ E1 −→ S dual ⊗k F −→ coker(α) −→ 0 satisﬁes the condition ii of the previous lemma. This identiﬁes UF with the moduli stack of all exact sequences (5) that satisfy ii. Hence both Bunr,d and Grh (Hom(E1univ , F )) contain as an open substack the moduli stack UF of all exact sequences (5) that satisfy both conditions i and ii. But UF is nonempty by our choice of F , so it is dense in both stacks; thus they are birational over Bunr1 ,d1 .

Still following [4], the proof of Theorem 2.2 can now be summarized in the following diagram; it is explained below. ρ μ˜1 Bunr,d _ _ _/ Grh (W) _ _ _/ Parhh1 ,0 J J J θ1 λF J J % Bunr1,d1 _ _ _/ Bunh1 ,0

θ2

/ Bunh1 ,−h _μ_2 _/ Bunh,0

μ1

Here μ1 and μ2 are the birationally linear maps given by the induction hypothesis. (θ1 , θ2 ) is the Hecke correspondence explained in the previous section; note that θ2 is birationally linear by Corollary 3.10. The square in this diagram is cartesian, so μ˜1 is the pullback of μ1 , and W := μ∗1 (Epuniv )dual is the pullback of the vector bundle (Epuniv )dual over Bunh1 ,0 to which θ1 is the associated Grassmannian bundle. Using Remark 2.4, we may assume that μ1 preserves scalar automorphisms, i.e., that W has the same weight −1 as (Epuniv )dual . Then we can apply Proposition 3.9 to obtain the birationally linear map ρ. Now we have the required birationally linear map μ := μ2 ◦ θ2 ◦ μ˜1 ◦ ρ : Bunr,d _ _ _/ Bunh,0; it satisﬁes the determinant condition in Theorem 2.2 due to equations (7), (2) and the corresponding induction hypothesis on μ1 , μ2 .

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A Moduli stacks of sheaves on curves This section summarizes some well-known basic properties of moduli stacks of vector bundles and more generally coherent sheaves on curves. For the general theory of algebraic stacks, we refer the reader to [5] or the appendix of [12]. We prove that the moduli stacks in question are algebraic, smooth, and irreducible. Then we discuss descent to the coarse moduli scheme. Finally, we deduce Hirschowitz’ theorem [2] and a reﬁnement by Russo and Teixidor [9] about morphisms between general vector bundles. Recall that we have ﬁxed an algebraically closed ﬁeld k and a connected smooth projective curve C/k of genus g. We say that a coherent sheaf F on C has type t = (r, d) if its rank rk(F ) (at the generic point of C) equals r and its degree deg(F ) equals d. If F and F are coherent sheaves of types t = (r, d) and t = (r , d ) on C, then the Euler characteristic χ(F , F ) := dimk Hom(F , F ) − dimk Ext1 (F , F ) satisﬁes the Riemann–Roch theorem χ(F , F ) = χ(t , t) with χ(t , t) := (1 − g)r r + r d − rd . Note that Extn (F , F ) vanishes for all n ≥ 2 since dim(C) = 1. We denote by Coht the moduli stack of coherent sheaves F of type t on C. More precisely, Coht (S) is for each k-scheme S the groupoid of all coherent sheaves on C × S which are ﬂat over S and whose ﬁber over every point of S has type t. Now assume t = t1 + t2 . We denote by Ext(t2 , t1 ) the moduli stack of exact sequences of coherent sheaves on C 0 → F1 → F → F2 → 0 where Fi has type ti = (ri , di ) for i = 1, 2. This means that Ext(t2 , t1 )(S) is for each k-scheme S the groupoid of short exact sequences of coherent sheaves on C × S which are ﬂat over S and ﬁberwise of the given types. Proposition A.1. The stacks Coht and Ext(t2 , t1 ) are algebraic in the sense of Artin and locally of ﬁnite type over k. Proof. Let O(1) be an ample line bundle on C. For n ∈ Z, we denote by Cohnt ⊆ Coht

(resp. Ext(t2 , t1 )n ⊆ Ext(t2 , t1 ))

the open substack that parametrizes coherent sheaves F on C (resp. exact sequences 0 → F1 → F → F2 → 0 of coherent sheaves on C) such that the twist F (n) = F ⊗ O(1)⊗n is generated by global sections and H1 (F (n)) = 0. By Grothendieck’s theory of Quot-schemes, there is a scheme Quotnt of ﬁnite type over k that parametrizes such coherent sheaves F together with

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a basis of the k-vector space H0 (F (n)). Moreover, there is a relative Quotscheme Flag(t2 , t1 )n of ﬁnite type over Quotnt that parametrizes such exact sequences 0 → F1 → F → F2 → 0 together with a basis of H0 (F (n)). Let N denote the dimension of H0 (F (n)). According to Riemann–Roch, N depends only on t, n, and the ample line bundle O(1), but not on F . Changing the chosen basis deﬁnes an action of GL(N ) on Quotnt , and Cohnt is precisely the stack quotient Quotnt /GL(N ). Similarly, Ext(t2 , t1 )n is precisely the stack quotient Flag(t2 , t1 )n /GL(N ). Hence these two stacks are algebraic and of ﬁnite type over k. By the ampleness of O(1), the Cohnt (resp. Ext(t2 , t1 )n ) form an open covering of Coht (resp. Ext(t2 , t1 )).

Remark A.2. In general, Coht is not quasi-compact because the family of all coherent sheaves on C of type t is not bounded. Proposition A.3. i) Coht is smooth of dimension −χ(t, t) over k. ii) If we assign to each exact sequence 0 → F1 → F → F2 → 0 the two sheaves F1 , F2 , then the resulting morphism of algebraic stacks Ext(t2 , t1 ) −→ Coh(t1 ) × Coh(t2 ) is smooth of relative dimension −χ(t2 , t1 ), and all its ﬁbers are connected. iii) Ext(t2 , t1 ) is smooth of dimension −χ(t2 , t2 ) − χ(t2 , t1 ) − χ(t1 , t1 ) over k. Proof. i) By standard deformation theory, Hom(F, F ) is the automorphism group of any inﬁnitesimal deformation of the coherent sheaf F , Ext1 (F, F ) classiﬁes such deformations, and Ext2 (F, F ) contains the obstructions against extending deformations inﬁnitesimally, cf. [3, 2.A.6]. Because Ext2 vanishes, deformations of F are unobstructed and hence Coht is smooth; its dimension at F is then dim Ext1 (F, F ) − dim Hom(F, F ) = −χ(t, t). ii) The ﬁber of this morphism over [F1 , F2 ] is the moduli stack of all extensions of F2 by F1 ; it is the stack quotient of the aﬃne space Ext1 (F2 , F1 ) modulo the trivial action of the algebraic group Hom(F2 , F1 ). Hence this ﬁber is smooth of dimension −χ(t2 , t1 ) and connected. More generally, consider a scheme S of ﬁnite type over k and a morphism S → Coh(t1 ) × Coh(t2 ); let F1 and F2 be the corresponding coherent sheaves over C × S. By EGA III, the object RHom(F2 , F1 ) in the derived category of coherent sheaves on S can locally be represented by a complex of length one δ V 0 −→ V 1 where V 0 , V 1 are vector bundles. This means that the pullback of Ext(t2 , t1 ) to S is locally the stack quotient of the total space of V 1 modulo & 0 the action of the algebraic group V S given by δ. Hence this pullback is smooth over S; this proves ii. iii) follows from i) and ii).

Proposition A.4. The stacks Coht and Ext(t2 , t1 ) are connected.

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Proof. Proposition A.3 implies that Ext(t2 , t1 ) is connected if Coht1 and Coht2 are. We prove the connectedness of the latter by induction on the rank (and on the degree for rank zero). Coht is connected for t = (0, 1) because there is a canonical surjection C → Coht ; it sends a point P to the skyscraper sheaf OP . Now consider t = (0, d) with d ≥ 2 and write t = t1 + t2 . By induction hypothesis and A.3, Ext(t1 , t2 ) is connected. But there is a canonical surjection Ext(t1 , t2 ) → Coht ; it sends an extension 0 → F1 → F → F2 → 0 to the sheaf F . This shows that Coht is also connected; now we have proved all connectedness assertions in rank zero. If F and F are two coherent sheaves on C of type t = (r, d) with r ≥ 1, then there is a line bundle L on C such that both Ldual ⊗F and Ldual ⊗F have a generically nonzero section. In other words, there are injective morphisms L "→ F and L "→ F . Let tL be the type of L; then F and F are both in the image of the canonical morphism Ext(t − tL , tL ) → Coht . But Ext(t − tL , tL ) is connected by the induction hypothesis and A.3. This shows that any two points F and F lie in the same connected component of Coht , i.e., Coht is connected.

Corollary A.5. The stacks Coht and Ext(t2 , t1 ) are reduced and irreducible. The moduli stack Bunt of vector bundles, the moduli stack Sstabt of semistable vector bundles, and the moduli stack Stabt of (geometrically) stable vector bundles on C of type t = (r, d) are open substacks Stabt ⊆ Sstabt ⊆ Bunt ⊆ Coht . Hence these stacks are all irreducible and smooth of the same dimension −χ(t, t) if they are nonempty. Stabt is known to be nonempty for g ≥ 2 and r ≥ 1. Moreover, Sstabt and Stabt are quasi-compact (and thus of ﬁnite type) because the family of (semi-)stable vector bundles of given type t is bounded. Proposition A.6. Assume g ≥ 2. Let Stabt −→ Bunt be the coarse moduli scheme of stable vector bundles of type t, and let V be a vector bundle of some weight w ∈ Z over an open substack U ⊆ Stabt . i) U descends to an open subscheme U ⊆ Bunt . ii) Grj (V) descends to a (twisted) Grassmannian scheme Grj (V) over U. iii) If V has weight w = 0, then it descends to a vector bundle over U. iv) More generally, any vector bundle of weight 0 over Grj (V) descends to a vector bundle over Grj (V). v) Any birationally linear map of stacks Stabt _ _ _/ Stabt induces a birationally linear map of schemes Bunt _ _ _/ Bunt . Proof. We continue to use the notation introduced in the proof of Proposition A.1. By boundedness, there is an integer n such that Stabt is contained in

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Cohnt ; hence Stabt = Quotstab /GL(N ) where Quotstab ⊆ Quotnt is the stable t t locus. Here the center of GL(N ) acts trivially; by Geometric Invariant Theory [6], Quotstab is a principal PGL(N )-bundle over Bunt . t be the inverse image of U. Then U is a PGL(N )i) Let U ⊆ Quotstab t stable open subscheme in the total space of this principal bundle and hence the inverse image of a unique open subscheme U ⊆ Bunt . ii) Let V be the pullback of V to U ; it is a vector bundle with GL(N )action. Hence its Grassmannian scheme Grj (V ) → U also carries an action of GL(N ). But here the center acts trivially: λ · id ∈ GL(N ) acts as the scalar λw on the ﬁbers of V and hence acts trivially on Grj (V ). Thus we obtain an action of PGL(N ) on our Grassmannian scheme Grj (V ) over U . Since this action is free, Grj (V ) descends to a Grassmannian bundle Grj (V) over U (which may be twisted, i.e., not Zariski-locally trivial). iii) The assumption w = 0 means that the scalars in GL(N ) act trivially on the ﬁbers of V . Hence PGL(N ) acts on V over U here. Again since this action is free, V descends to a vector bundle over U. iv) Here weight 0 means that the scalars in GL(N ) act trivially on the pullback of the given vector bundle to Grj (V ). Hence PGL(N ) acts on this pullback; but it acts freely on the base Grj (V ), so the vector bundle descends to Grj (V). v) Such a birationally linear map can be represented by an isomorphism ϕ : U → U between dense open substacks U ⊆ Stabt × A? and U ⊆ Stabt . By i, U corresponds to an open subscheme U ⊆ Bunt ; the proof of i shows that U is a coarse moduli scheme for the stack U. A straightforward generalization of this argument shows that U corresponds to an open subscheme U ⊆ Bunt × A? and that U is again a coarse moduli scheme for U . By the universal property of coarse moduli schemes, ϕ induces an isomorphism U → U and thus the required birationally linear map of schemes.

Theorem A.7 (Hirschowitz, Russo–Teixidor). Assume g ≥ 2. Let F1 and F2 be a general pair of vector bundles on C with given types t1 = (r1 , d1 ) and t2 = (r2 , d2 ). i) If χ(t1 , t2 ) ≥ 0, then dim Hom(F1 , F2 ) = χ(t1 , t2 ) and Ext1 (F1 , F2 ) = 0. ii) If χ(t1 , t2 ) ≥ 1 and r1 > r2 (resp. r1 = r2 , resp. r1 < r2 ), then a general morphism F1 → F2 is surjective (resp. injective, resp. injective with torsion-free cokernel). Proof. The cases r1 = 0 and r2 = 0 are trivial, so we may assume r1 , r2 ≥ 1; then Stabt1 = ∅ = Stabt2 . By semicontinuity, there is a dense open substack U ⊆ Stabt1 × Stabt2 of stable vector bundles F1 , F2 with dim Hom(F1 , F2 ) minimal, say equal to m. According to Riemann–Roch, m ≥ χ(t1 , t2 ); part i of the theorem precisely claims that we have equality here. Let Hom(F1univ , F2univ ) be the vector bundle of rank m over U whose ﬁber over F1 , F2 is Hom(F1 , F2 ). By generic ﬂatness (cf. EGA IV, §6.9), there is a dense open substack V in the total space of Hom(F1univ , F2univ ) such that the

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cokernel of the universal family of morphisms F1 → F2 is ﬂat over V. Then its image and kernel are also ﬂat over V; we denote the types of cokernel, image, and kernel by tQ = (rQ , dQ ), t = (r, d), and tK = (rK , dK ). If r = 0, then the theorem clearly holds: In this case, the general morphism ϕ : F1 → F2 has generic rank zero, so ϕ = 0; this means m = 0. Together with m ≥ χ(t1 , t2 ), this proves i and shows that the hypothesis of ii cannot hold here. Henceforth, we may thus assume r = 0. Note that t1 = tK + t and t2 = t + tQ ; moreover, we have a canonical morphism of moduli stacks Φ : V −→ Ext(t, tK ) ×Coht Ext(tQ , t) that sends a morphism ϕ : F1 → F2 to the extensions 0 → ker(ϕ) → F1 → im(ϕ) → 0 and 0 → im(ϕ) → F2 → coker(ϕ) → 0. Conversely, two extensions 0 → K → F1 → I → 0 and 0 → J → F2 → Q → 0 together with an isomorphism I → J determine a morphism ϕ : F1 → F2 . Thus Φ is an isomorphism onto the open locus in Ext ×Coh Ext where both extension sheaves F1 , F2 are stable vector bundles and dim Hom(F1 , F2 ) = m. Hence the stack dimensions coincide, i.e., m − χ(t1 , t1 ) − χ(t2 , t2 ) = −χ(t1 , tK ) − χ(t, t) − χ(tQ , t2 ). Since χ is biadditive, this is equivalent to m − χ(t1 , t2 ) = −χ(tK , tQ ).

(8)

In particular, χ(tK , tQ ) ≤ 0 follows. Now suppose that tK and tQ were both nonzero. Since the general vector bundles F1 and F2 are stable, we then have d2 d1 d dQ dK < < < < . rK r1 r r2 rQ Using the assumption χ(t1 , t2 ) ≥ 0, we get dK dQ d1 d2 χ(t1 , t2 ) χ(tK , tQ ) =1−g− + >1−g− + = ≥0 rK rQ rK rQ r1 r2 r1 r2 and hence χ(tK , tQ ) > 0. This contradiction proves tK = 0 or tQ = 0. (In some sense, this argument also covers the cases rK = 0 and rQ = 0. More precisely, rK = 0 implies tK = 0 because every rank zero coherent subsheaf of a vector bundle F1 is trivial. On the other hand, rK = 0 and tQ = (0, dQ ) = 0 would imply χ(tK , tQ ) = rK dQ > 0 which is again a contradiction.) In particular, we get χ(tK , tQ ) = 0; together with equation (8), this proves part i of the theorem.

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If r1 > r2 (resp. r1 ≤ r2 ), then rK > rQ (resp. rK ≤ rQ ) and hence rK = 0 = rQ (resp. rK = 0); we have just seen that this implies tQ = 0 (resp. tK = 0), i.e., the general morphism ϕ : F1 → F2 is surjective (resp. injective). Furthermore, the morphism of stacks V → CohtQ that sends a morphism ϕ : F1 → F2 to its cokernel is smooth (due to the open embedding Φ and Proposition A.3). If r1 < r2 , then rQ ≥ 1, so BuntQ is open and dense in CohtQ ; this implies that the inverse image of BuntQ is open and dense in V, i.e., the general morphism ϕ : F1 → F2 has torsion-free cokernel.

References 1. A. Grothendieck, Éléments de géométrie algébrique (EGA). Inst. Hautes Études Sci. Publ. Math., 4, 8, 11, 17, 20, 24, 28, 32, 1960–1967. 2. A. Hirschowitz, Problèmes de Brill-Noether en rang supèrieur. http://math.unice.fr/~ah/math/Brill/. 3. D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves. Friedr. Vieweg & Sohn, Braunschweig, 1997. 4. A. King and A. Schofield, Rationality of moduli of vector bundles on curves. Indag. Math. (N.S.), 10(4):519–535, 1999. 5. G. Laumon and L. Moret-Bailly, Champs algébriques. Springer-Verlag, Berlin, 2000. 6. D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory. Springer-Verlag, Berlin, 1994. 7. P. E. Newstead, Rationality of moduli spaces of stable bundles. Math. Ann., 215:251–268, 1975. 8. P. E. Newstead, Correction to: “Rationality of moduli spaces of stable bundles”. Math. Ann., 249(3):281–282, 1980. 9. B. Russo and M. Teixidor i Bigas, On a conjecture of Lange. J. Algebraic Geom., 8(3):483–496, 1999. 10. A. N. Tyurin, Classiﬁcation of vector bundles over an algebraic curve of arbitrary genus. Izv. Akad. Nauk SSSR Ser. Mat., 29:657–688, 1965. English translation: Amer. Math. Soc. Transl. 63, 1967. 11. A. N. Tyurin, Classiﬁcation of n-dimensional vector bundles over an algebraic curve of arbitrary genus. Izv. Akad. Nauk SSSR Ser. Mat., 30:1353–1366, 1966. English translation: Amer. Math. Soc. Transl. 73, 1968. 12. A. Vistoli, Intersection theory on algebraic stacks and on their moduli spaces. Invent. Math., 97(3):613–670, 1989.

Noether’s Problem for Some p-Groups Shou-Jen Hu1 and Ming-chang Kang 1

2

Department of Mathematics Tamkang University Tamsui, Taiwan Department of Mathematics National Taiwan University Taipei, Taiwan [email protected]

Summary. Let K be any ﬁeld and G be a ﬁnite group. Let G act on the rational function ﬁeld K(xg : g ∈ G) by K-automorphisms deﬁned by g · xh = xgh for any g, h ∈ G. Noether’s problem asks whether the ﬁxed ﬁeld K(G) = K(xg : g ∈ G)G is rational (=purely transcendental) over K. We will prove that if G is a non-abelian p-group of order pn containing a cyclic subgroup of index p and K is any ﬁeld containing a primitive pn−2 -th root of unity, then K(G) is rational over K. As a corollary, if G is a non-abelian p-group of order p3 and K is a ﬁeld containing a primitive p-th root of unity, then K(G) is rational.

Key words: Noether’s problem, rationality problem, inverse Galois problem, p-group actions 2000 Mathematics Subject Classiﬁcation codes: 12F12, 13A50, 11R32, 14E08

1 Introduction Let K be any ﬁeld and G be a ﬁnite group. Let G act on the rational function ﬁeld K(xg : g ∈ G) by K-automorphisms such that g · xh = xgh for any g, h ∈ G. Denote by K(G) the ﬁxed ﬁeld K(xg : g ∈ G)G . Noether’s problem asks whether K(G) is rational (=purely transcendental) over K. Noether’s problem for abelian groups was studied by Swan, Voskresenskii, Endo, Miyata and Lenstra, etc. See the survey article [Sw] for more details. Consequently we will restrict our attention to the non-abelian case in this article. First we will recall several results of Noether’s problem for non-abelian p-groups.

F. Bogomolov, Y. Tschinkel (eds.), Cohomological and Geometric Approaches to Rationality Problems, Progress in Mathematics 282, DOI 10.1007/978-0-8176-4934-0_6, © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

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Theorem 1.1 (Chu and Kang [CK], Theorem 1.6). Let G be a nonabelian p-group of order ≤ p4 and exponent pe . Assume that K is any ﬁeld such that either (i) char K = p > 0 or (ii)char K = p and K contains a primitive pe -th root of unity. Then K(G) is rational over K. Theorem 1.2 (Kang [Ka2], Theorem 1.5). Let G be a non-abelian metacyclic p-group of exponent pe . Assume that K is any ﬁeld such that either (i) char K = p > 0 or (ii)char K = p and K contains a primitive pe -th root of unity. Then K(G) is rational over K. Theorem 1.3 (Saltman [Sa1]). Let K be any ﬁeld with char K = p (in particular, K may be any algebraically closed ﬁeld with char K = p). There exists a non-abelian p-group G of order p9 such that K(G) is not rational over K. Theorem 1.4 (Bogomolov [Bo]). There exists a non-abelian p-group G of order p6 such that C(G) is not rational over C. All the above theorems deal with ﬁelds K containing enough roots of unity. For a ﬁeld K which does not have enough roots of unity, so far as we know, the only two known cases are the following Theorems 1.5 and 1.6. Theorem 1.5 (Saltman [Sa2], Theorem 1). Let G be a non-abelian pgroup of order p3 . Assume that K is any ﬁeld such that either (i) char K = p > 0 or (ii)char K = p and K contains a primitive p-th root of unity. Then K(G) is stably rational over K. Theorem 1.6 (Chu, Hu and Kang [CHK], [Ka1]). Let K be any ﬁeld. Suppose that G is a non-abelian group of order 8 or 16. Then K(G) is rational over K except when G = Q, the generalized quaternion group of order 16 (see Theorem 1.9 for its deﬁnition). When G = Q and K(ζ) is cyclic over K where ζ is a primitive 8th root of unity, then K(G) is also rational over K. We will remark that if G = Q is the generalized quaternion group of order 16, then Q(G) is not rational over Q by a theorem of Serre [GMS, Theorem 34.7, p. 92]. The main result of this article is the following. Theorem 1.7. Let G be a non-abelian p-group of order pn such that G contains a cyclic subgroup of index p. Assume that K is any ﬁeld such that either (i) char K = p > 0 or

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(ii)char K = p and [K(ζ) : K] = 1 or p where ζ is a primitive pn−1 -th root of unity. Then K(G) is rational over K. As a corollary of Theorems 1.1 and 1.7, we have Theorem 1.8. Let G be a non-abelian p-group of order p3 . Assume that K is any ﬁeld such that either (i) char K = p > 0 or (ii) char K = p and K contains a primitive p-th root of unity. Then K(G) is rational over K. Noether’s problem is studied for the inverse Galois problem and the construction of a generic Galois G-extension over K. See [DM] for details. We will describe the main ideas of the proof of Theorem 1.7 and Theorem 1.8. All the p-groups containing cyclic subgroups of index p are classiﬁed by the following theorem. Theorem 1.9 (Suzuki [Su], p. 107). Let G be a non-abelian p-group of order pn containing a cyclic subgroup of index p. (i) If p is an odd prime number, then G is isomorphic to M (pn ); and (ii)If p = 2, then G is isomorphic to M (2n ), D(2n−1 ), SD(2n−1 ) where n ≥ 4, and Q(2n ) where n ≥ 3 such that M (pn ) = D(2n−1 ) = SD(2n−1 ) = Q(2n ) =

< σ, τ < σ, τ < σ, τ < σ, τ

: : : :

n−1

σp n−1 σ2 n−1 σ2 n−1 σ2

= τp = τ2 = τ2 = τ4

= 1, = 1, = 1, = 1,

τ −1 στ = σ 1+p >, τ −1 στ = σ −1 >, n−2 τ −1 στ = σ −1+2 >, n−2 2 2 −1 σ = τ , τ στ = σ −1 > . n−2

The groups M (pn ), D(2n−1 ), SD(2n−1 ), Q(2n ) are called the modular group, the dihedral group, the quasi-dihedral group, and the generalized quaternion group, respectively. Thus we will concentrate on the rationality of K(G) for G = M (pn ), D(2n−1 ), SD(2n−1 ), Q(2n ) with the assumption that [K(ζ) : K] = 1 or p, where G is a group of exponent pe and ζ is a primitive pe -th root of unity. If ζ ∈ K, then Theorem 1.7 follows from Theorem 1.2. Hence we may assume that [K(ζ) : K] = p. If p is an odd prime number, the condition on [K(ζ) : K] implies that K contains a primitive pe−1 -th root of unity. If p = 2, the condition [K(ζ) : K] = 2 implies that λ(ζ) = −ζ, ±ζ −1 where λ is a generator of the Galois group of K(ζ) over K. (The case λ(ζ) = −ζ is equivalent to that the primitive 2e−1 -th root of unity belongs to K.) In case K contains a primitive pe−1 -th root of unity, we construct a faithful representation G −→ GL(V ) such that dim V = p2 and K(V ) is rational over K. For the remaining cases, i.e., p = 2, we will add the root ζ to the ground ﬁeld K and show that K(G) = K(ζ)(G)<λ> is rational

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over K. In the case p = 2 we will construct various faithful representations according to the group G = M (2n ), D(2n−1 ), SD(2n−1 ), Q(2n ), and the possible image λ(ζ) because it seems that a straightforward imitation of the case for K containing a primitive pe−1 -th root of unity does not work. We organize this article as follows. Section 2 contains some preliminaries which will be used subsequently. In Section 3 we ﬁrst prove Theorem 1.7 for the case when K contains a primitive pe−1 -th root of unity. This result will be applied to prove Theorem 1.8. In Section 4 we continue to complete the proof of Theorem 1.7. The case when char K = p > 0 will be covered by the following theorem due to Kuniyoshi. Theorem 1.10 (Kuniyoshi [CK], Theorem 1.7). If char K = p > 0 and G is a ﬁnite p-group, then K(G) is rational over K. Standing Notation: The exponent of a ﬁnite group, denoted by exp(G), is deﬁned as exp(G) = max{ord(g) : g ∈ G} where ord(g) is the order of the element g. Recall the deﬁnitions of modular groups, dihedral groups, quasidihedral groups, and generalized quaternion groups which are deﬁned in Theorem 1.9. If K is a ﬁeld with char K = 0 or char K m, then ζm denotes a primitive m-th root of unity in some extension ﬁeld of K. If L is any ﬁeld and we write L(x, y), L(x, y, z) without any explanation, we mean that these ﬁelds L(x, y), L(x, y, z) are rational function ﬁelds over K.

2 Generalities We list several results which will be used in the sequel. Theorem 2.1. [CK, Theorem 4.1] Let G be a ﬁnite group with an action on L(x1 , . . . , xm ), the rational function ﬁeld of m variables over a ﬁeld L such that (i) for any σ ∈ G, σ(L) ⊂ L; (ii) the restriction of the action of G to L is faithful; (iii) for any σ ∈ G, ⎞ ⎛ ⎞ ⎛ x1 σ(x1 ) ⎜ . ⎟ ⎜ .. ⎟ ⎝ . ⎠ = A(σ) ⎝ .. ⎠ + B(σ) σ(xm )

xm

where A(σ) ∈ GLm (L) and B(σ) is an m × 1 matrix over L. Then there exist z1 , . . . , zm ∈ L(x1 , . . . , xm ) so that L(x1 , . . . , xm ) = L(z1 , . . . , zm ) with σ(zi ) = zi for any σ ∈ G, any 1 ≤ i ≤ m.

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Theorem 2.2. [AHK, Theorem 3.1] Let G be a ﬁnite group acting on L(x), the rational function ﬁeld of one variable over a ﬁeld L. Assume that, for any σ ∈ G, σ(L) ⊂ L and σ(x) = aσ x + bσ for any aσ , bσ ∈ L with aσ = 0. Then L(x)G = LG (z) for some z ∈ L[x]. Theorem 2.3. [CHK, Theorem 2.3] Let K be any ﬁeld, K(x, y) the rational function ﬁeld of two variables over K, and a, b ∈ K \ {0}. If σ is a K-automorphism on K(x, y) deﬁned by σ(x) = a/x, σ(y) = b/y, then K(x, y)<σ> = K(u, v) where a x , u= ab xy − xy x−

b y v= . ab xy − xy y−

Moreover, x + (a/x) = (−bu2 + av 2 + 1)/v, y + (b/y) = (bu2 − av 2 + 1)/u, xy + (ab/(xy)) = (−bu2 − av 2 + 1)/(uv). Lemma 2.4. Let K be any ﬁeld whose prime ﬁeld is denoted by F. Let m ≥ 3 be an integer. Assume that char F = 2, [K(ζ2m ) : K] = 2, and λ(ζ2m ) = −1 ζ2−1 m (resp. λ(ζ2m ) = −ζ2m ) where λ is the non-trivial K-automorphism on ' K(ζ2m ). Then K(ζ2m ) = K(ζ4 ) and K F(ζ4 ) = F. Proof. Since m ≥ 3, it follows that λ(ζ4 ) = ζ4−1 no matter whether λ(ζ2m ) = −1 ζ2−1 m or −ζ2m . Hence λ(ζ4 ) = ζ4 . It follows that ζ4 ∈ K(ζ2m ) \ K. Thus / F. Since [K(ζ4 ) : K] = 2 and K(ζ2m ) = K(ζ4 ). In particular, ' ζ4 ∈ [F(ζ4 ) : F] = 2, it follows that K F(ζ4 ) = F.

3 Proof of Theorem 1.8 Because of Theorem 1.10 we will assume that char K = p for any ﬁeld K considered in this section. Theorem 3.1. Let p be any prime number, G = M (pn ) the modular group of order pn where n ≥ 3, and let K be any ﬁeld containing a primitive pn−2 -th root of unity. Then K(G) is rational over K. Proof. Let ξ be a primitive pn−2 -th root of unity in K. Step 1. Let g∈G K ·x(g) be the representation space of the regular representation of G. Deﬁne ξ −i [x(σ ip ) + x(σ ip τ ) + · · · + x(σ ip τ p−1 )]. v= 0≤i≤pn−2 −1

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Then σ p (v) = ξv and τ (v) = v. Deﬁne xi = σ i v for 0 ≤ i ≤ p − 1. We note that σ : x0 → x1 → · · · → xp−1 → ξx0 and τ : xi → η −i xi , where η = ξ p . Applying Theorem 2.1 we ﬁnd that if K(x0 , x1 , . . . , xp−1 )G is rational over K, then K(G) = K(x(g) : g ∈ G)G is also rational over K. n−3

Step 2. Deﬁne yi = xi /xi−1 for 1 ≤ i ≤ p − 1. Then K(x0 , x1 , . . . , xp−1 ) = K(x0 , y1 , . . . , yp−1 ) and σ : x0 → y1 x0 , y1 → y2 → · · · → yp−1 → ξ/(y1 · · · yp−1 ), τ : x0 → x0 ,

yi → η −1 yi .

By Theorem 2.2, if the ﬁeld K(y1 , . . . , yp−1 )G is rational over K, so is K(x0 , y1 , . . . , yp−1 )G over K. Deﬁne ui = yi /yi−1 for 2 ≤ i ≤ p − 1. Then K(y1 , . . . , yp−1 ) = K(y1 , u2 , . . . , up−1 ) and σ : y1 → y1 u2 , 2 u2 → u3 → · · · → up−1 → ξ/(y1 y2 · · · yp−2 yp−1 ) = ξ/(y1p up−1 up−2 · · · u2p−1 ), 2 3 −1 τ : y1 → η y1 , ui → ui , for 2 ≤ i ≤ p − 1. Thus K(y1 , u2 , . . . , up−1 )<τ > = K(y1p , u2 , . . . , up−1 ). Deﬁne u1 = ξ −1 y1p . Then u1 up2 , σ : u1 → u2 → u3 → · · · → 1/(u1 up−1 · · · u2p−1 ) → u1 up−2 up−3 · · · u2p−2 up−1 → u2 . 2 2 3 Deﬁne w1 = u2 , wi = σ i−1 (u2 ) for 2 ≤ i ≤ p − 1. Then K(u1 , u2 , . . . , up−1 ) = K(w1 , w2 , . . . , wp−1 ). It follows that K(y1 , . . . , yp−1 )G = {K(y1 , . . . , yp−1 )<τ > }<σ> = K(w1 , w2 , . . . , wp−1 )<σ> and

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σ : w1 → w2 → · · · → wp−1 → 1/(w1 w2 · · · wp−1 ). Step 3. Deﬁne T0 = 1 + w1 + w1 w2 + · · · + w1 w2 · · · wp−1 , T1 = (1/T0 ) − (1/p), Ti+1 = (w1 w2 · · · wi /T0 ) − (1/p) for 1 ≤ i ≤ p − 1. Thus K(w1 , . . . , wp−1 ) = K(T1 , . . . , Tp ), with T1 + T2 + · · · + Tp = 0 and σ : T1 → T2 → · · · → Tp−1 → Tp → T0 . Deﬁne si =

1≤j≤p

η −ij T j for 1 ≤ i ≤ p − 1. Then

K(T1 , T2 , . . . , Tp ) = K(s1 , s2 , . . . , sp−1 ) and σ : si → η i si . Clearly K(s1 , . . . , sp−1 )<σ> is rational over K.

Proof (of Theorem 1.8). If p ≥ 3, a non-abelian p-group of order p3 is either of exponent p or contains a cyclic subgroup of index p (see [CK, Theorem 2.3]). The rationality of K(G) of the ﬁrst group follows from Theorem 1.1 while that of the second group follows from Theorem 3.1. If p = 2, the rationality of K(G) is a consequence of Theorem 1.6.

The method used in the proof of Theorem 3.1 can be applied to other groups, e.g., D(2n−1 ), Q(2n ), SD(2n−1 ). The following results will be used in the proof of Theorem 1.7. Theorem 3.2. Let G = D(2n−1 ) or Q(2n ) with n ≥ 4. If K is a ﬁeld containing a primitive 2n−2 -th root of unity, then K(G) is rational over K. n−2 Proof. Let -th root of unity in K. ξ be a primitive 2 Let g∈G K ·x(g) be the representation space of the regular representation of G. Deﬁne ξ −i x(σ 2i ). v= 0≤i≤2n−2 −1

Then σ 2 (v) = ξv. Deﬁne x0 = v, x1 = σ · v, x2 = τ · v, x3 = τ σ · v. We ﬁnd that σ : x0 → x1 → ξx0 , x2 → ξ −1 x3 , x3 → x2 , τ : x0 → x2 → x0 , x1 → x3 → x1 where = 1 if G = D(2n−1 ), and = −1 if G = Q(2n ). By Theorem 2.1 it suﬃces to show that K(x0 , x1 , x2 , x3 )G is rational over K.

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Since σ 2 (xi ) = ξxi for i = 0, 1, σ 2 (xi ) = ξ −1 xj for j = 2, 3, it follows that 2 n−2 K(x0 , x1 , x2 , x3 )<σ > = K(yo , y1 , y2 , y3 ) where y0 = x20 , y1 = x1 /x0 , y2 = x0 x2 , y3 = x1 x3 . The actions of σ and τ are given by n−2

σ : y0 → y0 y12

τ : y0 → y0−1 y22

, y1 → ξ/y1 , y2 → ξ −1 y3 , y3 → ξy2 ,

n−2

, y1 → y1−1 y2−1 y3 , y2 → y2 , y3 → y3 .

Deﬁne n−3

z0 = y0 y12

y2−2

n−4

y3−2

n−4

, z1 = y1 , z2 = y2−1 y3 , z3 = y2 .

We ﬁnd that σ : z0 → −z0 , z1 → ξz1−1 , z2 → ξ 2 z2−1 , z3 → ξ −1 z2 z3 , τ : z0 → z0−1 , z1 → z1−1 z2 , z2 → z2 , z3 → z3 . By Theorem 2.2 it suﬃces to prove that K(z0 , z1 , z2 )<σ,τ > is rational over K. Now we will apply Theorem 2.3 to ﬁnd K(z0 , z1 , z2 )<σ> with a = 1 and b = z2 . Deﬁne a b z0 − z1 − z0 z1 u= , v= . ab ab z0 z1 − z0 z1 − z0 z1 z0 z1 By Theorem 2.3 we ﬁnd that K(z0 , z1 , z2 )<τ > = K(u, v, z2 ). The actions of σ on u, v, z2 are given by σ :z2 → ξ 2 z2−1 , a z0 u → z1 z0 , ξ( − ) bz0 z1 −z0 +

z1 1 − ) z1 b v→ z1 z0 . ξ( − ) bz0 z1 ξ(

Deﬁne w = u/v. Then σ(w) = bw/ξ = z2 w/ξ. Note that a a −z0 + z0 − bu b z0 z0 σ(u) = . = z1 z0 = ξ bz0 2 az ξ(bu − av 2 ) 1 ξ( − ) − bz0 z1 z1 z0 The last equality of the above formula is equivalent to the following identity: a u x = , 2 bx ay bu − av 2 − y x x−

(1)

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where x, y, u, v, a, b are the same as in Theorem 2.3. A simple way to verify Identity (1) goes as follows: The right-hand side of (1) is equal to (y + (b/y) − (1/u))−1 by Theorem 2.3. It is not diﬃcult to check that the left-hand side of (1) is equal to (y + (b/y) − (1/u))−1 . Thus σ(u) = bu/(ξ(bu2 − av 2 )) = z2 u/(ξ(z2 u2 − v 2 )) = z2 w2 /(ξu(z2 w2 − 1)). Deﬁne T = z2 w2 /ξ, X = w, Y = u. Then K(u, v, z2 ) = K(T, X, Y ) and σ : T → T, X → A/X, Y → B/Y where A = T, B = T /(ξT −1). By Theorem 2.3 it follows that K(T, X, Y )<σ> is rational over K(T ). In particular, it is rational over K.

Theorem 3.3. Let G = SD(2n−1 ) with n ≥ 4. If K is a ﬁeld containing a primitive 2n−2 -th root of unity, then K(G) is rational over K. Proof. The case n = 4 is a consequence of [CHK, Theorem 3.2]. Thus we may assume n ≥ 5 in the following proof. The proof is quite similar to that of Theorem 3.2. Deﬁne v, x0 , x1 , x2 , x3 by the same formulas as in the proof of Theorem 3.2. Then σ : x0 → x1 → ξx0 , x2 → −ξ −1 x3 , x3 → −x2 , τ : x0 → x2 → x0 , x1 → x3 → x1 . n−2 Deﬁne y0 = x20 , y1 = x1 /x0 , y2 = x0 x2 , and y3 = x1 x3 . Then K(x0 , x1 , 2 x2 , x3 )<σ > = K(y0 , y1 , y2 , y3 ) and n−2

σ : y0 → y0 y12

τ : y0 → y0−1 y22

, y1 → ξ/y1 , y2 → −ξ −1 y3 , y3 → −ξy2 ,

n−2

, y1 → y1−1 y2−1 y3 , y2 → y2 , y3 → y3 .

Note that the actions of σ and τ are the same as those in the proof of Theorem 3.2 except for the coeﬃcients. Thus we may deﬁne z0 , z1 , z2 , z3 by the same formulas as in the proof of Theorem 3.2. Using the assumption that n ≥ 5, we ﬁnd σ : z0 → −z0 , z1 → ξz1−1 , z2 → ξ 2 z2−1 , z3 → −ξ −1 z2 z3 , τ : z0 → z0−1 , z1 → z1−1 z2 , z2 → z2 , z3 → z3 . By Theorem 2.2 it suﬃces to prove that K(z0 , z1 , z2 )<σ,τ > is rational over K. But the actions of σ, τ on z0 , z1 , z2 are completely the same as those in the proof of Theorem 3.2. Hence the result.

4 Proof of Theorem 1.7 In this section we will complete the proof of Theorem 1.7. Let ζ be a primitive pn−1 -th root of unity. If ζ ∈ K, then Theorem 1.7 is a consequence of

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Theorem 1.2. Thus we may assume that [K(ζ) : K] = p from now on. Let Gal(K(ζ)/K) =< λ > and λ(ζ) = ζ a for some integer a. If p ≥ 3, it is easy to see that a = 1 (mod pn−2 ) and ζ p ∈ K. By Theorem 1.9, the p-group G is isomorphic to M (pn ). Apply Theorem 3.1. We are done. Now we consider the case p = 2. By Theorem 1.9, G is isomorphic to M (2n ), D(2n−1 ), SD(2n−1 ), or Q(2n ). If G is a non-abelian group of order 8, the rationality of K(G) is guaranteed by Theorem 1.6. Thus it suﬃces to consider the case G is a 2-group of order ≥ 16, i.e., n ≥ 4. n−1 =1 Recall that G is generated by two elements σ and τ such that σ 2 and τ −1 στ = σ k where (i) k = −1 if G = D(2n−1 ) or Q(2n ), (ii) k = 1 + 2n−2 if G = M (2n ), (iii) k = −1 + 2n−2 if G = SD(2n−1 ). As before, let ζ be a primitive 2n−1 -th root of unity and Gal(K(ζ)/K) =< λ >, with λ(ζ) = ζ a , where a2 = 1 (mod 2n−1 ). It follows that the only possibilities of a (mod 2n−1 ) are a = −1, ±1 + 2n−2 . It follows that we have four types of groups and three choices for λ(ζ) and thus we should deal with 12 situations. Fortunately many situations behave quite similar. And if we abuse the terminology, we may even say that some situations are “semi-equivariant” isomorphic (but it may not be equivariant isomorphic in the usual sense). Hence they obey the same formulas of changing the variables. After every situation is reduced to a ﬁnal form we may reduce the rationality problem of a group of order 2n (n ≥ 4) to that of a group of order 16. Let g∈G K ·x(g) be the representation space of the regular representation of G. We will extend the actions of G and λ to g∈G K(ζ) · x(g) by requiring ρ(ζ) = ζ and λ(x(g)) = x(g) for any ρ ∈ G. Note that K(G) = K(x(g) : g ∈ K(ζ)(x(g) : g ∈ G) G)G = {K(ζ)(x(g) : g ∈ G)<λ> }G = . We will ﬁnd a faithful subspace 0≤i≤3 K(ζ) · xi of g∈G K(ζ) · x(g) such that K(ζ)(x0 , x1 , x2 , x3 ) (y1 , . . . , y12 ) is rational over K where each yi is ﬁxed by G and λ. By Theorem 2.1, K(ζ)(x(g) : g ∈ G) = K(ζ)(x0 , x1 , x2 , x3 ) (X1 , . . . , XN ) where N = 2n − 4 and each Xi is ﬁxed by G and λ. It follows that K(G) is rational provided that K(ζ)(x0 , x1 , x2 , x3 ) (y1 , . . . , y12 ) is rational over K. Deﬁne v1 =

0≤j≤2n−1 −1

ζ −j x(σ j ),

v2 =

0≤j≤2n−1 −1

ζ −aj x(σ j )

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159

where a is the integer with λ(ζ) = ζ a . We ﬁnd that σ : v1 → ζv1 , v2 → ζ a v2 , λ : v1 → v2 → v1 . Deﬁne x0 = v1 , x1 = τ · v1 , x2 = v2 , x3 = τ · v2 . It follows that σ : x0 → ζx0 , x1 → ζ k x1 , x2 → ζ a x2 , x3 → ζ ak x3 , λ : x0 → x2 → x0 , x1 → x3 → x1 , ζ → ζ a , x1 → x0 , x2 → x3 → x2 , τ : x0 → τ λ : x0 → x3 → x0 , x1 → x2 , x2 → x1 , ζ → ζ a where (i) = 1 if G = Q(2n ), and (ii) = −1 if G = Q(2n ). Case 1. k = −1, i.e., G = D(2n−1 ) or Q(2n ). Throughout the discussion of this case, we will adopt the convention that

= 1 if G = D(2n−1 ), while = −1 if G = Q(2n ). Subcase 1.1. a = −1, i.e., λ(ζ) = ζ −1 . It is easy to see that n−1

K(ζ)(x0 , x1 , x2 , x3 )<σ> = K(ζ)(x20 Deﬁne

n−1

y0 = x20

, x0 x1 , x0 x2 , x1 x3 ).

, y1 = x0 x1 , y2 = x0 x2 , y3 = x1 x3 .

It follows that λ : y0 → y0−1 y22

, y1 → y1−1 y2 y3 , y2 → y2 , y3 → y3 , ζ → ζ −1 ,

τ : y0 → y0−1 y12

, y1 → y1 , y2 → y3 → y2 .

n−1

n−1

Deﬁne z0 = y0 y1−2

n−2

y2−2

n−3

n−3

y32

, z1 = y2 y3 , z2 = y2 , z3 = y1 .

We ﬁnd that λ : z0 → 1/z0 , z1 → z1 , z2 → z2 , z3 → z1 /z3 , ζ → ζ −1 , 1/z0 , z1 → z1 , z2 → z1 /z2 , z3 → z3 . τ : z0 → It turns out the parameter n does not come into play in the actions of λ and τ on z0 , z1 , z2 , z3 . By Theorem 2.1 K(G) = K(ζ)(z0 , z1 , z2 , z3 )<λ,τ > (X1 , . . . , XN ) where N = 2n − 4 and λ(Xi ) = τ (Xi ) = Xi for 1 ≤ i ≤ N . By Lemma 2.4 K(ζ) = K(ζ4 ) where λ(ζ4 ) = ζ4−1 . Thus K(G) = K(ζ4 )(z0 , z1 , z2 , z3 )<λ,τ > (X1 , . . . , XN ). Denote G4 = D(8) or Q(16). Then K(G4 ) = K(ζ4 )(z0 , z1 , z2 , z3 )<λ,τ > (X1 , . . . , X12 ).

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Since K(G4 ) is rational over K by Theorem 1.6 (see [Ka1, Theorem 1.3]), it follows that K(ζ4 )(z0 , . . . , z3 )<λ,τ > (X1 , . . . , X12 ) is rational over K. Thus K(ζ4 ) (z0 , . . . , z3 )<λ,τ > (X1 , . . . , XN ) is rational over K for N = 2n − 4. The last ﬁeld is nothing but K(G). Done. Subcase 1.2. a = −1 + 2n−2 , i.e., λ(ζ) = −ζ −1 . The actions of σ, τ, λ, τ λ are given by σ : x0 → ζx0 , x1 → ζ −1 x1 , x2 → −ζ −1 x2 , x3 → −ζx3 , x2 → x0 , x1 → x3 → x1 , ζ → −ζ −1 , λ : x0 → x1 → x0 , x2 → x3 → x2 , τ : x0 →

.

τ λ : x0 → x3 → x0 , x1 → x2 , x2 → x1 , ζ → −ζ −1 x3 . Then Deﬁne y0 = x20 , y1 = x0 x1 , y2 = x2 x3 , y3 = x−1−2 0 K(ζ)(x0 , . . . , x3 )<σ> = K(ζ)(y0 , . . . , y3 ). Consider the actions of τ λ and τ on K(ζ)(y0 , . . . , y3 ). We ﬁnd that n−2

n−1

n−2

τ λ : y0 → y01+2

τ : y0 → y0−1 y12

n−1

y32

n−1

, y1 → y2 → y1 , y3 → y0−1−2

n−3

y3−1−2

n−2

, y1 → y1 , y2 → y2 , y3 → y1−1−2

n−2

, ζ → −ζ −1 ,

y2 y3−1 .

Deﬁne z0 = y1 , z1 = y1−1 y2 , z2 = y0 y1 y2−1 y32 , z3 = y01+2

n−4

y1−2

n−4

y2−2

n−4

n−3

y31+2

.

We ﬁnd τ λ : z0 → z0 z1 , z1 → 1/z1 , z2 → 1/z2, z3 → z1−1 z2−1 z3 , ζ → −ζ −1 , τ : z0 → z0 , z1 → z1 , z2 → 1/z2 , z3 → z1 /z3 . By Lemma 2.4 we may replace K(ζ) in K(ζ)(z0 , z1 , z2 , z3 )<τ λ,τ > by K(ζ4 ) where τ λ(ζ4 ) = ζ4−1 . Then we may proceed as in Subcase 1.1. The details are omitted. Subcase 1.3. a = 1 + 2n−2 , i.e., λ(ζ) = −ζ. Note that ζ 2 ∈ K and ζ 2 is a primitive 2n−2 -th root of unity. Thus we may apply Theorem 3.2. Done. Case 2. k = 1 + 2n−2 , i.e., G = M (2n ). Subcase 2.1. a = −1, i.e., λ(ζ) = ζ −1 . The actions of σ, τ, λ, τ λ are given by σ : x0 → ζx0 , x1 → −ζx1 , x2 → ζ −1 x2 , x3 → −ζ −1 x3 , x2 → x0 , x1 → x3 → x1 , ζ → ζ −1 , λ : x0 → x1 → x0 , x2 → x3 → x2 , τ : x0 → τ λ : x0 → x3 → x0 , x1 → x2 → x1 , ζ → ζ −1 .

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Deﬁne X0 = x0 , X1 = x2 , X2 = x3 , X3 = x1 . Then the actions of σ, τ, λ on X0 , X1 , X2 , X3 are the same as those of σ, τ λ, τ on x0 , x1 , x2 , x3 in Subcase 1.2 for D(2n−1 ) except on ζ. Thus we may considerK(ζ)(X0 , X1 , X2 , X3 )<σ,τ,λ> (Y1 , . . . , Y12 ). Hence the same formulas of changing the variables in Subcase 1.2 can be copied and the same method can be used to prove that K(ζ)(X0 , X1 , X2 , X3 )<σ,τ,λ> (Y1 , . . . , Y12 ) is rational over K. Subcase 2.2. a = −1 + 2n−2 , i.e., λ(ζ) = −ζ −1 . The actions of σ, τ, λ, τ λ are given by σ : x0 → ζx0 , x1 → −ζx1 , x2 → −ζ −1 x2 , x3 → ζ −1 x3 , x2 → x0 , x1 → x3 → x1 , ζ → −ζ −1 , λ : x0 → x1 → x0 , x2 → x3 → x2 , τ : x0 → τ λ : x0 → x3 → x0 , x1 → x2 → x1 , ζ → −ζ −1 . Deﬁne X0 = x0 , X1 = x3 , X2 = x2 , X3 = x1 . Then the actions of σ, τ, τ λ on X0 , X1 , X2 , X3 are the same as those of σ, τ λ, τ on x0 , x1 , x2 , x3 in Subcase 1.2 for D(2n−1 ). Hence the result. Subcase 2.3. a = 1 + 2n−2 , i.e., λ(ζ) = −ζ. Apply Theorem 3.1. Case 3. k = −1 + 2n−2 , i.e., G = SD(2n−1 ). Subcase 3.1. a = −1, i.e. λ(ζ) = ζ −1 . The actions of σ, τ, λ, τ λ are given by σ : x0 → ζx0 , x1 → −ζ −1 x1 , x2 → ζ −1 x2 , x3 → −ζx3 , λ : x0 → x2 → x0 , x1 → x3 → x1 , ζ → ζ −1 , τ : x0 → x1 → x0 , x2 → x3 → x2 , τ λ : x0 → x3 → x0 , x1 → x2 → x1 , ζ → ζ −1 . Deﬁne X0 = x0 , X1 = x2 , X2 = x1 , X3 = x3 . Then the actions of σ, τ λ, λ on X0 , X1 , X2 , X3 are the same as those of σ, τ λ, τ on x0 , x1 , x2 , x3 in Subcase 1.2 for D(2n−1 ) except on ζ. Done. Subcase 3.2. a = −1 + 2n−2 , i.e., λ(ζ) = −ζ −1 . n−2 n−1 −1 x1 , y2 = x−1 Deﬁne y0 = x20 , y1 = x1+2 1 x2 , y3 = x0 x3 . Then 0 K(ζ)(x0 , x1 , x2 , x3 )<σ> = K(ζ)(y0 , y1 , y2 , y3 ) and

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τ : y0 → y0−1−2

n−2

n−1

τ λ : y0 → y0 y32

n−1

y12

, y1 → y0−1−2

n−3

n−2

, y1 → y1 y2 y31+2

Deﬁne z0 = y01+2 y1−2 y2−2 y2 , z3 = y2−1 y3 . It follows that n−3

n−2

n−3

, y2 → y3 → y2 ,

, y2 → y2−1 , y3 → y3−1 , ζ → −ζ −1 .

n−3

y32

n−2

y11+2

n−4

, z1 = y02

n−3

y11−2

y2−2

n−4

n−4

y32

, z2 =

K(ζ)(y0 , y1 , y2 , y3 ) = K(ζ)(z0 , z1 , z2 , z3 ) and τ : z0 → 1/z0 , z1 → z1 /z0 , z2 → z2 z3 , z3 → 1/z3 , τ λ : z0 → z0 , z1 → z1 z22 z3 , z2 → 1/z2 , z3 → 1/z3 , ζ → −ζ −1 . Thus we can establish the rationality because we may replace K(ζ) by K(ζ4 ) as in Subcase 1.2. Subcase 3.3. a = 1 + 2n−2 , i.e., λ(ζ) = −ζ. It suﬃces to apply Theorem 3.3. Thus we have ﬁnished the proof of Theorem 1.7.

References [AHK] H. Ahmad, M. Hajja, M. Kang, Rationality of some projective linear actions, J. Algebra 228 (2000) 643–658. [Bo] F. A. Bogomolov, The Brauer group of quotient spaces by linear group actions, Math. USSR Izv. 30 (1988) 455–485. [CHK] H. Chu, S. J. Hu, M. Kang, Nother’s problem for dihedral 2-groups, Comment. Math. Helv. 79 (2004) 147–159. [CK] H. Chu, M. Kang, Rationality of p-group actions, J. Algebra 237 (2001) 673–690. [DM] F. DeMeyer, T. McKenzie, On generic polynomials, J. Algebra 261 (2003) 327–333. [GMS] S. Garibaldi, A. Merkurjev, J. P. Serre, Cohomological invariants in Galois cohomology, AMS Univ. Lecture Series Vol. 28, Amer. Math. Soc., Providence, 2003. [Ka1] M. Kang, Noether’s problem for dihedral 2-groups II, to appear in Paciﬁc J. Math.. [Ka2] M. Kang, Noether’s problem for metacyclic p-groups, Advances in Math. 203 (2006) 554–567. [Sa1] D. J. Saltman, Noether’s problem over an algebraically closed ﬁeld, Invent. Math. 77 (1984) 71–84. [Sa2] D. J. Saltman, Galois groups of order p3 , Comm. Algebra 15 (1987) 1365– 1373. [Su] M. Suzuki, Group theory II, Grundlehren Math. Wiss. Vol. 248, SpringerVerlag, Berlin, 1986. [Sw] R. G. Swan, Noether’s problem in Galois theory, in “Emmy Noether in Bryn Mawr”, edited by B. Srinivasan and J. Sally, Springer-Verlag, Berlin, 1983.

Generalized Homological Mirror Symmetry and Rationality Questions Ludmil Katzarkov Department of Mathematics University of Miami Coral Gables, FL 33124, USA [email protected] Summary. We study geometric consequences of Homological Mirror Symmetry, with special regard to rationality questions.

Key words: Homological Mirror Symmetry, birational geometry, rationality 2000 Mathematics Subject Classiﬁcation codes: 14J32, 14E08

1 Introduction The goal of this paper is to geometrize Homological Mirror Symmetry (HMS) in order to get some geometric consequences. We introduce the idea of Landau–Ginzburg models with several potentials and show how to exchange the role of a divisor and a potential. The novelty of the paper is in introducing several new approaches for studying the connection of HMS with coverings, linear systems, taking divisors out. We also suggest ways to employ even further Hodge theory and higher categories in HMS. The paper is somewhat of an appendix to [KKP08a]—we try to run some algebro-geometric constructions through HMS to extract information for both the A and B sides. We explain in some detail how Birational Geometry interacts with HMS with hope that we or others will look at possible applications of the constructions outlined here in the future. The geometric applications we have in mind are rationality questions. We outline a new approach of studying nonrationality of algebraic varieties based on HMS, categories, and vanishing cycles. We give several examples. Acknowledgment. We are grateful to D. Auroux, V. Golyshev, M. Gross, T. Pantev, P. Seidel, D. Orlov, M. Kontsevich, A. Kuznetsov, V. Przyjalkowski

F. Bogomolov, Y. Tschinkel (eds.), Cohomological and Geometric Approaches to Rationality Problems, Progress in Mathematics 282, DOI 10.1007/978-0-8176-4934-0_7, © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

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for many useful conversations. Many thanks go to V. Boutchaktchiev without whom this paper would not have been written. We are grateful to IHES, ESI, and EPFL for the support, This paper came out of a talk given in Augsburg in 2007. More details will appear elsewhere. This work was partially supported by NSF Grant DMS0600800, by NSF FRG DMS-0652633, FWF grant P20778, and ERC grant.

2 Generalized HMS Mirror symmetry was introduced as a duality between two N = 2 superconformal ﬁeld theories. Historically, the ﬁrst version of the HMS conjecture was formulated for Calabi–Yau manifolds. Generalizations to symplectic manifolds with nonzero ﬁrst Chern class, and the role of Landau–Ginzburg models as their mirrors, appeared soon afterward, ﬁrst in mathematics and then in physics. From a mathematical point of view, this generalization covers a variety of issues of vastly diﬀerent complexity. We ﬁrst recall some basic notions and facts about the categories we are going to use. These categories and their modiﬁcations will play an important role throughout the paper. The following deﬁnition is due to P. Seidel. Historically the idea was introduced ﬁrst by M. Kontsevich and later studied by K. Hori. We begin by brieﬂy reviewing Seidel’s construction of a Fukaya-type A∞ -category associated to a symplectic Lefschetz pencil. Let (X, ω) be an open symplectic manifold of dimension dimR X = 2n. Let f : X → C be a symplectic Lefschetz ﬁbration, i.e., f is a C ∞ complexvalued function with isolated nondegenerate critical points p1 , . . . , pr near which f is given in adapted complex local coordinates by f (z1 , . . . , zn ) = f (pi ) + z12 + · · · + zn2 , and whose ﬁbers are symplectic submanifolds of X. Fix a regular value λ0 of f , and consider an arc γ ⊂ C joining λ0 to a critical value λi = f (pi ). Using the horizontal distribution which is symplectic orthogonal to the ﬁbers of f , we can transport the vanishing cycle at pi along the arc γ to obtain a Lagrangian disc Dγ ⊂ X ﬁbered over γ, whose boundary is an embedded Lagrangian sphere Lγ in the ﬁber Σ0 = f −1 (λ0 ). The Lagrangian disc Dγ is called the Lefschetz thimble over γ, and its boundary Lγ is the vanishing cycle associated to the critical point pi and to the arc γ. Let γ1 , . . . , γr be a collection of arcs in C joining the reference point λ0 to the various critical values of f , intersecting each other only at λ0 , and ordered in the clockwise direction around p0 . Each arc γi gives rise to a Lefschetz thimble Di ⊂ X, whose boundary is a Lagrangian sphere Li ⊂ Σ0 . We can assume that these spheres intersect each other transversely inside Σ0 - see [Sei07]. Definition 2.1 (Seidel). The directed Fukaya category F S(f ; {γi }) is the following A∞ -category (over a coeﬃcient ring R): the objects of F S(f ; {γi })

Mirror Symmetry and Rationality Questions

165

are the Lagrangian vanishing cycles L1 , . . . , Lr ; the morphisms between the objects are given by ⎧ ∗ |Li ∩Lj | ⎪ if i < j ⎨CF (Li , Lj ; R) = R Hom(Li , Lj ) = R · id if i = j ⎪ ⎩ 0 if i > j; and the diﬀerential m1 , composition m2 , and higher order products mk are deﬁned in terms of Lagrangian Floer homology inside Σ0 . More precisely, mk : Hom(Li0 , Li1 ) ⊗ · · · ⊗ Hom(Lik−1 , Lik ) → Hom(Li0 , Lik )[2 − k] is trivial when the inequality i0 < i1 < · · · < ik fails to hold (i.e., it is always zero in this case, except for m2 where composition with an identity morphism is given by the obvious formula). When i0 < · · · < ik , mk is deﬁned by ﬁxing a generic ω-compatible almostcomplex structure on Σ0 and counting pseudo-holomorphic maps from a disc with k + 1 cyclically ordered marked points on its boundary to Σ0 , mapping the marked points to the given intersection points between vanishing cycles, and the portions of boundary between them to Li0 , . . . , Lik , respectively. Let X be a complex algebraic variety (or a complex manifold). Denote by OX the sheaf of regular functions (or the sheaf of holomorphic functions). Coherent sheaves form an abelian category which will be denoted by Coh(X). Following [Orl04] we deﬁne: Definition 2.2 (Orlov). The derived category Db (Y, w) of a holomorphic potential w : Y → C is deﬁned as the direct sum ( b Db (Y, w) := Dsing (Yt ) t b (Yt ) of all ﬁbers Yt := w−1 (t) of of the derived categories of singularities Dsing w. b (Yt ) = Db (Yt )/ Perf(Yt ) is deﬁned as the quotient catThe category Dsing egory of derived category of coherent sheaves Db (Coh(Yt )) on Yt modulo the b (Yt ) full triangulated subcategory of perfect complexes on Yt . Note that Dsing is nontrivial only for singular ﬁbers Yt .

Here is a schematic picture of the version of HMS discussed above. The table below lists all geometric objects involved, and the categories associated to them:

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A-models (symplectic)

B-models (algebraic)

X = (X, ω) a closed symplectic manifold

X a smooth projective variety

Fukaya category Fuk(X): Derived category Db (X): objects are Lagrangian submanobjects are complexes of coherifolds L (maybe equipped with ent sheaves E, morphisms are ﬂat line bundles), morphisms Ext∗ (E0 , E1 ) are given by Floer cohomology HF ∗ (L0 , L1 ) 7 gOOO OOO ooooo O o O o ooo OOOOO wooo ' Y noncompact symplectic manifold, with a proper map W : Y → C which is a symplectic ﬁbration with singularities Relative Fukaya category Fuk(W ): objects are Lagrangian submanifolds L ⊂ Y which, at inﬁnity, are ﬁbered over R+ ⊂ C. The morphisms are HF ∗ (L+ 0 , L1 ), where the superscript + indicates a perturbation removing intersection points at inﬁnity

Y smooth quasi-projective variety, with a proper holomorphic map W : Y → C b The category Dsing (W ) of algebraic B-branes, obtained by considering the singular ﬁbers Yz = W −1 (z), and dividing Db (Yz ) by the subcategory of perfect complexes Perf (Yz ), then taking the direct sum over all such z

The smooth quasi-projective variety Y , with a proper holomorphic map W : Y → C, is often called the Landau–Ginzburg model or Landau–Ginzburg mirror of X. We adapt the setting of Landau–Ginzburg models to the toric degeneration approach studied by Gross and Siebert in the Calabi–Yau case [Gro04]. We outline a construction of Landau–Ginzburg models in the Fano setting by introducing some geometric ideas. Step 1. We consider a toric degeneration of the Fano manifold X to a conﬁguration X of toric varieties intersecting along toric divisors. This degenerate conﬁguration can be viewed as a Lagrangian ﬁbration X → B, whose base carries a natural aﬃne structure encoding the monodromy of the ﬁbration [Gro04]. Step 2. Following [Gro04] we apply a Legendre transform to X → B. This procedure roughly consists of taking the dual intersection complex and spec-

Mirror Symmetry and Rationality Questions

167

ifying normal fan structures and the corresponding monodromies at vertices. In this way we obtain a dual Lagrangian ﬁbration Y → B , where Y is a conﬁguration of toric varieties and additionally carries a complex-valued function W . Step 3. We smooth and partially compactify (Y , W ) in order to obtain the Landau–Ginzburg mirror W : Y → C. Recall that the mirror Y can in many cases be viewed as a moduli space of (complexes of) Lagrangians in X, and the superpotential can be interpreted as a Floer theory obstruction. Roughly speaking, the above procedure is motivated by the observation that the toric degeneration X determines a distinguished family of Lagrangian tori in X; the behavior of pseudo-holomorphic discs in X under this degeneration should then determine the behavior of the superpotential W under the mirror degeneration—see [Aur07]. This construction allows us to see HMS in purely geometric terms. 2.1 HMS for pairs We enhance now HMS by supplementing the above setup with additional categorical and geometric structures. B side (X, D)

A side (Y, Y∞ , W )

Db (D) i∗ ↑↓ i∗ Db (X) ∪ b Dcompact (X \ D)

Fuk(Y∞ ) i∗ ↑↓ i∗ F S(Y, W ) ∪ F S compact (Y, W )

∩ Db (X \ D)

∩ Fukwrapped (Y )

support

Lagrangians

Table 1. HMS for Fanos

In Table 1 we make HMS depend on additional geometric data: a divisor D on the B side and a modiﬁed LG mirror on the A side. This modiﬁcation will be discussed in the next section. The wrapped Fukaya category appearing in the table has its Lagrangians going many times around given, vertical with respect to W , divisor and in such a way it records an additional ﬁltration. On the B side this corresponds to the log mixed Hodge structure associated

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L. Katzarkov

with X \ D . These additional geometric data and categorical correspondences allow us to relate HMS to Hodge theory and draw geometric conclusions—see the next sections. Finally Y∞ is a marked smooth ﬁber of f : Y → C but not the ﬁber at inﬁnity of the Landau Ginzburg model. Table 2 reverses some of the correspondences in Table 1: A side (X, D) Fuk

compact (X Lagrangians

B side (Y, W ) b Dcompact (Y )

\ D)

support

b Dsing (Y )

Fukwrapped (X \ D) Table 2. HMS revisited

The degeneration procedure described in Steps 1–3 allows us to extend the construction of Landau Ginzburg Mirrors beyond the usual settings of complete intersections in toric varieties. Also the enhanced HMS sketched in Table 1 allows us to extract new information about these Landau Ginzburg Mirrors. In particular, the smoothing procedure in Step 3 can be seen from Table 3. Z P + Q = 0 Desingularised LG

A

ωα → ∞

P Q

Smoothing LG

Smoothing the aﬃne structure

A

Compactifying

Table 3. Smoothing in diﬀerent moduli directions

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169

To the pencil P + Q representing a smoothing on the B side we associate changing behavior at inﬁnity under compactiﬁcation of the symplectic form on the A side. Here the choice of direction of deformation Q on the B side corresponds to the volume ) ω α → ∞. A

X\D W1 +W2 W2 W1

C

2

Fig. 1. Iterated Landau-Ginzburg models

Figure 1 demonstrates how HMS and Landau–Ginzburg models with a single potential can be iterated as a sequence of Landau–Ginzburg models with many potentials. In particular the Landau–Ginzburg model for a projective space (and many Fanos) can be represented as a sequence of pencils of elliptic curves and then smoothed as in Step 3. Table 4 represents the iterated LG model for P3 as a sequence of two quadrics. The symplectic meaning of the procedure is as follows. We take ﬁrst P3 \Q1 and then ﬁnd a special Lagrangian submanifold in it. The moduli space of special Lagrangian deformations of this special Lagrangian submanifold produces an LG model which has ﬁbers mirrors to Q2 , the potential is given by all holomorphic discs with boundary on Q2 . The mirrors of Q2 have Kodaira ﬁbers I8 at ∞. As a result we have eight double points and two I8 which produce the mirror of the quartic K3 in P3 - the singular ﬁber of the potential W1 + W2 —see Table 4. In the case of nonrational Fanos the picture is much more complicated—see the next sections. Remark 2.3. This construction is an attempt to ﬁnd a geometric explanation of the work of Golyshev [Gol05]. Table 5 represents a triple iterated procedure for P3 , where we use hyperplanes instead of quadrics. We proceed with several other examples demonstrating and extending the HMS correspondences. We start with a generic toric correspondence—a component on the boundary corresponds to a monomial in the potential—see Tables 6 and 7.

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L. Katzarkov LG(P3 ) → C2

(P3 , Q1 ∩ Q2 = E) W2 I8

I8 W1

Table 4. New cube

(P3 , H1 ∪ H2 ∪ H3 ∪ H4 ) (P3 , 4P2 )

(P2 , 3H)

LG(P3 ) → C4 4 C∗3 → C « „ c (x, y, z) → x, y, z, xyz

3 C∗2 → C « „ c (x, y) → x, y, xy

C∗ → “ C2 ” c z → z, z

(P1 , 2 pts)

Table 5. Triple iterated LG procedure for P3

“ X ” PΔ , Di

“

C∗ , W =

X

zi∨

Table 6. Boundaries: monomial correspondences

”

Mirror Symmetry and Rationality Questions

171

V(Y) V(X)

(C2 , V (X) ∪ V (Y ))

(C2 , W = X + Y )

2

(C2 , X, V (Y ))

2

(C2 , V (X), Y )

(C , V (X), Y ) (C , X, V (Y )) Table 7. C2 self mirror

Table 7 demonstrates how C2 mirrors itself and the monomials X, Y correspond to boundary divisors V (X), V (Y ).

A

1

P1 A1

(C2 , A1 ∪ P1 ∪ A1 , W = X + Y )

(C2 , V (X) ∪ V (Y ), W = X + Y + XY )

Table 8. The blowup of C2 and its mirror

Table 8 demonstrates how blowing up C2 and adding an additional boundary component leads to adding an additional term in the potential. In fact, Tables 6,7, and 8 demonstrate how divisors and additional potentials exchange role under HMS. We concentrate on the case of HMS for CP2 . ˜ on CP2 \ D leads to removing Table 9 explains how changing the potential h divisors from E(1)—an elliptic ﬁbration with 12 singular ﬁbers. Here D is the divisor of three lines in CP2 . Now suppose that D splits as in Figure 2. This leads to several diﬀerent HMS correspondences summarized in Table 10.

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L. Katzarkov

(CP2 \ D, h) “

9 [

E(1),

! Cj

j=1

” ˜ CP2 \ D, h

E(1) \

ﬁber = 8 − k punctured C∗

k [

Cj ,

j=1

!

9 [

Cj

j=1

Table 9. Changing the potential on CP2 \ D leads to removing divisors from E(1)

D1 D D2

Fig. 2. D = D1 + D2

P2 \ D

w =x+y+

1 on C∗2 xy

(P2 \ D1 , D2 )

w = x + y on C∗2

D = D1 + D2

ﬁber F1

(P2 \ D2 , D1 )

w = x on C∗2

D = D1 + D2

ﬁber F2

Table 10. Mirror symmetry for CP2 with diﬀerent divisors

As before, taking diﬀerent divisors from CP2 leads to changing the potentials and having diﬀerent ﬁbers of the Landau–Ginzburg model—see Figure 3. Table 11 demonstrates one more time how removing diﬀerent divisors on the B side leads to diﬀerent potentials on the A side. We also conjecture an isomorphism of Hochshild homology on side B and Symplectic Homology (see [Sei07]) on the A side.

Mirror Symmetry and Rationality Questions

173

F2

F1

Fig. 3. The ﬁbers F1 and F2

(X \ D1 ) rel D2

W2 - discs hitting D2 . ﬁbers mirror to D2

(X \ D2 ) rel D1

W1 - discs hitting D1 . ﬁbers mirror to D1

Db (X \ D) H H (Db (X \ D))

Fukwrapped (Y ) SH 0 (Y )

Table 11. X \ D, D = D1 ∪ D2 , W = W1 + W2

2.2 Linear systems and divisors In this section we explain how to extend the idea of ﬁnding a mirror for X \ D further. Consider a Calabi–Yau manifold X and a strict normal crossing divisor D in it. To follow the situation we have studied before we assume X is an anticanonical section in a Fano manifold M . Then we take the blowup of M × C along X × 0 , which gives an LG mirror Y . Now we can remove an anticanonical divisor from M , to get an open manifold M and look at the blowup of M ×C along X ×0, where X is the part of X that lies in M . Then the mirror Y is the same space but with a diﬀerent superpotential with fewer monomial terms, and in particular the singular locus of the superpotential, i.e., X \ D, will be more degenerate since the superpotential itself is more degenerate. The new singular locus Y is the mirror to X —see Table 12. To demonstrate the above discussion we analyze some examples. First take the mirror of CP2 \ (at most 3 lines). It has a (partially wrapped) mirror LG model, in which (R+ )2 is an element of the Fukaya category that corresponds to the structure sheaf on CP2 \ (at most 3 lines). Observe that the more lines

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L. Katzarkov Y4

K3

Y2

Y3

Y1

Base

Mirror

D

Db (X \ D)

Fukwrapped (Yi )

Fuk(X \ D)

b Dsing (Yi )

lim Db (X \ D) −→

lim Fukwrapped (Yi )

AllD

Yi

Table 12. K3 and mirror

we remove in CP2 , the more terms we delete from the superpotential. Then we have functors between these Fukaya–Seidel categories and can take limits. The algebra End((R+ )2 ) gets more and more localized: it is just C when the superpotential is W = x + y + 1/xy, i.e., the mirror to CP2 ; it becomes C[x, y] when the superpotential is W = x + y i.e., the mirror to CP2 \ line; then C[x±1 , y] when the superpotential is W = x, i.e., the mirror to CP2 \ (2 lines) and ﬁnally C[x±1 , y ±1 ] when W = 0, i.e., mirror to CP2 \ (3 lines)— see Figures 6 and 7. We can also build a mirror of CP2 \ (4 lines) in which case we get a four dimensional LG model W = xyzt on C4 . Instead we concentrate on the example of CP1 minus points, which will be treated in many details leading to many diﬀerent Hodge theoretic observations. We start with the simplest example—the mirror of CP1 \ (2 points) is C∗ . The mirror of CP1 \ 3 points is a three-dimensional LG model W = xyz on C3 , and the mirror of CP1 \ (more points) is a more complicated LG model on a 3-fold. Geometrically it could be realized as follows: ﬁrst we embed CP1 into CP1 × CP1 as a (1, n)-curve and then intersect with (C∗ )2 to get a CP1 \ (2n + 2 points). Then we blow up (C∗ )2 × C along this punctured rational curve in order to get a a three-dimensional LG mirror. The mirror of the structure sheaf of the punctured CP1 is going to be a thimble ﬁbering over many intervals, which because of wrapping will have a very large endomorphism ring—C[x] localized at many points, see Table 13. In general one can build mirrors to any smooth variety V that embeds in a toric variety X by blowing up X × C along V × {0}. We can ﬁnd mirrors to some open subset V0 inside V obtained by intersecting it with X0 , where X0 is the complement of some toric strata in X. It is easy to see how to take limits of the corresponding F S categories in these situations.

Mirror Symmetry and Rationality Questions

P 1 \ 3 pts

175

w=xyz

P 1 \ 4 pts

Table 13. P1 minus points

Remark 2.4. It seems certain that following the procedure in Table 12 one should always be able to construct mirrors for pairs X – Fano, D – divisor or even in more general situations—see SYZ for open Fanos in Section 2.3. Now we restrict ourselves to the case of Fano manifolds with rank of the Picard group equal to one. Recall that if X is a Fano manifold with Pic(X) = Z, then the index nX of X is the biggest positive integer n such that −KX is divisible by n in Pic(X) = Z. Theorem 2.5. ([Kat08b]) Let X1 , X2 be Fano manifolds with Pic(X1 ) = Pic(X2 ) = Z with indexes nX1 and nX2 , respectively. Suppose that nX2 divides nX1 . Then the corresponding LG mirrors (Y1 , W1 ) and (Y2 , W2 ) have the same general ﬁber but the monodromy of (Y2 , W2 ) is more complicated. More precisely the local system on the middle dimensional cohomology of the smooth ﬁbers of W1 : Y1 → C is a pullback of the corresponding local system for W2 : Y2 → C. We consider some known examples, which demonstrate the above theorem. First we look at: Example 2.6. We discuss the LG model mirror to CP3 and CP3 blown up at a point. The LG mirror of CP3 given by the equation w = x+y+z+

1 xyz

is a family of quartic K3 surfaces and has four singular ﬁbers. The mirror of the blown-up CP3 is a family of K3 surfaces with six singular ﬁbers. Four of these ﬁbers correspond to the LG mirror of CP3 and they are situated near zero. The two remaining ﬁbers are sitting over the second roots of unity in the local chart around ∞—see Figure 4. Now we look at the LG mirror of the three-dimensional quartic. It is given by the equation (x1 + x2 + x3 + x4 )4 = λx1 x2 x3 x4 . The Fano manifold CP3 has index four. The LG mirror of CP3 descends to LG mirror of the three-dimensional quartic. As can be seen from the equations

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L. Katzarkov

Fig. 4. LG model of CP3 blown-up at a point

the ﬁbers are still mirrors of quartics. The diﬀerence is a new singular ﬁber with deeper singularity and nonunipotent monodromy. Example 2.7. Let us look at the Landau–Ginzburg model mirror of the manifold X which is the blow-up of CP3 in a genus two curve embedded as a (2, 3) curve on a quadric surface. We describe the mirror of X. It is a deformation of the image of the compactiﬁcation D of the map (t : u1 : u2 : u3 ) → (t : u1 : u2 : u3 : u1 · u2 : u2 · u3 : u1 · u3 ) of CP3 in CP7 . The function w gives a pencil of degree 8 on the compactiﬁed D. We can interpret this pencil as obtained by ﬁrst taking the Landau–Ginzburg mirror of CP3 and then adding to it a new singular ﬁber over zero consisting of three rational surfaces. The potential in the equation above is w = t + u1 + u2 + u3 +

1 1 1 + + . u1 · u2 u2 · u3 u1 · u3

Clearing denominators and substituting t = u3 = 0 and u21 = u32 we get a singularity. Its modiﬁcation produces the LG mirror of intersection of two quadrics in CP5 , a Fano manifold of index two. Its Landau Ginzburg mirror produces by descent a Landau Ginzburg mirror of intersection of three quadrics in CP6 a Fano manifold of index one. Its LG mirror is given by (x1 + x2 )2 · (x3 + x4 )2 · (x5 + x6 )2 = λx1 x2 x3 x4 x5 x6 . As can be seen from the equations the ﬁbers are in both cases mirrors of K3’s of degree 8. The diﬀerence is that the singular ﬁber over zero has deeper singularity and nonunipotent monodromy. We will use this to deﬁne a nonrationality invariant.

Mirror Symmetry and Rationality Questions

177

Fig. 5. LG of blowup of CP3 in a genus 2 curve

Example 2.8. In the table below we discuss one more example to make a point of our covering construction and HMS—the example of two-dimensional cubic. The details of HMS for two-dimensional cubic can be found in [AKO06]. Here we ﬁrst deform the two-dimensional cubic to weighted projective P2w which covers three to one P2 then we do the HMS construction described at the beginning of the section and then we deform back to get the Landau– Ginzburg mirror for a generic two-dimensional cubic. Similar procedure works for three and four-dimensional cubics and we will make use of it in Section 4.

cubic

/o /o /o /o / P2w

` ´ ⊂ LG P2

3:1

P2

0

0

3:1

WP2

` ´ ⊂ LG P2w

C

CP2w Moduli of cubics

Moduil

0

3:1

0

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L. Katzarkov quasi-unipotent X4 ⊂ P4 P6 (2, 2, 2) V10

unipotent pullback P3 5 P (2, 2) V5

Table 14. Fanos and monodromies

2.3 Localized categories In this section we explain how to get birational invariants from the symplectic A-side. We use the idea of “localizations”. On the B side we can deﬁne a new “localized” category D (X) as the limit with respect to all divisors D of Db (X\ D). The “limit” is taken in a sense that we can restrict from the complement of some divisor to the complement of another divisor that contains it. In such a way they form a direct system. The result amounts to localizing with respect to all complexes with support in codimension one and higher; see [MePa08]. Example 2.9. For CPn we are localizing with respect to deﬁning equation of any divisor and ending with a category of modules over the algebra of rational functions C(x1 , . . . , xn ). On the mirror side we are deﬁning the direct system over which we take a limit, and functors that mirror the restriction from X \ D1 to X \ D2 , where D2 is a “larger” divisor than D1 . The mirror Y2 of X \ D2 is a degeneration of Y1 , We have a restriction functor from the Fukaya Seidel wrapped category of Y1 to the one of Y2 . It is a mirror of the restriction functor on the B side—see the table below. From this perspective the examples from Table 10 demonstrate in a way restriction functor on the level of wrapped Fukaya–Seidel categories. The following theorem is a consequence of HMS. Db (X \ D) and limYi Fukwrapped (Yi ) are Theorem 2.10. The categories lim −→ All D

birational invariants of X. This theorem allows us to deﬁne the following computable birational invariants. First we restrict ourselves to the situation when X is a nonsimple three-dimensional Fano manifold. Theorem 2.11. Let X be a nonsimple three-dimensional Fano manifold and Y be its LG mirror. If the monodromy Y around zero is not unipotent then X is not rational. Table 14 illustrates Theorem 2.11 - nonrational Fanos with quasi-unipotent LG mirror monodromies produce after a pullback rational Fano manifolds with unipotent LG mirror monodromies. In this table P6 (2, 2, 2) is the intersection of three quadrics in P6 and P5 (2, 2) and is the intersection of two quadrics in P5 . V10 and V5 are Fano threefolds from the Fano threefold classiﬁcation.

Mirror Symmetry and Rationality Questions

179

R 2+

W 2(R 2 +)

Arg

W(R 2 +)

X

R 2+

Fig. 6. Wrapping

x2 y 2

y 2 x2

Fig. 7. Computing Floer homologies

Remark 2.12. In the above theorem the term “nonsimple” refers to nonexistence of an exceptional collection in the corresponding derived category. We outline the idea behind this theorem in the next section after introducing some cohomology structures. The last part of this section is dedicated to construction mirror of X \ D. In the table on p. 181 we illustrate the procedure introduced at the beginning of Section 2 applied to the example of P1 \ (n points).

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L. Katzarkov

As before we ﬁrst embed CP1 into CP1 × CP1 as a (1, n)-curve and then intersect with (C∗ )2 to get a CP1 \ (2n + 2 points). The novelty of the procedure described pictorially on p. 181 is that we think of the embedded CP1 as a projective tropical curve: we add aﬃne curves C∗ at the points of intersection and degenerate tropically (1, n) curve. The categories F ukayawrapped(Yi ) b require some modiﬁcations—this is done in more detail in [GK07]. and Dsing Depending on the purpose we can leave some potential or completely ﬁnish the wrapping—see the SYZ procedure for Fanos. The construction below has some clear advantages: 1. The dimension of the mirror Y is the same as of initial variety X. 2. It allows us to follow many geometric notions and constructions through, e.g., coverings, linear systems, monodromy. In the table below we build the mirror of a genus two curve as a two-sheeted covering of CP1 ramiﬁed at two points. 3. It works for all X \ D. SYZ for Fanos and open Fanos. Let us work out the procedure of the beginning of the section in the case of the Landau Ginzburg model for three dimensional cubic X. The proof of Theorem 4.3 (see [Kat08b]) will be based on these considerations, which in a way are giving the basics of SYZ construction for Fano manifolds. (1) We ﬁrst take X as a hypersurface in CP4 , and draw its amoeba in tropicalization T P4 of CP4 . (2) Then we take a tropical limit of X (a tropical hypersurface T X in T P4 ), i.e., a piecewise linear cell complex of dimension 3—with some bounded cells and some unbounded cells. (3) Observe that each of these cells is polyhedral and hence determines a toric variety which has this cell as its moment polytope; these toric 3-folds (some compact, some noncompact) are glued together along toric strata according to the combinatorics of the tropical 3-fold T X. (For a curve in P1 × P1 , this would be a union of P1 ’s and A1 glued together at trivalent vertexes.) (4) The mirror of the open cubic X = X ∩ (C∗ )4 is a singular 3-fold in this singular conﬁguration of glued mirror toric 3-folds without a superpotential. So when studying its Fukaya category we need to wrap at inﬁnity—the divisor corresponding in the mirror to the divisor at inﬁnity of X = X ∩ (C∗ )4 . (5) A diﬀerent way to think about this is the singular 3-fold Y actually being the critical set of a 5-dimensional Landau–Ginzburg model, which would be mirror to the blowup of C × (C∗ )4 along 0 × X . This ﬁve-dimensional Landau–Ginzburg model has one very singular ﬁber which is a union of toric 4-folds intersecting along a three-dimensional critical locus which is precisely the singular 3-fold Y described above—see the previous construction for P1 minus points and [AAK08].

Mirror Symmetry and Rationality Questions

P1 \ (3 points)

P1 \ (4 points)

P1 \ (6 points)

W :

181

C∗3

/C

(x, y, z)

/ xyz

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L. Katzarkov

(6) The mirror to the closed cubic 3-fold X is obtained by adding terms to the superpotential of the singular 3-fold Y . On each of the open (aﬃne) components above we essentially have the potential W being equal to the sum of the linear coordinates along the aﬃne directions (the exact meaning of this is given by the manner in which the tropical hypersurface T X sits inside tropical T P4 ). The mirror to something intermediate between the closed X and the open X , e.g. X \D is obtained by gluing in only 3 of the 5 hyperplane sections that were missing from X . Equivalently by removing 2 of the 5 toric hyperplane sections from X we equip Y with a superpotential that only has some of the terms (namely, on each open component the sum of the aﬃne variables which correspond to directions in T P4 that are not the 2 removed hyperplane sections). One can think of this as of 5-dimensional Landau–Ginzburg model instead - the blowup construction. Similarly we take the Landau–Ginzburg model mirror to (C∗ )4 × C blown up at X ×0 and add ﬁve terms to the superpotential if we want the mirror to P4 × C blown up at X × 0 or only three of these ﬁve terms if we want to only add 3 hyperplanes instead of all ﬁve. The eﬀect of adding these terms to the superpotential is that the noncompact components of the singular 3-fold Y are no longer inside the central ﬁber of the Landau–Ginzburg model—in fact the critical locus of the 5-dimensional Landau–Ginzburg model with these extra superpotential terms becomes simpler. To demonstrate our point let us look once more at the mirror to the genus two curve in P1 × P1 —see Figure 8. For the open curve in (C∗ )2 we get a three-dimensional Landau–Ginzburg model whose critical locus is equal to the conﬁguration of many of P1 ’s and A1 ’s, but upon adding four terms to the superpotential to get the mirror to the closed genus 2 curve, the A1 ’s deform outside of the central ﬁber and the remaining singular conﬁguration is just a theta-conﬁguration of three P1 ’s. If we had only glued in some of the toric divisors, to get something that lives in a toric manifold larger than (C∗ )2 but smaller than (P1 )2 , then we only add some of those four terms to the superpotential, and the critical locus still has some A1 ’s in addition to the P1 ’s. If in the construction above - the Landau–Ginzburg model of the threedimensional cubic we want to remove ﬁve hyperplane sections, then we embed the cubic into something larger than P4 , and proceed in a similar way. Remark 2.13. The above method is very well equipped for following the behavior of subvarieties and Lefschetz pencils under Homological Mirror Symmetry - we will make use of it in the proof of Theorem 4.3. Remark 2.14. The above construction allows us also to study the changes of Landau Ginzburg models and HMS in general when we move in the moduli space. In particular we can follow the behavior of the Landau Ginzburg models moving from one Brill–Noether loci to another—the so called earthquakes— see, e.g., [Kat08a]. Table 15 brieﬂy demonstrates this phenomenon on the

Mirror Symmetry and Rationality Questions

Contracting Lagrangian Initial

Smooth Cubic

S

183

Small Resolution

3

Singular Algebraic Contraction

LG Mirror

Degenerate g=4 curve Table 15. Earthquake for Fano 3 dim cubic

example of three-dimensional cubic. Following the Hori–Vafa procedure one can check that contracting an S 3 Lagrangian and small resolution on the B side lead to a ﬂop (we call it an earthquake) on the A side. Now we brieﬂy discuss a new deﬁnition of the wrapped Fukaya Seidel category. It is based on the above SYZ construction for open Fano’s. We start with the example of Landau Ginzburg mirror of P1 minus three points considered above - see Table 13. Once again this is a singular space formed by 3 aﬃne lines intersecting at one point (in other words the coordinate axes the critical locus of W = xyz). When doing Floer theory we will be wrapping inside the ﬁbers of W i.e., in the three aﬃne lines there is wrapping about the ends. Below we deﬁne wrapped Fukaya category of the singular conﬁguration of three A1 ’s which should be the same as Fukaya–Seidel category of W . Here are some obvious objects: (1) The easiest objects of the Fukaya category are circles inside each of the aﬃne lines (or, when talking about the Fukaya–Seidel of the LG model W = xyz, the thimbles built from such circles); these objects are mirrors to points of the pair of pants P1 minus three points which lie “in the cylindrical ends” of the pair of pants. (2) Points of P1 minus points which lie in the “core” of the pair of pants (a theta graph or a slight thickening of it) correspond to the point at the origin in some way (i.e., take the thimble from the origin in W = xyz, which is a singular Lagrangian equal to a cone over T 2 : The last has a nontrivial moduli space of smoothings, which gives the points completing the B part.

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L. Katzarkov

(3) The mirror to the structure sheaf of P1 minus three points corresponds to the Lagrangian graph Γ (Lefschetz cycle) formed by the three real positive axes R+ inside the singular union of the three aﬃne lines (in the LG model of W = xyz, the corresponding element of the Fukaya Seidel category is the real positive part (R+ )3 inside C3 ). The homomorphisms from Γ to an object of type (1) above (a circle) are one-dimensional (the circle is hit once by one leg of the graph); the endomorphisms of Γ are inﬁnite-dimensional because there is wrapping in the ends (looks like C[x, y, z] with relations xy = 0, yz = 0, xz = 0). What corresponds to O(1), O(2), . . . on P1 is a wrapping of Γ i-times around in each A1 . This example suggests the following general deﬁnition: Definition 2.15 (Local definition for Fukaya - Seidel wrapped). A Lefschetz cycle L is a (S 1 )i D-module over a tropical scheme in the base B ( see [Gro04]) of the aﬃne variety satisfying the following conditions: 1) the balance condition—this means that the base of the D-module is a welldeﬁned tropical scheme; see [Kat08a]. 2) the phase conditions—this means that the (S 1 )i ﬁbrations are well behaved 3) the superabundance conditions—this means that L has the deformations of a cycle on the B side; see [Kat08a].

3 Cohomologies The constructions in the previous section give interesting correspondences in HMS on a cohomological level. What follows should be seen as algebrogeometrization of Deﬁnition 2.15 - the Lefschetz cycle L is the symplectic image under HMS of the perverse sheaf of vanishing cycles. Suppose M is a complex manifold equipped with a proper holomorphic map f : M → Δ onto the unit disc, which is submersive outside of 0 ∈ Δ. For simplicity we will assume that f has connected ﬁbers. In this situation there is a natural deformation retraction r : M → M0 of M onto the singular ﬁber M0 := f −1 (0) of f . The restriction rt : Mt → M0 of the retraction r to a smooth ﬁber Mt := f −1 (t) is the “specialization to 0” map in topology. The complex of nearby cocycles associated with f : M → Δ is by deﬁnition the complex of sheaves Rrt∗ ZMt ∈ D− (M0 , Z). Since for the constant sheaf we have ZMt = rt∗ ZM0 , we get (by adjunction) a natural map of complexes of sheaves sp : ZM0 → Rrt∗ rt∗ ZM0 = Rrt∗ ZMt . The complex of vanishing cocycles for f is by deﬁnition the complex cone(sp) and thus ﬁts in an exact triangle sp

ZM0 → Rrt∗ ZMt → cone(sp) → ZM0 [1]

Mirror Symmetry and Rationality Questions

185

of complexes in D− (M0 , Z)). Note that Hi (M0 , Rrt∗ ZMt ) ∼ = H i (Mt , Z), and so if we pass to hypercohomology, the exact triangle above induces a long exact sequence r∗

t . . . H i (M0 , Z) → H i (Mt , Z) → Hi (M0 , cone(sp)) → H i+1 (M0 , Z) → . . . (1)

Since M0 is a projective variety, the cohomology spaces H i (M0 , C) carry the canonical mixed Hodge structure deﬁned by Deligne. Also, the cohomology spaces H i (Mt , C) of the smooth ﬁber of f can be equipped with the Schmid– Steenbrink limiting mixed Hodge structure which captures essential geometric information about the degeneration Mt /o /o /o / M0 . With these choices of Hodge structures it is known from the works of Scherk and Steenbrink that the map rt∗ in (1) is a morphism of mixed Hodge structures and that Hi (M0 , cone(sp)) can be equipped with a mixed Hodge structure so that (1) is a long exact sequence of Hodge structures. Now given a proper holomorphic function w : Y → C we can perform above construction near each singular ﬁber of w in order to obtain a complex of vanishing cocycles supported on the union of singular ﬁbers of w. We will write Σ ⊂ C for the discriminant of w, YΣ := Y ×C Σ for the union of all singular ﬁbers of w, and F • ∈ D− (YΣ , Z) for the complex of vanishing cocycles. Slightly more generally, for any subset Φ ⊂ Σ we can look at the union YΦ of singular ﬁbers of w sitting over points of Φ and at the corresponding complex • F|Y ∈ D− (YΦ , Z) FΦ• = σ σ∈Φ

of cocycles vanishing at those ﬁbers. In the following we take the hypercohomology Hi (YΣ , F • ) and Hi (YΦ , FΦ• ) with their natural Scherk–Steenbrink mixed Hodge structure. For varieties with anti-ample canonical class we have: Theorem 3.1. Let X be a d-dimensional Fano manifold realized as a complete intersection in some toric variety. Consider the mirror Landau–Ginzburg model w : Y → C. Suppose that Y is smooth and that all singular ﬁbers of w are either normal crossing divisors or have isolated singularities. Then the Deligne ip,q numbers for the mixed Hodge structure on H• (YΣ , F • ) satisfy the identity ip,q (H• (YΣ , F • )) = hd−p,q−1 (X). Similarly we have: Theorem 3.2. Suppose that X is a variety with an ample canonical class realized as a complete intersection in a toric variety. Let (Y, w) be the mirror Landau–Ginzburg model. Suppose that all singular ﬁbers of w are either normal crossing divisors or have isolated singularities. Then there exists a Zariski open set U ⊂ C so that Deligne’s ip,q numbers of the mixed Hodge structure on H• (YΣ∩U , FU• ) satisfy the identity • ip,q (Hi (YΣ∩U , FΣ∩U )) = hd−p,q−i+1 (X).

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L. Katzarkov

(Y, f, F )

(X, D, h)

H • (Yλ , F, d + ∧df )

H • (X \ D, d + ∧dh)

Table 16. Hodge theoretic correspondences for Table 1

(Y, KY ) → D

(LG(Y, f ), g)

D ⊂ LG(F, h), H

F : LG(Y, f ) → C

H • (LG(F, h), H)

H • (Y, d ∧ df, d ∧ dg)

Table 17. Hodge theoretic correspondences for Table 2

We summarize the information in Tables 16, 17, and 18: Theorem 3.3. Let X be Fano manifold and D an anticanonical normal crossing divisor. Theorems 3.1 and 3.2 hold for open varieties X \ D as well. The proof can be found in [GK07] and it is based on chasing Mirror Symmetry through aﬃne structures described in Table 18. Similar statements hold in the case of LG mirrors with many potentials - see Table 17. Most of what follows can be found in [KKP08a]. We summarize it brieﬂy in Table 19. In fact the data parametrize the following: 1) 2) 3) 4)

A pair of derived categories CD and CX . A restriction functor φ∗ : CX → CD . A right adjoint functor φ∗ : CD → CX . The isomorphism Cone(IdX → φ∗ φ∗ ) = SX [1 − n] where SX is the Serre functor.

Conjecture 3.4. The deformations of the triple (CD , CX , φ∗ ) are unobstructed.

Mirror Symmetry and Rationality Questions X◦ = X \ D

X

o

X∨ = Y

?_

/

H p (B ◦ , Rq f∗ C)

∼ =

Hc∗ (X◦)

B◦

H p (B ◦ , Rn−q f∗ C)

H ∗ (X◦)

Hcp (B ◦ , Rcq f∗ C)

f∨

B◦

B

187

H ∗ (Y )

∼ =

H p (B ◦ , Rcn−q f∗ C)

Hc∗ (Y )

Table 18. Cohomological correspondences for open varieties

We also have Conjecture 3.5. Cone(φ∗ φ∗ − IdS ) is an automorphism of CD . Observe that the tangent space to the moduli of deformations is given by Cone (HH• (CD ) → HH• (CX )) [2 − n]. So we can associate to this situation an extension of noncommutative Hodge structures, i.e., a mixed noncommutative Hodge structure deﬁned as follows. a) u2 ∇∂/∂u : H → H is everywhere holomorphic diﬀerential operator on H of order ≤ 1. b) For every locally deﬁned vector ﬁeld v ∈ TM we have that u∇v : H → H is everywhere holomorphic diﬀerential operator on H of order ≤ 1. •

Eb is a constructible sheaf on D × M . In the Landau–Ginzburg case Eb is given by H = H i (W −1 (D), Yt , Q).

•

An isomorphism iso : DR(H, ∇) → Eb ⊗ C of constructible sheaves on D × M.

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X — Fano, D — CY

f : Y → C,

Y∞ = f −1 (∞)

Cohomologies ⊕H 1 (X, Ω n−i (Log(D))) — coordi- k H (Y ) — coordinates nates Theorem. (Y, Y∞ ) has unobstructed Theorem. (X, D) has unobstructed deformations. deformations. Categories

• • • •

Db (X) = CX , CD = Db (D) φ∗ : CX → CD restriction φ∗ : CD → CX right adjoint Cone(IdX → φ∗ φ∗ ) ∼ = S[1 − n]

• CD = Fuk(Y∞ ) • CX = F S(Y, f ) • φ∗ : CX → CD restriction • Tangent space H • (Y )[2] Conjecture The deformation theory • HH• = HH • (Y, Y∞ )[n] of (CX , CD , φ∗ ) is unobstructed. Tan- • HH• (CD ) = H • (Y∞ )[n − 1] gent space: Conjecture The deformation theory φ (CX , CD , φ) is unobstructed. ConeHH• →∗ HH• (CX )[2 − n] Table 19. Categories and Deformations

The formulas for the connections can be found in [KKP08a]. Let us start with a CY manifold X and a smooth curve D in it. We formulate and prove the following: Theorem 3.6. The deformations of the pair (X, D) are unobstructed. Proof. See [KKP08a]. This deformation framework provides the proper moduli space over which, the limit MHS we discuss below, vary—these are the so-called canonical coordinates. The cusp points live in these moduli spaces. Now we outline the idea behind Theorem 2.11. The category limit limYi F S(Yi ) comes with a limit of MHS on H p (B ◦ , Rcn−q f∗ C). This limit MHS encodes the monodromy of the LG model and in particular determines when this monodromy is unipotent. In fact this limit MHS encodes the monodromy of the MHS of a degeneration to the corresponding cusp point. The limit MHS corresponds to a degeneration of a Jacobian of a curve iﬀ the monodromy of the LG model is unipotent which happens only in the case when X is rational.

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189

F3

P 2(1,1,3)

f =x+y

braking

Table 20. Mirror of the complement of a ﬁber in the Hirzebruch surface F3

We end this section with two remarks. Remark 3.7. One can consider the category limit limn F (X \ n · D) and the noncommutative MHS associated with this limit. In case D = KX we get some interesting symplectic invariants. We plan to investigate this in the future and give here one more example. Consider the Hirzebruch surface F3 and a curve C of genus two embedded in F3 as a two-sheeted covering of the base. The surface F3 maps to a weighted projective plane P2 (1, 1, 3)—a singular Calabi Yau, and a ﬁber of F3 maps to a conic in P2 (1, 1, 3). The mirror of P2 (1, 1, 3) minus a conic is the broken LG mirror of P2 (1, 1, 3); see Table 20. Here we get that the category Fukwrapped (C \ (2 points)) is equivalent to b Dsing (ﬁber of LGbroken ). (Here broken means that we take a section of the LG model of F3 deﬁned in [AKO08].) A similar construction should work for the Horikawa surfaces - the role of the ﬁber is played by a ruled surface over the canonical class of Horikawa’s. One needs to notice that the limit noncommutative MHS has a symplectic meaning—it measures precisely the braid monodromy of the ﬁber of LGbroken ; compare with [ADKY04].

Remark 3.8. The above cohomological correspondences lead to some Chern class information. In particular:

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c1 (X) is Poincare dual to the divisor that corresponds to the boundary of the aﬃne structure on X. c2 (X) is Poincare dual to the codimension two subvarieties that correspond to: (1) the corners of the aﬃne structure, (2) the singularities of the aﬃne structure. The connection between Hodge structures and HMS goes further. We demonstrate this on very simple examples P1 \ 3 pts and P1 \ 3 pts. We ﬁrst demonstrate Theorem 3.1—Tables 21, 22. We assign constant sheaf Z to every line of the mirror of P1 \3 pts and the complex Z[2] = Z+Z[2] to the edge. This forms a perverse sheaf of vanishing cycles whose hypercohomologies coincide with the cohomology of P1 \3 pts. In particular H 1 (Y, Re(W ) >> 0, F ) = Z2 . The above approach allows us to compute the monodromy acting on Htrans (Yt ), the transcendental part of H(Yt ). The procedure can be described as follows. To each edge we assign number 1 and to each vertex number 2, see Table 22, similarly, this monodromy can be computed in many cases. On the A side these numbers represent monodromy rotations of corresponding thimbles; see also Tables 21 and 22. It leads to the following: Theorem 3.9. [Kat08b] Let X be three-dimensional Fano and Y → C be its Landau–Ginzburg Mirror. Suppose that there exists a singular ﬁber with nonisolated singularities in Y → C. Then the Serre functor of all subcategories in the semiorthogonal decomposition on the B side can be computed from the combinatorics of the singular set on the A side. This theorem will play an important role in the next section. In Table 22 we demonstrate a similar Serre functor calculation for P1 \ 4 pts.. The proof of this theorem is connected with the following observation: the Serre functor measures the monodromy of the Picard–Fuchs operators. The toric degenerations of the LG model allow us to obtain Picard–Fuchs operators when we move in the moduli space. When the Picard–Fuchs operators change, the monodromy changes and so do the Serre functors; for more see [Kat08b].

4 Applications to Birational Geometry After building the HMS theory of linear systems, open manifolds, and coverings we move to Birational Geometry. We suggest the following conjectural dictionary relating Birational Geometry and HMS: Birational Geometry Homological Mirror Symmetry X w : Y → CP1 Blow-up Adding a singular ﬁber Blow-down Taking a singular ﬁber to ∞

Mirror Symmetry and Rationality Questions H 1 (Y, Re(W ) >> 0, F ) = Z2

Z[1] Z[1] Z[2] Z[1]

w

Serre functor for LG(P1 \ pts): 1+1+1−2= 1 −2 1

1

1 Serre functor for LG of genus 2 curve:

1+1 = 2

1 −2

1 1

1

1

1

Table 21. Serre functor calculations 1

−2

191

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L. Katzarkov

Z[1] Z[1]

Z[2]

Z[2]

Z[1]

w

H 1 (X, Re(w) >> 0, Z3 ) Serre functor for P \ 4pts.: 1

−2 + 1 + 1 + 1 + 1 = 2

−1

−1 +2

−1

−1

Table 22. Serre Functor Calculations 2

It is based on the following theorems: Theorem 4.1. (Orlov) Let X be a smooth projective variety and XZ be a blow up of X in a smooth subvariety Z of codimension k. Then Db (XZ ) has a semiorthogonal decomposition (Db (X), Db (Z)k−1 , . . . , Db (Z)1 ). Here Db (Z)i are corresponding twists by O(i). This B-side statement has an A-model counterpart, which relates suitable Landau–Ginzburg mirrors LG(X), LG(Y ), and LG(XY ). In a joint work in progress with M. Abouzaid and D. Auroux we consider the following: Theorem 4.2. ([AAK08]) XY has a Landau–Ginzburg mirror LG(XY ) such that, for a suitable value of the constant R > 0, the region {|W | < R} is topologically equivalent to LG(X), while the region {|W | > R} contains k − 1 clusters of critical values, each of which is topologically equivalent to a stabilization of LG(Y ). In particular, F(LG(XY )) admits a semiorthogonal decomposition F(LG(X)), F(LG(Y ))k−1 , . . . , F(LG(Y ))1 . Figures 4 and 5 are good illustrations of this theorem. We illustrate further the above theorem on some examples. We ﬁrst consider the example of the intersection of two quadics in P3 . As we can see from

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193

Fig. 8. Blowing up a genus two curve

Fig. 9. Blowing down quadric

Figures 8 and 9, blowing up a curve of genus two aﬀects the aﬃne structure by creating new singularities over a tropical genus two curve. Blowing down the quadric surface on which the genus two curve lives leads to sending some singularities to the boundary and makes the ﬁbers of the new LG mirror surfaces of degree 8. We consider next the example of three-dimensional cubic—all theorems from now on are modulo HMS. Applying the standard Hori–Vafa procedure we get after smoothing the following Landau–Ginzburg Mirror: xyuvw = (u + v + w)3 · t with potential W = x + y. Here (u : v : w) ∈ CP2 and (x, y) ∈ A2 . The singular set W of this Landau–Ginzburg model looks as in Figure 10 - see also Section 2.3 and Table 15. The above Landau–Ginzburg model represents a family of K3 surfaces. The ﬁber over zero consists of six surfaces and if we compute the monodromy we get that it is quasi-unipotent. A calculation as in Theorem 3.9 implies that S 3 = [5]. We formulate the following: Theorem 4.3 (Nonrationality criteria for 3-dimensional Fano’s). Let X be three-dimensional Fano and Y → C its Landau–Ginzburg Mirror. Sup-

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Fig. 10. The singular set for the LG of a 3d cubic

pose that there exists a singular ﬁber with nonisolated singularities in Y → C such that corresponding monodromy is not unipotent. Then X is not rational. The proof is a consequence of Theorem 2.11; see also [Gol05]. If X is rational, as follows from Theorem 4.2, the monodromy of all singular ﬁbers with nonisolated singularities in Y → C is unipotent. Observe that in this case the monodromy of limYi F S(Yi ) is trivial. From the SYZ construction for open Fanos it is clear that if the monodromy of all singular ﬁbers with nonisolated singularities in Y → C is unipotent it stays unipotent for the limit Landau–Ginzburg model and for the limit category limYi F S(Yi ). Indeed as Theorem 3.5 shows, all local monodromies in all singular ﬁbers are nothing else but the Serre funtors of the subcategories in the given semiorthogonal decomposition. If we remove a divisor we get the Serre functor well deﬁned b (X \ D) and F S compact (Y, W ) (see Table 1) and for the categories Dcompact support

Lagrangians

after taking a limit we still preserve the unipotency property. Similarly if there exists a singular ﬁber with nonisolated singularities in Y → C such that corresponding monodromy is not unipotent, it stays nonunipotent if we take the monodromy of limYi F S(Yi ). So X is not rational. We give several known applications of the above criteria, here R stands for rational, M for monodromy, q for quasi-unipotent, and u for unipotent. Also we use standard notations from Iskovskikh classiﬁcation of three-dimensional Fanos.

Mirror Symmetry and Rationality Questions M

R

M

195

R

P4 ⊃ X4

q

−

P3

u

+

P6 ⊃ Q1 ∩ Q2 ∩ Q3

q

−

P5 ⊃ Q1 ∩ Q2

u

+

V10

q

−

V5

u

+

Guided by the analysis in [AAK08] we formulate the following criterion for nonrationality of conic bundles. Conjecture 4.4. (Nonrationality criteria for three-dimensional conic bundles) Let X be three-dimensional conic bundle and Y → C be its Landau– Ginzburg Mirror. Suppose there exists a singular ﬁber with nonisolated singularities in Y → C such that corresponding monodromy is not unipotent. Then X is not rational. One can try to formulate the above criteria in terms of derived category on the B side, a joint project with A. Kuznetsov. It is expected that there should be a connection between the monodromy of the Landau–Ginzburg model and the local monodromy of the Lefschetz cycles; see Deﬁnition 2.15. We formulate the following conjecture: Conjecture 4.5. Let Y be a three-dimensional Landau–Ginzburg model. Then the unipotency of the monodromy in any singular ﬁber implies that the local Lefschetz cycle at this ﬁber produces an algebraic tropical curve; compare with [Kat08a]. As Theorem 3.9 suggests, in dimension three we have a correspondence: Nonunipotent monodromy of some ﬁber in LG(Y ) O

The Serre functor does not behave as a Serre functor of category ofO a curve

The MHS on H3 (X ) is not a MHS of a curve degeneration

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L. Katzarkov

The last part of the table means that there exists a degeneration X of X so that MHS on H3 (X ) is not a MHS of a curve degeneration. Questions. Does similar correspondence hold in higher dimension? Is it possible to have Serre functor of a singular ﬁber behaving as a Serre functor of a smooth variety and still have nonrationality? Is it possible to have nonunipotent monodromy and still have rationality? What other criteria do we apply then? We suggest possible approaches to these questions in the next section. An interesting direction to pursue is the connection between monodromy (local and of the Landau Ginzburg model) and the theory of Hodge cycles compare with [Kat08a]. The idea of multiple potentials seems a possible way to go, see, e.g., [Kat08b]. Observe that the earthquakes discussed in Table 15 are a powerful method to change monodromies—local and of the Landau Ginzburg model. In particular, it allows to jump from rational to nonrational loci. Its interaction with Hodge cycles is studied in [Kat08b]. A diﬀerent direction to pursue is operations with LG mirrors—cross products, quotients, and gluing— see Table 23. In some cases these operations create interesting Hodge cycles. Table 23 and 24 also suggest that by gluing and smoothing we can compute the Picard Fucks systems in the same way we compute monodromy—the Serre functor. Procedure in Tables 21 and 22 suggests the same for the calculations of the monodromy of the perverse sheaf of vanishing cycles. In particular, degenerating the aﬃne structures ([Gro04]) for the threedimensional quartic implies its nonrationality; indeed, the monodromy is not unipotent in this case as well. The LG model for a three-imensional quartic is obtained by gluing LG model for three-dimensional cubic and LG model for the three-dimensional projective space, see Table 28. Observe that a calculation as in Theorem 3.9 implies that S 4 = [5] and thefore nonrationality of a three-dimensional quartic; for more details see [KP08].

5 Quadric bundles and the four-dimensional cubic 5.1 The four-dimensional cubic As in the previous subsection we start by degenerating the four-dimensional cubic X to three four-dimensional projective spaces and applying the procedure from Section 2. We get after smoothing the following Landau–Ginzburg Mirror: xyzuvw = (u + v + w)3 · t with a potential x + y + z. Here (u : v : w) ∈ CP2 and (x, y, z) ∈ A3 . The singular set W of this Landau–Ginzburg model can be seen in Figure 11. It consists of an elliptic surface E and nine other pairs of P2 intersecting the elliptic surfaces in 9 triple lines li , a complete study can be found in [KP08].

Mirror Symmetry and Rationality Questions

197

Procedures: 1. Gluing

Y1 Y2 Y3 Yi = CY ;

Yi = LG

2. Product

Y1 Y2 3. Quotient

Y/

Y Table 23. Operations with LG

P3

Aﬃne structure for the quadric in P4 .

cubic

Degenerated aﬃne structure for P3 and cubic.

Table 24. Degeneration of LG model for three-dimensional quartic

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L. Katzarkov

Fig. 11. Singular set for four-dimensional cubic

Conjecture 5.1. The generic four-dimensional cubic is not rational. We brieﬂy discuss the idea behind the above conjecture. It comes from the fact that for generic cubic, the perverse sheaf of vanishing cycles Fi has nontrivial monodromy when restricted on li . The calculation of Serre functor, i.e., monodromy of the LG model at zero in this case shows that S = [2], it is just shift by two on cohomology of every cubic. In other words the Serre functor behaves just as a Serre functor of an usual K3 surface and cannot be used to study rationality. The singular set above is obtained as follows. We ﬁrst do HMS of the Fermat cubic in P5 and then deform. The singular set for the Fermat cubic is a degeneration of E × E/Z3 . (Fermat cubic is rational and the perverse sheaf of vanishing cycles F has nontrivial monodromy.) Further degeneration produces conﬁguration in Figure 11 and leads to nontrivilaity of the monodromy of Fi . Here E is the Fermat elliptic curve and Z3 acts via multiplication by powers of third root of unity. Theorem 2.10 and examples of Tables 12 and 13 suggest that the complexity of the monodromy of the perverse sheaf of vanishing cycles F could be made a birational invariant. Many examples of four dimensional cubics have been studied by Beauville, Donagi [BD], Voisin [VOI], Hassett [HASS] , Iliev [ILI], Kuznetsov [KUZ], Zucker [ZUCK]. We will analyze some of these examples now. Example 5.2. Examples of cubics containing two planes P1 and P2 . In this case the ﬁber Y0 of the LG model Y looks as on Figure 13. The monodromy of Fi is trivial and the Serre functor behaves just as a Serre functor of an usual K3 surface. These cubics are known to be rational see also [KP08].

Mirror Symmetry and Rationality Questions

199

0 Fig. 12. Singular set for cubics containing two planes

Example 5.3. Examples of cubics containing a plane P1 and being pfaﬃan. In this particular case we get a section in the projection from P , which is a singular del Pezzo surface consisting of elliptic quintic curves, see Figure 14. Recall that pfaﬃan cubics are obtained from CP4 by blowing a K3 surface and blowing down a scroll over a Del Pezzo surface of degree 5. The monodromy of Fi is trivial and the Serre functor behaves just as a Serre functor of a commutative K3 surface. Such a generic cubic is rational - see [KP08]. In the discussion above we have used HMS in full - we will brieﬂy discuss how one can try proving HMS for cubics - we will extend Seidel’s ideas from the case of the quintic. We ﬁrst have: Conjecture 5.4. HMS holds for the total space of OCP4 (3). The next step is: Conjecture 5.5. The Hochschild cohomology HH • (D0b (tot(OCP4 (3)))) are isomorphic to the space all homogeneous polynomials of degree three in ﬁve variables. Here D0b stands for category with support at the zero section.

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L. Katzarkov

0 Fig. 13. Singular set for a Pfaﬃan cubic containing a plane

The Hochschild cohomology HH • (D0b (tot(OCP4 (3)))) produce all B side deformations of categories of the three-dimensional cubic and we have similar phenomenon on the A side as well. Remark 5.6. The above argument should work for any degree and for any dimension toric hypersurfaces. 5.2 Quadric bundles In this section we discuss a generic approach for studying rationality questions for quadric bundles using HMS. This approach is based on the examples we have considered: three- and four-dimensional cubics. For quadric bundles we outline more connections of HMS with Hodge theory, rationality questions and higher categories. We start with the simplest example in X ∨ ⊂ C2(u,v) × C∗ 2(x,y) , ft (x, y) = uv + xy + x + y + t. This is a conic bundle depending on a parameter t. Similarly we can extend this to quadric bundles over P2 by considering ft (x, y) = Q(v1 , v2 , . . . , vm ) + xy + x + y + t where Q(v1 , v2 , . . . , vm ) is a quadric polynomial. The example of four-dimensional cubic with a plane in it produces an example of a quadric bundle. We can consider quadric bundles over Pn as well. All of them can be considered in the following framework:

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201

Definition 5.7. (Local CY manifold) Let X be a smooth quasiprojective variety of dimension n with a nonzero everywhere on X form Ω ∈ Γ (X, KX ) and a chosen compactiﬁcation X such that on D = X \ X Ω has only poles. It is easy to check that all quadric bundles satisfy this deﬁnition. We have the following: Theorem 5.8. ([KKP08b] ) The deformations of the local CY X are unobstructed.

OP1 (−1) ⊕ OP1 (−1)

X ∨ ⊂ C2(u,v) × C∗2(x,y) ft (x, y) = uv + xy + x + y + t

Table 25. Local MS for OP1 (−1) ⊕ OP1 (−1)

Before we look at how quadric bundles relate to rationality questions we look at a very simple example of local CY, OP1 (−1)⊕OP1 (−1), and the quadric bundle with a degeneration curve ft (x, y) = uv + xy + x + y + t, with a volume parameter t, Table 25. The degeneration curve of it is P1 \ 4 pts and its mirror is given in Tables 26 and 27 with diﬀerent degenerations; see Table 28.

γ

I — period – canonical coordinate Z t∂t f dx − t∂t I = y∂y f x γ

I

Table 26. Canonical coordinates for P1 \ 4 pts

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L. Katzarkov

In Table 26 we demonstrate how one computes the canonical coordinate on the example of P1 \ 4 pts. In this particular case we compute the integral ) t∂t f dx t∂t I = − y∂y f x γ over cycle γ in P1 \ 4 pts chosen as shown on Table 30. H 1 (P1 \{P1 , P2 , P3 , P4 }) is one dimensional and after some change we can use I as a coordinate. We can put a MHS on H 1 (P1 \ {P1 , P2 }, {P3 , P4 }) and using HMS identify it with H1 (Z, U, F ). This allows us to put an interplay of polylogs (in this case logs) and HMS. In order to see polylogs (Table 29) we consider powers of P1 \ {P1 , P2 , P3 , P4 } and its mirror and then apply procedures from Theorem 2.5 and Table 29.

P1

P2

U

Z

P3 P4

H (P \ {P1 , P2 }, {P3 , P4 }) 1

1

H1 (Z, U, F )

Log = Ext1

MHS

Table 27. HMS and Log

We discuss a diﬀerent type of MHS (Table 30), which allows us to see canonical coordinates and degenerations of MHS as well as some higher categories related to HMS. On a rational curve in P2 we consider the following limit of MHS for pairs: lim H 1 (P1 \ {P1 , P2 , P5 , P6 }, (P3 , P 4))

P3 →P1 P4 →P2

We prove in [KKP08b]: Theorem 5.9. The extension class in lim H 1 (P1 \ {P1 , P2 , P5 , P6 }, (P3 , P 4))

P3 →P1 P4 →P2

Mirror Symmetry and Rationality Questions

t=1 1 + x + y + xy = 0

t=0 x + y + xy = 0

t=∞

Table 28. Three diﬀerent degenerations

H 1 ((P1 \ 4 pts.)n )

H1 (Z n , U, F )

n-polylog

n-polylog Table 29. Polylogs

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L. Katzarkov

deﬁnes a canonical coordinate. The MHS deﬁned above is a good way to approach rationality questions for quadric bundles, see Table 31.

P3 P1

P2 P4

γ

P6 P5

lim H 1 (P1 \{P1 , P2 , P5 , P6 }, P3 , P 4)

P3 →P1 P4 →P2

lim H(Y, F )

γ→0

I A side realization

Ext

Table 30. MHS of the pair

Let us ﬁrst consider a three-dimensional conic bundle. We have an MHS associated to the natural double cover of the curve of degeneration for the conic bundle in question: lim H 1− (C˜ \ {P˜1 , P˜2 , P˜3 , P˜4 }).

P3 →P1 P4 →P2

Here H 1− is the anti-invariant part of the cohomologies. In the previous section we have shown that rationality of the threedimensional cubic is based on the nontriviality of the monodromy of the perverse sheaf of vanishing cycles. This, we have shown, also relates to the type of Serre functor we have. In limP3 →P1 H 1− (C˜ \ {P˜1 , P˜2 , P˜3 , P˜4 }) we deﬁne the P4 →P2

˜ Z). Deforming Z structure via integrating over the noninvariant part of H 1 (C, the conic bundle leads to a degeneration of MHS: lim H 1− (C˜ \ {P˜1 , P˜2 , P˜3 , P˜4 })

P3 →P1 P4 →P2

which is not an MHS of a curve degeneration. Through the HMS procedure described in Section 2 we can check that this relates to nontriviality of the monodromy of the perverse sheaf of vanishing cycles and to the Serre functor not being a Serre functor of a curve. Similar observation can be made in the case of four-dimensional cubic but there the Serre functor calculation leads to a Serre functor of a K3 surface and it is not clear what characterizes an MHS

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205

of a degeneration of a surface. What remains true is that from HMS we get nontriviality of the monodromy of the perverse sheaf of vanishing cycles - see [KP08]. According to our consideration this leads to nonrationality of generic four-dimensional cubic, see Table 31 below. Observe that in order to describe all canonical coordinates of the four-dimensional cubic we need to add some more coordinates to the ones coming from the sixtic. One of them is obtained as an integral over the whole tropical limit of the two-sheeted covering - a section which degenerates when we move away from cubics with two planes. This integral relates to 2,0 forms of the two-sheeted covering above. The other integral is connected with a preimage of a line in P2 . When the quadric bundle above has a section these integrals become dependent. Existence of a section is equivalent to the fact that the perverse sheaf of vanishing cycles on the corresponding LG model has a trivial monodromy. B side

A side

γ2 γ1 γ

˜→C C is a 2:1 cover in A2 ˜ \ {P˜1 , P˜2 , P˜3 , P˜4 }) lim H 1 (C

P3 →P1 P4 →P2

K3 ﬁbration H i (Y, F )

A2 sixtic

4−dim cubic

quadric bundles for 4-dimensional cubic with a plane Table 31. HMS for quadric bundles

We will brieﬂy mention here an approach to nonrationality based on higher categories. More can be found in [Kat08b]. Let us consider a four-dimensional cubic X with a plane P in it. Following the procedure of Table 4 we have an LG model with two potentials for four-dimensional cubic. This LG model, a ﬁbration over C2 , can be thought of as a family of LG models associated to the mirror to the family of quadrics we get by projecting the cubic X from P . So we have a family of LG models Y1 for each line in C2 . Here we introduce a two category with objects Fukaya–Seidel categories Y1 and

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L. Katzarkov

morphisms are Fukaya–Seidel categories of the cross-product LG model Y1 × Y2 , see [Kat08b]. The nonrationality of the monodromy of the perverse sheaf of vanishing cycles associated with the mirror of X relates to the Fukaya– Seidel category of the cross-product LG model Y1 × Y2 being equivalent to F S(LG(P1 )) × F S(LG(P1 )). [Kat08b]. The ﬁndings are brieﬂy summarized in Table 32.

H1 (Z n , U, F )

Y1

Nontrivial monodromy of the perverse sheaf of vanishing cycles. FS of LG model Y1 × Y2 being a product of FS categories.

LG

Y

P1 Table 32. Higher Categorical Structures

Remark 5.10. Similar higher categories approach might be useful in studying algebraic cycles, see [Kat08b]. Conclusion. As we have mentioned, Tables 23 and 24 suggest that by gluing and smoothing we can compute the Picard–Fuchs systems in the same way we compute monodromy—the Serre functor. Procedures described in Tables 21 and 22 suggest the same for the calculations of the perverse sheaf of vanishing cycles. The example of four-dimensional cubic shows that in some cases we get Serre functors no diﬀerent from the Serre functors of smooth algebraic surfaces and therefore the monodromy of the Picard-Fuchs systems cannot be used for distinguishing rationality, but the monodromy of the perverse sheaf of vanishing cycles—the strongest criteria so far still can be used. Similar arguments, see [Kat08b], suggest that there might exist conic bundles of dimension three with Intermediate Jacobians—Jacobians of curves which are not rational. Such an example can be obtained from the double solid by putting on it seven or more singular points in general position. One can use the idea behind the proofs of Theorems 2.10 and 2.11 to prove that the monodromy of the perverse sheaf of vanishing cycles is a birational invariant after taking all divisors away. It is known that in higher dimensions there are Landau Ginzburg models of rational varieties for which the perverse sheaf of vanishing cycles has nontrivial monodromy, e.g., P5 blown up in a

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cubic and some six-dimensional cubics. The important thing is what monodromy we are left with after we take all divisors away, a hard to control issue in dimension higher than four.

References M. Abouzaid, D. Auroux, L. Katzarkov, Homological mirror symmetry for blowups, preprint, 2008. [Aur07] D. Auroux, Mirror symmetry and T- duality in the complement of the anticanonical divisor , preprint ArXiv math 07063207. [ABH01] V. Alexeev, C. Birkenhake, K. Hulek, Degenerations of Prym varieties, AG 0101241. [ADKY04] D. Auroux, S. Donaldson, L. Katzarkov, M. Yotov, Fundamental groups of complements of plane curves and symplectic invariants, Topology 43 (2004), no. 6, 1285–1318. [AKO06] D. Auroux, L. Katzarkov, D. Orlov, Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves, Invent. Math. 166 (2006), 537–582. [AKO08] D. Auroux, L. Katzarkov, D. Orlov, Mirror symmetry for noncommutative P2 , Ann. Math. 167 (2008), 867–943. [BD] A. Beauville, R. Donagi, La vari´et´e des droites d’une hypersurface cubique de dimension 4, C. R. Acad. Sci. Paris S´er. I Math. 301 (1985), no. 14, 703–706. [Cle02] H. Clemens, Cohomology and obstructions II, AG 0206219. [Gol05] V. Golyshev, Classification Problems and Mirror Symmetry, AG 051028. [Gro04] M. Gross, Toric Degenerations and Batyrev-Borisov Duality, AG 0406171. [GK07] M. Gross, L. Katzarkov, Mirror Symmetry and Vanishing Cycles, preprint. [CK99] D. Cox, S. Katz, Mirror symmetry and algebraic geometry, volume 68 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1999. [HASS] B. Hassett, Some rational cubic fourfolds, J. Algebraic Geom. 8 (1999), no. 1, 103–114. [HKK+ 03] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, E. Zaslow, Mirror symmetry, volume 1 of Clay Mathematics Monographs. American Mathematical Society, Providence, RI, 2003. With a preface by Vafa. [HV00] K. Hori, C. Vafa, Mirror symmetry, 2000, hep-th/0002222. [ILI] A. Iliev, L. Manivel, Cubic hypersurfaces and integrable systems, AG 0606211. [KKOY04] A. Kapustin, L. Katzarkov, D. Orlov, M. Yotov, Homological mirror symmetry for manifolds of general type, preprint, 2004. [KKP08a] L. Katzarkov, M. Kontsevich, T. Pantev, Hodge theoretic aspects of mirror symmetry, preprint, arXiv:0806.0107, 124 pp.. [KKP08b] L. Katzarkov, M. Kontsevich, T. Pantev, Hodge theoretic aspects of mirror symmetry, II, in preparation. [AAK08]

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L. Katzarkov, V. Przyjalkowski, Generalized homological mirror symmetry and nonrationality and cubics, preprint, 2008. [Kat08a] L. Katzarkov, Homological Mirror Symmetry and Algebraic Cycles, Conference in honor of C. Boyer, Birkh¨ auser, 2008. [Kat08b] L. Katzarkov, Homological Mirror Symmetry, Monodromies and Cycles, AIP, 2008, to appear. [KUZ] A. Kuznetsov, Derived categories of cubic fourfolds, AG 08083351. [MePa08] S. Meinhardt, H. Partsch, Quotient Categories, Stability Conditions and Birational Geometry, AG 08050492. [Orl04] D. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Tr. Mat. Inst. Steklova 246 (Algebr. Geom. Metody, Svyazi i Prilozh.) (2004), 240–262. [Sei07] P. Seildel, Fukaya categories, book available at Seidel’s MIT homepage. [VOI] C. Voisin, Torelli theorems for cubics in CP5 , Invent. Math. 86, 1986, 3, 577–601. [ZUCK] S. Zucker, The Hodge conjecture for four dimensional cubic, Compos. Math. 34 (1977), 199–209. [KP08]

The Bogomolov Multiplier of Finite Simple Groups Boris Kunyavski˘ı Department of Mathematics Bar-Ilan University 52900 Ramat Gan, Israel [email protected] Summary. The subgroup of the Schur multiplier of a ﬁnite group G consisting of all cohomology classes whose restriction to any abelian subgroup of G is zero is called the Bogomolov multiplier of G. We prove that if G is quasisimple or almost simple, its Bogomolov multiplier is trivial except for the case of certain covers of P SL(3, 4).

Key words: Rationality, Brauer group, Bogomolov multiplier 2000 Mathematics Subject Classiﬁcation codes: 14E08, 14L30, 14F20

Introduction A common method for proving that a given algebraic variety X over a ﬁeld k is not rational is as follows. We consider some easily computable object (usually of algebraic nature), which can be deﬁned functorially on a suﬃciently large class of algebraic varieties and is known to be preserved under birational transformations (birational invariant). We calculate its value for X (or for some Y birationally equivalent to X). If this value is not trivial, i.e., does not coincide with the value of this birational invariant on the aﬃne or projective space, X is not rational. The Brauer group Br(X) = H´e2t (X, Gm ), whose birational invariance in the class of smooth projective varieties has been established by Grothendieck, turned out to be a very convenient tool (the Artin–Mumford counterexample to L¨ uroth’s problem, based on using this invariant, conﬁrms its power). Moreover, even if X is not projective, this invariant can be useful: embed X as an open subset into a smooth projective variety Y (if the ground ﬁeld is of characteristic zero, this is always possible by Hironaka) and compute Br(Y ). If the latter group is not zero, Y cannot be birational to Pn , and thus X

F. Bogomolov, Y. Tschinkel (eds.), Cohomological and Geometric Approaches to Rationality Problems, Progress in Mathematics 282, DOI 10.1007/978-0-8176-4934-0_8, © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

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is not rational. Note that Br(Y ) depends only on X (and not on the choice of a smooth projective model Y ); it is called the unramiﬁed Brauer group of X and denoted by Brnr (X). (The reader interested in historical perspective and geometric context, including more general invariants arising from higherdimensional cohomology, is referred to [Sh], [CTS], [GS, 6.6, 6.7], [Bo07].) In concrete cases, it may be diﬃcult to construct Y explicitly, and thus it is desirable to express Brnr (X) in intrinsic terms, i.e., get a formula not depending on Y . This approach was realized by Bogomolov [Bo87] in the case of the quotient variety X = V /G where V stands for a faithful linear representation of a linear algebraic group G over C. It turns out that in this case Brnr (X) depends solely on G (but not on V ). In the present paper we focus on the case where G is a ﬁnite group. The birational invariant Brnr (V /G) has been used by Saltman to give a negative answer to Noether’s problem [Sa]. In [Bo87] Bogomolov established an explicit formula for Brnr (V /G) in terms of G: this group is isomorphic to B0 (G), the subgroup of the Schur multiplier M(G) := H 2 (G, Q/Z) consisting of all cohomology classes whose restriction to any abelian subgroup of G is zero. We call B0 (G) the Bogomolov multiplier of G. In [Sa], [Bo87] one can ﬁnd examples of groups G with nonzero B0 (G) (they are all p-groups of small nilpotency class). In contrast, in [Bo92] Bogomolov conjectured that B0 (G) = 0 when G is a ﬁnite simple group. In [BMP] it was proved that B0 (G) = 0 when G is of Lie type An . In the present paper we prove Bogomolov’s conjecture in full generality. Acknowledgment. The author’s research was supported in part by the Minerva Foundation through the Emmy Noether Research Institute of Mathematics, the Israel Academy of Sciences grant 1178/06, and a grant from the Ministry of Science, Culture and Sport (Israel) and the Russian Foundation for Basic Research (the Russian Federation). This paper was mainly written during the visit to the MPIM (Bonn) in August–September 2007 and completed during the visit to ENS (Paris) in April–May 2008. The support of these institutions is highly appreciated. My special thanks are due to O. Gabber who noticed a gap in an earlier version of the paper and helped to ﬁll it. I am also grateful to M. Conder, D. Holt, and A. Hulpke for providing representations of P SL(3, 4) necessary for MAGMA computations, and to N. A. Vavilov for useful correspondence.

1 Results We maintain the notation of the introduction and assume throughout the paper that G is a ﬁnite group. We say that G is quasisimple if G is perfect and its quotient by the center L = G/Z is a nonabelian simple group. We say that G is almost simple

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if for some nonabelian simple group L we have L ⊆ G ⊆ Aut L. Our ﬁrst observation is Theorem 1.1. If G is a ﬁnite quasisimple group other than a 4- or 12-cover of P SL(3, 4), then B0 (G) = 0. Corollary 1.2. If G is a ﬁnite simple group, then B0 (G) = 0. This corollary proves Bogomolov’s conjecture. From Corollary 1.2 we deduce the following: Theorem 1.3. If G is a ﬁnite almost simple group, then B0 (G) = 0. Remark 1.4. Following [GL, Ch. 2, §§ 6,7], we call quasisimple groups Q (as in Theorem 1.1) and almost simple groups A (as in Theorem 1.3), as well as the extensions of A by Q, decorations of ﬁnite simple groups. It is most likely that one can complete the picture given in the above theorems, allowing both perfect central extensions and outer automorphisms, by deducing from Theorems 1.1 and 1.3 that B0 (G) = 0 for all nearly simple groups G (see the deﬁnition in Section 2.3 below) excluding the groups related to the above listed exceptional cases. In particular, this statement holds true for all ﬁnite “reductive” groups such as the general linear group GL(n, q), the general unitary group GU (n, q), and the like. Our notation is standard and mostly follows [GLS]. Throughout below “simple group” means “ﬁnite nonabelian simple group”. Our proofs heavily rely on the classiﬁcation of such groups.

2 Preliminaries In order to make the exposition as self-contained as possible, in this section we collect the group-theoretic information needed in the proofs. All groups are assumed ﬁnite (although some of the notions discussed below can be deﬁned for inﬁnite groups as well). 2.1 Schur multiplier The material below (and much more details) can be found in [Ka]. The group M(G) := H 2 (G, Q/Z), where G acts on Q/Z trivially, is called the Schur multiplier of G. It can be identiﬁed with the kernel of some central extension # → G → 1. 1 → M(G) → G # is deﬁned uniquely up to isomorphism provided G is The covering group G perfect (i.e., coincides with its derived subgroup [G, G]).

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We will need to compute M(G) in the case where G is a semidirect product of a normal subgroup N and a subgroup H. If A is an abelian group on which G acts trivially, the restriction map ResH : H 2 (G, A) → H 2 (H, A) gives rise to a split exact sequence [Ka, Prop. 1.6.1] 1 → K → H 2 (G, A) → H 2 (H, A) → 1. The kernel K can be computed from the exact sequence [Ka, Th. 1.6.5(ii)] Res

1 → H 1 (H, Hom(N, A)) → K →N H 2 (N, A)H → H 2 (H, Hom(N, A)). If N is perfect and A = Q/Z, we have Hom(N, A) = 1 and thus [Ka, Lemma 16.3.3] (1) M(G) ∼ = M(N )H × M(H). 2.2 Bogomolov multiplier * The following properties of B0 (G) := ker[H 2 (G, Q/Z) → A H 2 (A, Q/Z)] are taken from [Bo87], [BMP]. * (1) B0 (G) = ker[H 2 (G, Q/Z) → B H 2 (B, Q/Z)], where the product is taken over all bicyclic subgroups B = Zm × Zn of G [Bo87], [BMP, Cor. 2.3]. (2) For an abelian group A denote by Ap its p-primary component. We have B0 (G) = B0,p (G), p

where B0,p (G) := B0 (G) ∩ M(G)p . For any Sylow p-subgroup S of G we have B0,p (G) ⊆ B0 (S). In particular, if all Sylow subgroups of G are abelian, B0 (G) = 0 [Bo87], [BMP, Lemma 2.6]. (3) If G is an extension of a cyclic group by an abelian group, then B0 (G) = 0 [Bo87, Lemma 4.9]. (4) For γ ∈ M(G) consider the corresponding central extension: # γ → G → 1, 1 → Q/Z → G i

and denote Kγ := {h ∈ Q/Z | i(h) ∈

ker(χ)}.

eγ ,Q/Z) χ∈Hom(G

Then γ does not belong to B0 (G) if and only if some nonzero element of # γ [BMP, Kγ can be represented as a commutator of a pair of elements of G Cor. 2.4].

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(5) If 0 = γ ∈ M(G), we say that G is γ-minimal if the restriction of γ to all proper subgroups H ⊂ G is zero. A γ-minimal group must be a p-group. We say that a γ-minimal nonabelian p-group G is a γ-minimal factor if for any quotient map ρ : G → G/H there is no γ ∈ B0 (G/H) such that γ = ρ∗ (γ ) and γ is G/H-minimal. A γ-minimal factor G must be a metabelian group (i.e., [[G, G], [G, G]] = 0) with central series of length at most p, and the order of γ in M(G) equals p [Bo87, Theorem 4.6]. Moreover, if G is a γ-minimal p-group which is a central extension of Gab := G/[G, G] and Gab = (Zp )n , then n = 2m and n ≥ 4 [Bo87, Lemma 5.4]. 2.3 Finite simple groups We need the following facts concerning ﬁnite simple groups (see, e.g., [GLS]) believing that the classiﬁcation of ﬁnite simple groups is complete. (1) Classiﬁcation. Any ﬁnite simple group L is either a group of Lie type, or an alternating group, or one of 26 sporadic groups. # and (2) Schur multipliers. As L is perfect, it has a unique covering group L, ∼ # L = L/ M(L). The Schur multipliers M(L) of all ﬁnite simple groups L are given in [GLS, 6.1]. (3) Automorphisms. The group of outer automorphisms Out(L) := Aut(L)/L is solvable. It is abelian provided L is an alternating or a sporadic group. For groups of Lie type deﬁned over a ﬁnite ﬁeld F = Fq the structure of Out(L) can be described as follows. If L comes from a simple algebraic group L deﬁned over F , we denote by T a maximal torus in L. Every automorphism of L is a product idf g where i is an inner automorphism (identiﬁed with an element of L), d is a diagonal automorphism (induced by conjugation by an element h of the normalizer NT (L), see [GLS, 2.5.1(b)]), f is a ﬁeld automorphism (arising from an automorphism of the ﬁeld F ), and g is a graph automorphism (induced by an automorphism of the Dynkin diagram corresponding to L); see [GL, Ch. 2, § 7] or [GLS, 2.5] for more details. The group Out(L) is a split extension of Outdiag(L) := Inndiag(L)/L by the group ΦΓ , where Inndiag(L) is the group of inner-diagonal automorphisms of L (generated by all i’s and d’s as above), Φ is the group of ﬁeld automorphisms, and Γ is the group of graph automorphisms of # by the L. The group O = Outdiag(L) is isomorphic to the center of L isomorphism preserving the action of Aut(L) and is nontrivial only in the following cases (where (m, n) stands for the greatest common divisor of m and n): L is of type An (q); O = Z(n+1,q) ; L is of type 2 An (q); O = Z(n+1,q−1) ; L is of type Bn (q), Cn (q), or 2 D2n (q); O = Z(2,q−1) ; L is of type D2n (q); O = Z(2,q−1) × Z(2,q−1) ;

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L is of type 2 D2n+1 (q); O = Z(4,q−1) ; L is of type 2 E6 (q); O = Z(3,q−1) ; L is of type E7 (q); O = Z(2,q−1) . If L is of type d Σ(q) for some root system Σ (d = 1, 2, 3), the group Φ is isomorphic to Aut(Fqd ). If d = 1, then Γ is isomorphic to the group of symmetries of the Dynkin diagram of Σ and ΦΓ = Φ × Γ provided Σ is simply-laced; otherwise, Γ = 1 except if Σ = B2 , F4 , or G2 and q is a power of 2, 2, or 3, respectively, in which cases ΦΓ is cyclic and [ΦΓ : Φ] = 2. If d = 1, then Γ = 1. The action of ΦΓ on O is described as follows. If L ∼ D2n (q), then Φ acts = on the cyclic group O as Aut(Fqd ) does on the multiplicative subgroup of Fqd of the same order as O; if L ∼ = D2n (q), then Φ centralizes O. If L∼ = An (q), D2n+1 (q), or E6 (q), then Γ = Z2 acts on O by inversion; if L∼ = D2n (q) and q is odd, then Γ , which is isomorphic to the symmetric group S3 (for m = 2) or to Z2 (for m > 2), acts faithfully on O = Z2 × Z2 . (4) Decorations. It is often useful to consider groups close to ﬁnite simple groups, namely, quasisimple and almost simple groups, as in the statements of Theorems 1.1 and 1.3 above. As an example, if the simple group under consideration is L = P SL(2, q), the group SL(2, q) is quasisimple and the group P GL(2, q) is almost simple. More generally, one can consider semisimple groups (central products of quasisimple groups) and nearly simple groups G, i.e., such that the generalized Fitting subgroup F ∗ (G) is quasisimple. F ∗ (G) is deﬁned as the product E(G)F (G) where E(G) is the layer of G (the maximal semisimple normal subgroup of G) and F (G) is the Fitting subgroup of G (or the nilpotent radical, i.e., the maximal nilpotent normal subgroup of G). The general linear group GL(n, q) is an example of a nearly simple group.

3 Proofs Proof (of Theorem 1.1). As G is perfect, there exists a unique universal central # of G whose center Z(G) # is isomorphic to M(G) and any other covering G # So we can argue exactly as perfect central extension of G is a quotient of G. in [Bo87, Remark after Lemma 5.7] and [BMP]. Namely, B0 (G) coincides with the collection of classes whose restriction to any bicyclic subgroup of G is zero, see 2.2(1). Therefore, to establish the assertion of the theorem, it is enough # can be represented as a commutator z = [a, b] to prove that any z ∈ Z(G) # Moreover, it is enough to prove that such a representation of some a, b ∈ G. exists for all elements z of prime power order, see 2.2(2). It remains to apply the results of Blau [Bl] who classiﬁed all elements z # having a ﬁxed point in the natural action on the set of conjugacy classes of G (such elements evidently admit a needed representation as a commutator):

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Theorem 3.1. ([Bl, Theorem 1]) Assume that G is a quasisimple group and let z ∈ Z(G). Then one of the following holds: (i) order(z) = 6 and G/Z(G) ∼ = A6 , A7 , F i22 , P SU (6, 22 ), or 2 E6 (22 ); (ii) order(z) = 6 or 12 and G/Z(G) ∼ = P SL(3, 4), P SU (4, 32), or M22 ; ∼ (iii) order(z) = 2 or 4, G/Z(G) = P SL(3, 4), and Z(G) is noncyclic; (iv) there exists a conjugacy class C of G such that Cz = C. This theorem implies that the only possibility for an element of G of prime power order to act on the set of conjugacy classes without ﬁxed points is case # # ∼ (iii) where G/Z( G) = P SL(3, 4) and z is an element of order 2 or 4. So the classes γ ∈ H 2 (G, Q/Z) corresponding to such z’s are the only candidates for nonzero elements of B0 (G). # ∼ A more detailed analysis of the case P SL(3, 4), where Z := Z(G) = Z3 × Z4 ×Z4 , is sketched in [Bl, Remark (2) after Theorem 1]. The result (rechecked by MAGMA computations) looks as follows: all elements of Z of orders 2 and # all elements of orders 6 and 12 act without 3 ﬁx some conjugacy class of G, ﬁxed points, and of the twelve elements of order 4 exactly six ﬁx a conjugacy # class of G. First note that this description implies B0 (P SL(3, 4)) = 0. Indeed, the criterion given in 2.2(4) can be rephrased for a quasisimple group G as follows: # has a system of generators each of which can be B0 (G) = 0 if and only if Z(G) # It remains to notice represented as a commutator of a pair of elements of G. that if G = P SL(3, 4), each 5-tuple of elements of order 4 in Z generates Z4 × Z4 because the subgroup of the shape Z4 × Z2 contains only 4 elements of order 4 (I am indebted to O. Gabber for this argument). (Another way to prove that B0 (P SL(3, 4)) = 0 was demonstrated in [BMP] where Lemma 5.3 establishes a stronger result: vanishing of B0 (S), where S is a 2-Sylow subgroup of P SL(3, 4).) The only cases where the condition of the above-mentioned criterion breaks down are those where G is a 12- or 4-cover of P SL(3, 4). Indeed, in these cases # where Z is generated either by an element of order 4 (which we have G = G/Z may not be representable as a commutator) or of order 12 (which cannot be representable as a commutator). Thus in these cases we have B0 (G) = 0. More precisely, in both cases we have B0 (G) = Z2 because the subgroup generated by commutators is of index 2 in G (I thank O. Gabber for this remark). Remark 3.2. It is interesting to compare [BMP, Lemma 3.1] with a theorem from the PhD thesis of Robert Thompson [Th, Theorem 1]. Proof (of Theorem 1.3). Let L ⊆ G ⊆ Aut(L) where L is a simple group. Clearly, it is enough to prove the theorem for G = Aut(L). The group Out(L) = Aut(L)/L of outer automorphisms of L acts on M(L), and since L is perfect, we have an isomorphism M(G) ∼ = M(L)Out(L) × M(Out(L)) (see (1)).

(2)

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Lemma 3.3. B0 (Out(L)) = 0. Proof (of Lemma 3.3). We maintain the notation of Section 2.3. If Out(L) is abelian, the statement is obvious. This includes the cases where L is an alternating or a sporadic group. So we may assume L is of Lie type. If O = 1, i.e., L is of type E8 , F4 , or G2 , the result follows immediately. If the group ΦΓ is cyclic, the result follows because O is abelian (see 2.3(3)). This is the case for all groups having no graph automorphisms, in particular, for all groups of type Bn or Cn (n ≥ 3), E7 , and for all twisted forms. For the groups of type B2 , the group ΦΓ is always cyclic. It remains to consider the cases An , Dn , and E6 . In the case L = E6 all Sylow p-subgroups of Out(L) are abelian, and the result holds. Let L = D2m (q). If q is even, we have O = 1, Γ = Z2 (if m > 2) or S3 (if m = 2); in both cases the Sylow p-subgroups of Out(L) are abelian, and we are done. If q is odd, we have O = Z2 × Z2 , and Φ centralizes O (see Section 2.3), so every Sylow p-subgroup of Out(L) can be represented as an extension of a cyclic group by an abelian group, and we conclude as above. Finally, let L be of type An (q) or D2m+1 (q). Then we have O = Zh , h = (n + 1, q − 1) or h = (4, q − 1), respectively, Γ = Z2 , Φ = Aut(Fq ). The action of both Γ and Φ on O may be nontrivial: Γ acts by inversion, Φ acts on O as Aut(Fq ) does on the multiplicative subgroup of Fq of the same order as O. Hence we can represent the metabelian group Out(L) in the form 1 → V → Out(L) → A → 1,

(3)

where V , the derived subgroup of Out(L), is isomorphic to a cyclic subgroup Zc of O, and the abelian quotient A is of the form Za ×Zb ×Z2 for some integers a, b, c. Since it is enough to establish the result for a Sylow 2-subgroup, we may assume that a, b, and c are powers of 2. Then the statement of the lemma follows from the properties of γ-minimal elements described in Section 2.2. Indeed, if γ is a nonzero element of B0 (G) and G is γ-minimal, then G is metabelian, both V and A are of exponent p, and in any representation of G in the form (3) the group A must have an even number s = 2t of direct summands Zp with t ≥ 2. However, if G is a Sylow 2-subgroup of Out(L), this is impossible because A contains only three direct summands. Thus B0 (Syl2 (Out(L))) = 0, and so B0 (Out(L)) = 0. The lemma is proved. We can now ﬁnish the proof of the theorem. Let γ be a nonzero element of B0 (G). Using the isomorphism (2), we can represent γ as a pair (γ1 , γ2 ) where γ1 ∈ M(L), γ2 ∈ M(Out(L)). Restricting to the bicyclic subgroups of G, we see that γ1 ∈ B0 (L), γ2 ∈ B0 (Out(L)), and the result follows from Theorem 1.1 and Lemma 3.3.

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[Bo87] F. A. Bogomolov, The Brauer group of quotient spaces by linear group actions, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987) 485–516; English transl. in Math. USSR Izv. 30 (1988) 455–485. [Bo92] F. A. Bogomolov, Stable cohomology of groups and algebraic varieties, Mat. Sb. 183 (1992) 1–28; English transl. in Sb. Math. 76 (1993) 1–21. [Bo07] F. Bogomolov, Stable cohomology of finite and profinite groups, “Algebraic Groups” (Y. Tschinkel, ed.), Universit¨ atsverlag G¨ ottingen, 2007, 19–49. [BMP] F. Bogomolov, J. Maciel, T. Petrov, Unramified Brauer groups of finite simple groups of Lie type A , Amer. J. Math. 126 (2004) 935–949. [CTS] J.-L. Colliot-Th´ el` ene, J.-J. Sansuc, The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group), Proc. Int. Colloquium on Algebraic Groups and Homogeneous Spaces (Mumbai 2004) (V. Mehta, ed.), TIFR Mumbai, Narosa Publishing House, 2007, 113–186. [GS] P. Gille, T. Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge University Press, Cambridge, 2006. [GL] D. Gorenstein, R. Lyons, The Local Structure of Finite Groups of Characteristic 2 Type, Mem. Amer. Math. Soc., vol. 276, Providence, RI, 1983. [GLS] D. Gorenstein, R. Lyons, R. Solomon, The Classification of the Finite Simple Groups, Number 3, Math. Surveys and Monographs, vol. 40, Amer. Math. Soc., Providence, RI, 1998. [Ka] G. Karpilovsky, Group Representations, vol. 2, North-Holland Math. Studies 177, North-Holland, Amsterdam, 1993. [Sa] D. J. Saltman, Noether’s problem over an algebraically closed field, Invent. Math. 77 (1984) 71–84. [Sh] I. R. Shafarevich, The L¨ uroth problem, Tr. Mat. Inst. Steklov 183 (1990) 199–204; English transl. in Proc. Steklov Inst. Math. 183 (1991) 241–246. [Th] R. C. Thompson, Commutators in the special and general linear groups, Trans. Amer. Math. Soc. 101 (1961) 16–33.

Derived Categories of Cubic Fourfolds Alexander Kuznetsov Algebra Section Steklov Mathematical Institute 8 Gubkin street Moscow 119991, Russia [email protected] Summary. We discuss the structure of the derived category of coherent sheaves on cubic fourfolds of three types: Pfaﬃan cubics, cubics containing a plane, and singular cubics, and discuss its relation to the rationality of these cubics.

Key words: Rationality, derived categories, cubic hypersurfaces 2000 Mathematics Subject Classiﬁcation codes: 14E08, 14J35

1 Introduction In this paper we are going to discuss one of the classical problems of birational algebraic geometry, the problem of rationality of a generic cubic hypersurface in P5 . Our point of view will be somewhat diﬀerent from the traditional approaches. We will use the derived category of coherent sheaves on the cubic hypersurface (more precisely, a certain piece of this category) as an indicator of nonrationality. To be more precise, let V be a vector space of dimension 6, so that P(V ) = P5 . Let Y ⊂ P(V ) be a hypersurface of degree 3, a cubic fourfold. By Lefschetz hyperplane section theorem Pic Y = Z and is generated by H, the restriction of the class of a hyperplane in P(V ). By adjunction KY = −3H. So, Y is a Fano variety. By Kodaira vanishing we can easily compute the cohomology of line bundles OY , OY (−1), and OY (−2). ! 1, for p = t = 0 p dim H (Y, OY (−t)) = (1) 0, for −2 ≤ t ≤ 0 and (p, t) = (0, 0). As a consequence, we see that the triple (OY , OY (1), OY (2)) is an exceptional collection in Db (Y ), the bounded derived category of coherent sheaves on Y . We denote by AY the orthogonal subcategory to this exceptional collection:

F. Bogomolov, Y. Tschinkel (eds.), Cohomological and Geometric Approaches to Rationality Problems, Progress in Mathematics 282, DOI 10.1007/978-0-8176-4934-0_9, © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

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AY = OY , OY (1), OY (2) ⊥ •

(2) •

•

= {F ∈ D (Y ) | H (Y, F ) = H (Y, F (−1)) = H (Y, F (−2)) = 0}. b

Then we have a semiorthogonal decomposition Db (Y ) = AY , OY , OY (1), OY (2)

(3)

(see Section 2 for the deﬁnition of semiorthogonal decompositions). This triangulated category AY is the piece of Db (Y ) discussed above. By many features it looks like the derived category of a K3 surface (for example, its Serre functor equals the shift by two functor [2], and its Hochschild homology is very similar to that of a K3 surface). Moreover, as we shall see, for some cubic fourfolds AY is equivalent to the derived category of a K3 surface. We expect that this happens if and only if Y is rational. Conjecture 1.1. The cubic fourfold Y is rational if and only if the subcategory AY ⊂ Db (Y ) deﬁned by (2) is equivalent to the derived category of a K3 surface. The “only if” part of the above is a special case of a more general conjecture suggested in [K4], where a new birational invariant, the Clemens–Griﬃths component of the derived category, is constructed. While it is very hard (and not completely clear how) to check that the deﬁnition of this invariant is correct, it is more or less straightforward that it is preserved under smooth blowups, which implies that it is a birational invariant. In the case of the cubic fourfold Y the Clemens–Griﬃths component is either 0, if AY is equivalent to the derived category of a K3 surface, or AY itself, in the opposite case. So, Conjecture 1.1 can be reformulated as stating that Y is rational if and only if the Clemens–Griﬃths component of its derived category is zero. Our goal is to give some evidence for Conjecture 1.1. More precisely, we analyze the category AY for all cubic fourfolds that are known to be rational, and show that in these cases AY is equivalent to the derived category of some K3 surface. Moreover, we give examples of cubic fourfolds Y for which AY is not equivalent to the derived category of any K3 surface. So, we expect these cubic fourfolds to be nonrational. Basically, there are three known series of rational cubic fourfolds: 1. Pfaﬃan cubic fourfolds [Tr1, Tr2]; 2. cubic fourfolds Y containing a plane P ∼ = P2 and a two-dimensional algebraic cycle T such that T · H 2 − T · P is odd [Ha1, Ha2]; 3. singular cubic fourfolds. We are dealing with these in Sections 3, 4, and 5, respectively, after introducing some necessary material in Section 2. Moreover, in Section 4 we investigate a more general case of a cubic fourfold containing a plane without any additional conditions. We show that in this case the category AY is equivalent to the twisted derived category of a K3 surface S (the twisting is given by a Brauer

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class of order 2) and in the Appendix we argue that this twisted derived category is not equivalent to the derived category of any surface if Pic S ∼ =Z and the Brauer class is nontrivial, which is true for general cubic fourfold with a plane. So, we expect such cubic fourfolds to be nonrational. Acknowledgment. I am very grateful to A. Bondal and D. Orlov for useful discussions and to L. Katzarkov for inspiring and stimulating questions, and especially for attracting my attention to the case of singular cubic fourfolds. I am also grateful to D. Huybrechts for clariﬁcations on the twisted Chern character and to F. Bogomolov and D. Kaledin for discussions on the Brauer group of K3 surfaces. I was partially supported by RFFI grants 05-01-01034, 07-01-00051, and 07-01-92211, INTAS 05-1000008-8118, NSh-9969.2006.1, the Russian Science Support Foundation, and gratefully acknowledge the support of the Pierre Deligne fund based on his 2004 Balzan prize in mathematics.

2 Preliminaries Let k be an algebraically closed ﬁeld of zero characteristic. All algebraic varieties will be over k and all additive categories will be k-linear. By Db (X) we denote the bounded derived category [Ve] of coherent sheaves on an algebraic variety X. This category is triangulated. For any morphism f : X → X of algebraic varieties we denote by f∗ : Db (X) → Db (X ) and by f ∗ : Db (X ) → Db (X) the derived pushforward and pullback functors (in ﬁrst case we need f to be proper, and in the second to have ﬁnite Tor-dimension; these assumptions ensure the functors to preserve both boundedness and coherence). Similarly, ⊗ stands for the derived tensor product. For a proper morphism of ﬁnite Tor-dimension f : X → X we will also use the right adjoint f ! of the pushforward functor ([H, N]), which is given by the formula f ! (F ) ∼ = f ∗ (F ) ⊗ ωX/X [dim X − dim X ], where ωX/X is the relative canonical line bundle. 2.1 Semiorthogonal decompositions Let T be a triangulated category. Definition 2.1 ([BK, BO]). A semiorthogonal decomposition of a triangulated category T is a sequence of full triangulated subcategories A1 , . . . , An in T such that HomT (Ai , Aj ) = 0 for i > j and for every object T ∈ T there exists a chain of morphisms 0 = Tn → Tn−1 → · · · → T1 → T0 = T such that the cone of the morphism Tk → Tk−1 is contained in Ak for each k = 1, 2, . . . , n.

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We will write T = A1 , A2 , . . . , An for a semiorthogonal decomposition of a triangulated category T with components A1 , A2 , . . . , An . An important property of a triangulated subcategory A ⊂ T ensuring that it can be extended to a semiorthogonal decomposition is admissibility. Definition 2.2 ([BK, B]). A full triangulated subcategory A of a triangulated category T is called admissible if for the inclusion functor i : A → T there is a right adjoint i! : T → A, and a left adjoint i∗ : T → A functor. Lemma 2.3 ([BK, B]). (i) If A1 , . . . , An is a semiorthogonal sequence of admissible subcategories in a triangulated category T (i.e., HomT (Ai , Aj ) = 0 for i > j), then + , + , , + A1 , . . . , Ak , ⊥ A1 , . . . , Ak ∩ Ak+1 , . . . , An ⊥ , Ak+1 , . . . , An is a semiorthogonal decomposition. , + (ii) If Db (X) = A1 , A2 , . . . , An is a semiorthogonal decomposition of the derived category of a smooth projective variety X, then each subcategory Ai ⊂ Db (X) is admissible. Actually the second part of the lemma holds for any saturated (see [BK]) triangulated category. Definition 2.4 ([B]). An object F ∈ T is called exceptional if Hom(F, F ) = k and Extp (F, F ) = 0 for all p = 0. A collection of exceptional objects (F1 , . . . , Fm ) is called exceptional if Extp (Fl , Fk ) = 0 for all l > k and all p ∈ Z. Assume that T is Hom-ﬁnite (which means that for any G, G ∈ T the vector space ⊕t∈Z Hom(G, G [t]) is ﬁnite-dimensional). + , Lemma 2.5 ([B]). The subcategory F of Db (X) generated by an exceptional object F is admissible and is equivalent to the derived category of vector spaces Db (k). Proof. Consider the functor Db (k) → Db (X) deﬁned by V → V ⊗ F , where V is a complex of vector + , spaces. It is fully faithful since F is exceptional, hence the subcategory F of Db (X) is equivalent to Db (k). The adjoint functors

are given by G → RHom(G, F )∗ and G → RHom(F, G), respectively. As a consequence of 2.3 and of 2.5 one obtains the following: Corollary 2.6 ([BO]). If X is a smooth projective algebraic variety, then any exceptional collection F1 , . . . , Fm in Db (X) induces a semiorthogonal decomposition Db (V ) = A, F1 , . . . , Fm where A = F1 , . . . , Fm ⊥ = {F ∈ Db (X) | Ext• (Fk , F ) = 0 for all 1 ≤ k ≤ m}. An example of such a semiorthogonal decomposition is given by (3). Indeed, by (1) the triple of line bundles (OY , OY (1), OY (2)) is an exceptional collection in Db (Y ), which gives a semiorthogonal decomposition with the ﬁrst component deﬁned by (2).

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2.2 Mutations If a triangulated category T has a semiorthogonal decomposition, then usually it has quite a lot of them. More precisely, there are two groups acting on the set of semiorthogonal decompositions — the group of autoequivalences of T , and a certain braid group. The action of the braid group is given by the so-called mutations. Roughly speaking, the mutated decomposition is obtained by dropping one of the components of the decomposition and then extending the obtained semiorthogonal collection by inserting new component at some other place as in Lemma 2.3. More precisely, the basic two operations are deﬁned as follows. Lemma 2.7 ([B]). Assume that A ⊂ T is an admissible ,subcategory, + so that , + we have two semiorthogonal decompositions T = A⊥ , A and T = A, ⊥ A . Then there are functors LA , RA : T → T vanishing on A and inducing mutually inverse equivalences ⊥ A → A⊥ and A⊥ → ⊥ A, respectively. The functors LA and RA are known as the left and the right mutation functors. Proof. Let i : A → T be the embedding functor. For any F ∈ T we deﬁne LA (F ) = Cone(ii! F → F ),

RA (F ) = Cone(F → ii∗ F )[−1].

Note that the cones in this triangles are functorial due to the semiorthogonality. All the properties are veriﬁed directly.

Remark 2.8. If A is generated by an exceptional object E, we can use explicit formulas for the adjoint functors i! , i∗ of the embedding functor i : A → T . Thus we obtain the following distinguished triangles RHom(E, F ) ⊗ E → F → LE (F ),

RE (F ) → F → RHom(F, E)∗ ⊗ E. (4)

It is easy to deduce from Lemma 2.7 the following: + , Corollary 2.9 ([B]). Assume that T = A1 , A2 , . . . , An is a semiorthogonal decomposition with all components being admissible. Then for each 1 ≤ k ≤ n − 1 there is a semiorthogonal decomposition + , T = A1 , . . . , Ak−1 , LAk (Ak+1 ), Ak , Ak+2 , . . . , An and for each 2 ≤ k ≤ n there is a semiorthogonal decomposition + , T = A1 , . . . , Ak−2 , Ak , RAk (Ak−1 ), Ak+1 , . . . , An . There are two cases when the action of the mutation functors is particularly simple.

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, + Lemma 2.10. Assume that T = A1 , A2 , . . . , An is a semiorthogonal decomposition with all components being admissible. Assume also that the components Ak and Ak+1 are completely orthogonal, i.e., Hom(Ak , Ak+1 ) = 0 as well as Hom(Ak+1 , Ak ) = 0. Then LAk (Ak+1 ) = Ak+1

and

RAk+1 (Ak ) = Ak ,

so that both the left mutation of Ak+1 through Ak and the right mutation of Ak through Ak+1 boil down to just a permutation and , + T = A1 , . . . , Ak−1 , Ak+1 , Ak , Ak+2 , . . . , An is the resulting semiorthogonal decomposition of T . b +Lemma , 2.11. Let X be a smooth projective algebraic variety and D (X) = A, B a semiorthogonal decomposition. Then

LA (B) = B ⊗ ωX

and

−1 RB (A) = A ⊗ ωX .

An analogue of this lemma holds for any triangulated category which has a Serre functor (see [BK]). In this case tensoring by the canonical class should be replaced by the action of the Serre functor. We will also need the following evident observation. Lemma 2.12. Let Φ be an autoequivalence of T . Then Φ ◦ RA ∼ = RΦ(A) ◦ Φ,

Φ ◦ LA ∼ = LΦ(A) ◦ Φ.

In particular, if L is a line bundle on X and E is an exceptional object in Db (X), then TL ◦ RE ∼ = RE⊗L ◦ TL ,

TL ◦ LE ∼ = LE⊗L ◦ TL ,

where TL : Db (X) → Db (X) is the functor of tensor product by L.

3 Pfaﬃan cubic fourfolds Let W be a vector space of dimension 6. Consider P(Λ2 W ∗ ), the space of skew-forms on W , and its closed subset Pf(W ) = {ω ∈ P(Λ2 W ∗ ) | ω is degenerate}. It is well known that a skew form is degenerate if and only if its Pfaﬃan is zero, so Pf(W ) is a hypersurface in P(Λ2 W ∗ ) and its equation is given by the Pfaﬃan in homogeneous coordinates of P(Λ2 W ∗ ). In particular, deg Pf(W ) =

1 dim W = 3. 2

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We will say that Pf(W ) is the Pfaﬃan cubic hypersurface. It is easy to check that Pf(W ) is singular, its singularity is the image of the Grassmannian Gr(2, W ∗ ) under the Plücker embedding. So, . codimP(Λ2 W ∗ ) sing(Pf(W )) = dim P(Λ2 W ∗ ) − dim Gr(2, W ∗ ) = 14 − 8 = 6. Let now V be a 6-dimensional subspace of Λ2 W ∗ such that P(V ) ⊂ Pf(W ∗ ). Then YV = P(V ) ∩ Pf(W ∗ ) is a cubic hypersurface in P(V ), its equation being the restriction of the Pfaﬃan in coordinates of P(Λ2 W ∗ ) to P(V ). We will say that YV is the Pfaﬃan cubic fourfold associated with the subspace V ⊂ Λ2 W ∗ . Note that due to the big codimension of sing(Pf(W ∗ )), the Pfaﬃan cubic fourfold YV is smooth for suﬃciently generic V . With each V ⊂ Λ2 W ∗ we can also associate a global section sV of the vector bundle V ∗ ⊗ OGr(2,W ) (1) on the Grassmannian Gr(2, W ) and its zero locus XV ⊂ Gr(2, W ). The following was proved in [K3]. Theorem 3.1. Let YV be a smooth Pfaﬃan cubic fourfold. Then XV is a smooth K3 surface and there is an equivalence of categories AYV ∼ = Db (XV ). Remark 3.2. The same holds true even for singular Pfaﬃan cubic fourfolds as long as XV is a surface.

4 Cubic fourfolds with a plane Let Y ⊂ P(V ) be a cubic fourfold containing a plane. In other words we assume that there is a vector subspace A ⊂ V of dimension 3 such that P(A) ⊂ Y . Let σ : Y# → Y be the blowup of P(A). Let also B = V /A be the quotient space. Then the linear projection from P(A) gives a regular map π : Y# → P(B) = P2 . Lemma 4.1. The map π is a ﬁbration in two-dimensional quadrics over P(B) with the degeneration curve of degree 6. Let D be the exceptional divisor of σ. Let i : D → Y# be the embedding and p : D → P(A) the projection. D { p {{ { {{ }{ { /Y P(A)

i σ

/ Y# CC CC π CC CC ! P(B)

Let H and h denote the pullbacks to Y# of the classes of hyperplanes in P(V ) and P(B), respectively. Then we have linear equivalences D = H − h,

h = H − D,

KYe = −3H + D = −2H − h.

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) of P(V ) in P(A). It is naturally projected Proof. Consider the blowup P(V ) with PP(B) (E) for a vector bundle to P(B) and this projection identiﬁes P(V E = A ⊗ OP(B) ⊕ OP(B) (−1) ) → P(V ) decomon P(B). The complete preimage of Y under the map P(V poses as the union of the proper preimage and the exceptional divisor. It ) of the follows that Y# = 3H − D , where H stands for the pullback to P(V class of a hyperplane in P(V ) and D stands for the exceptional divisor of ), where h stands for the ). Note also that we have h = H − D on P(V P(V ) of the class of a hyperplane in P(B). Thus Y# = 2H + h , pullback to P(V so it follows that the ﬁbers of Y# over P(B) are the quadrics in the ﬁbers ) = PP(B) (E). Moreover, it follows that the degeneration curve is the of P(V vanishing locus of the determinant of the morphism E → E ∗ ⊗ OP(B) (1), given by the equation of Y# . So, the degeneration curve is the zero locus of a section of the line bundle (det E ∗ )2 ⊗ OP(B) (4) ∼ = OP(B) (6). Restricting the relation h = H − D to Y# we deduce h = H − D. As for the canonical class, the ﬁrst equality is evident since Y# is a smooth blowup, and replacing D by H − h we get the last equality.

Now we will need the results of [K1] on the structure of the derived category of coherent sheaves on a ﬁbration in quadrics. According to loc. cit. there is a semiorthogonal decomposition Db (Y# ) = Φ(Db (P(B), B0 )), π ∗ (Db (P(B))) ⊗ OYe , π ∗ (Db (P(B))) ⊗ OYe (H) , (5) where B0 is the sheaf of even parts of Cliﬀord algebras corresponding to this ﬁbration. The functor Φ is described below. As a coherent sheaf on P(B) the sheaf B0 is given by the formula B0 = OP(B)

⊕

Λ2 E ⊗ OP(B) (−1)

⊕

Λ4 E ⊗ OP(B) (−2)

and the multiplication law depends on the equation of Y# . Further, Db (P(B), B0 ) is the bounded derived category of sheaves of coherent right B0 -modules on P(B), and the functor Φ : Db (P(B), B0 ) → Db (Y# ) is deﬁned as follows. Consider also the sheaf of odd parts of the corresponding Cliﬀord algebras, B1 = E

⊕

Λ3 E ⊗ OP(B) (−1).

Denote by α the embedding Y# → PP(B) (E), and by q : PP(B) (E) → P(B) the projection. Then there is a canonical map of left q ∗ B0 -modules q ∗ B0 →

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q ∗ B1 (H ), which is injective and its cokernel is supported on Y# . So, twisting by O(−2H ) for later convenience we obtain an exact sequence 0 → q ∗ B0 (−2H ) → q ∗ B1 (−H ) → α∗ E → 0, where E is a sheaf of left π ∗ B0 -modules on Y# . The functor Φ is just the kernel functor given by E, that is, Φ(F ) = π ∗ F ⊗π∗ B0 E. At ﬁrst glance, the category Db (P(B), B0 ) has nothing to do with K3 surfaces. However, after a small consideration it is easy to see that it has. Let us explain this. From now on we assume that the ﬁbers of Y# over P(B) are quadrics of rank ≥ 3 (i.e., they do not degenerate into a union of two planes). This condition holds if Y is suﬃciently generic. Consider the moduli space M of lines contained in the ﬁbers of Y# over P(B). Since on a smooth two-dimensional quadric the moduli space of lines is a disjoint union of two P1 , while on a singular quadric the moduli space of lines is one P1 , it follows that M is a P1 ﬁbration over the double cover S of P(B) ramiﬁed in the degeneration curve. Note that S is a K3 surface, and M produces an element of order 2 in the Brauer group of S. Let B be the corresponding sheaf of Azumaya algebras on S. Denote by f : S → P(B) the double covering. Lemma 4.2. We have f∗ B ∼ = B0 . In particular, Db (S, B) ∼ = Db (P(B), B0 ). The composition of the equivalence with the functor Φ is given by F → π ∗ f∗ F ⊗π∗ B0 E. Proof. The ﬁrst claim is the classical property of the Cliﬀord algebra. Indeed, M is nothing but the isotropic Grassmannian of half-dimensional linear subspaces in the quadrics, which always embeds into the projectivizations of the half-spinor modules, and for a two-dimensional quadric this embedding is an isomorphism. So (locally in the étale topology) M is the projectivization of the half-spinor module. On the other hand, the even part of the Cliﬀord algebra acts on the half-spinor modules and this action identiﬁes it with the product of their matrix algebras. Hence (locally in the étale topology) the Cliﬀord algebra is isomorphic to the pushforward of the endomorphism algebra of the half-spinor module which is precisely the Azumaya algebra corresponding to its projectivization. The second claim is evident (the equivalence is given by the pushforward

functor f∗ ). Theorem 4.3. There is an equivalence of categories AY ∼ = Db (S, B). Proof. Consider the semiorthogonal decomposition (5). Replacing the ﬁrst instance of Db (P(B)) by the exceptional collection

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(OP(B) (−1), OP(B) , OP(B) (1)) and the second instance of Db (P(B)) by the exceptional collection (OP(B) , OP(B) (1), OP(B) (2)) we obtain a semiorthogonal decomposition of Db (Y# ) as Φ(Db (P(B), B0 )), OYe (−h), OYe , OYe (h), OYe (H), OYe (h + H), OYe (2h + H) . (6) On the other hand, since Y# is the blowup of Y in P(A), we have by [Or] the following semiorthogonal decomposition: Db (Y# ) = σ ∗ (Db (Y )), i∗ p∗ (Db (P(A))) . Replacing Db (P(A)) with the standard exceptional collection (OP(A) , OP(A) (1), OP(A) (2)), and Db (Y ) with its decomposition (3), we obtain the following semiorthogonal decomposition Db (Y# ) = σ ∗ (AY ), OYe , OYe (H), OYe (2H), i∗ OD , i∗ OD (H), i∗ OD (2H) . (7) Now we are going to make a series of mutations, transforming decomposition (6) into (7). This will give the required equivalence Db (S, B) ∼ = Db (P(B), B0 ) ∼ = AY . Now let us describe the series of mutations. We start with decomposition (6). Step 1. Right mutation of Φ(Db (P(B), B0 )) through OYe (−h). After this mutation we obtain the following decomposition of Db (Y# ) as OYe (−h), Φ (Db (P(B), B0 )), OYe , OYe (h), OYe (H), OYe (h + H), OYe (2h + H) , (8) where Φ = ROYe (−h) ◦ Φ : Db (P(B), B0 ) → Db (Y# ). Step 2. Right mutation of OYe (−h) through the orthogonal subcategory ⊥ OYe (−h) . Applying Lemma 2.11 and taking into account equality KYe = −2H − h, we obtain as decomposition of Db (Y# ): Φ (Db (P(B), B0 )), OYe , OYe (h), OYe (H), OYe (h + H), OYe (2h + H), OYe (2H) . (9) Lemma 4.4. We have Ext• (OYe (2h + H), OYe (2H)) = 0, so that the pair (OYe (2h + H), OYe (2H)) is completely orthogonal.

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Proof. We have Ext• (OYe (2h + H), OYe (2H)) = H • (Y# , OYe (H − 2h))

= H • (P(B), E ∗ ⊗OP(B) (−2)) = H • (P(B), A∗ ⊗OP(B) (−2)⊕OP(B)(−1)) = 0.

By Lemma 2.10 the transposition of the pair (OYe (2h + H), OYe (2H)) gives a semiorthogonal decomposition. Step 3. Transpose the pair (OYe (2h + H), OYe (2H)). After the transposition we obtain the following decomposition of Db (Y# ): Φ (Db (P(B), B0 )), OYe , OYe (h), OYe (H), OYe (h + H), OYe (2H), OYe (2h + H) . (10) Step 4. Left mutation of OYe (2h + H) through the orthogonal subcategory OYe (2h + H) ⊥ . Applying Lemma 2.11 and taking into account equality KYe = −2H − h, we obtain that Db (Y# ) is OYe (h − H), Φ (Db (P(B), B0 )), OYe , OYe (h), OYe (H), OYe (h + H), OYe (2H) . (11) Step 5. Left mutation of Φ (Db (P(B), B0 )) through OYe (h − H). After this mutation we obtain the following decomposition of Db (Y# ): Φ (Db (P(B), B0 )), OYe (h − H), OYe , OYe (h), OYe (H), OYe (h + H), OYe (2H) , (12) where Φ = LOYe (h−H) ◦ Φ : Db (P(B), B0 ) → Db (Y# ). Step 6. Simultaneous right mutation of OYe (h − H) through OYe , of OYe (h) through OYe (H), and of OYe (h + H) through OYe (2H). Lemma 4.5. We have ROYe (OYe (h − H)) ∼ = i∗ OD [−1], ROYe (H) (OYe (h)) ∼ = i∗ OD (H)[−1], and also ROYe (2H) (OYe (h + H)) ∼ = i∗ OD (2H)[−1]. Proof. Note that Ext• (OYe (h − H), OYe ) = H • (Y# , OYe (H − h))

= H • (P(B), E ∗ ⊗ OP(B) (−1)) = H • (P(B), A∗ ⊗ OP(B) (−1) ⊕ OP(B) ).

It follows that

! dim Ext (OYe (h − H), OYe ) = p

1, for p = 0 0, otherwise

By (4) we have the following distinguished triangle: ROYe (OYe (h − H)) → OYe (h − H) → OYe . Since H − h = D, the right map in the triangle is given by the equation of D, so it follows that the ﬁrst vertex is i∗ OD [−1]. The same argument proves the second and the third claim.

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We conclude that the following semiorthogonal decomposition is obtained: Db (Y# )=Φ (Db (P(B), B0 )), OYe , i∗ OD , OYe (H), i∗ OD (H), OYe (2H), i∗ OD (2H) . (13) Lemma 4.6. We have Ext• (i∗ p∗ F, π ∗ G) = 0 for any F ∈ Db (P(A)), G ∈ Db (P(B)), so that i∗ OD (H) is completely orthogonal to OYe (2H) and so that i∗ OD is completely orthogonal to OYe (H), OYe (2H) . Proof. Since i is a divisorial embedding, we have ωD/Ye ∼ = OD (D), so i! (−) ∼ = ∗ i (−) ⊗ OD (D)[−1]. Therefore we have Ext• (i∗ p∗ F, π ∗ G) = Ext• (p∗ F, i! π ∗ G) = Ext• (p∗ F, i∗ π ∗ G(D)[−1]) = Ext• (p∗ F, p∗ j ∗ G(D)[−1]) = Ext• (F, p∗ (p∗ j ∗ G(D))[−1]) = Ext• (F, j ∗ G ⊗ p∗ OD (D)[−1]) = 0; the last equality is satisﬁed because p : D → P(A) is a P1 -ﬁbration and OD (D) = OD (H − h) restricts as O(−1) to all its ﬁbers.

By Lemma 2.10 the transposition of i∗ OD (H) to the right of OYe (2H), and of i∗ OD to the right of the subcategory OYe (H), OYe (2H) in (13) gives a semiorthogonal decomposition. Step 7. Transpose i∗ OD (H) to the right of OYe (2H), and i∗ OD to the right of OYe (H), OYe (2H) . After the transposition we obtain the following decomposition: Db (Y# )=Φ (Db (P(B), B0 )), OYe , OYe (H), OYe (2H), i∗ OD , i∗ OD (H), i∗ OD (2H) . (14) Now we are done. Comparing (14) with (7), we see that Φ (Db (P(B), B0 )) = σ ∗ (AY ), hence the functor σ∗ ◦ Φ : Db (P(B),B0 ) → AY , / F→ Hom(OYe (h−H),Φ(F ))⊗JP(A)→ Φ(F )→Hom(Φ(F ),OYe (−h))∗ ⊗OY (−H) is an equivalence of categories. Combining this with the equivalence of

Lemma 4.2 we obtain an equivalence Db (S, B) ∼ = AY . The category Db (S, B) can be considered as a twisted derived category of the K3 surface S, the twisting being given by the class of B in the Brauer group of S. However, sometimes the twisting turns out to be trivial. Let us describe when this happens. For a two-dimensional cycle T on a cubic fourfold Y containing a plane P consider the intersection index δ(T ) = T · H · H − T · P.

(15)

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Note that δ(P) = −2, and that δ(H 2 ) = 2. So, if the group of 2-cycles on Y modulo numerical equivalence is generated by P and H 2 , then δ takes only even values. Proposition 4.7. The sheaf of Azumaya algebras B on S splits if and only if there exists a 2-dimensional cycle T on Y such that δ(T ) is odd. Proof. By deﬁnition the P1 -ﬁbration over S corresponding to the sheaf of Azumaya algebras B is given by the moduli space M of lines in the ﬁbers of Y# over P(B), hence B splits if and only if the map M → S has a rational multisection of odd degree. Note that δ(T ) equals the intersection index of the proper preimage of T in Y# with the ﬁber of Y# over P(B). Hence, the set of lines in the ﬁbers of Y# that intersect T gives a multisection of M → S of degree δ(T ). To prove the converse, consider the component M(2) of M ×P(B) M lying over the graph of the involution of S over P(B) in S ×P(B) S. The points of M(2) correspond to pairs of lines in the ﬁbers of Y# over P(B) lying in diﬀerent families. Associating with such pair of lines the point of their intersection, we obtain a rational map M(2) → Y# → Y . If Z ⊂ M is a rational multisection of M → S of odd degree d, then we take T to be (the closure of) the image of Z (2) (deﬁned analogously to M(2) ) in Y . Then it is easy to see that δ(T ) = d2 is odd.

Note that the cubic fourfolds containing a plane P(A) and a 2-dimensional cycle T such that δ(T ) is odd are rational by results of Hassett [Ha1, Ha2]. So, if Conjecture 1.1 is true, the category AY for these cubic fourfolds should be equivalent to the derived category of a K3 surface. Combining Theorem 4.3 with Proposition 4.7 we see that this is the case. Now we want to see whether the Conjecture works the other way around. Assume that a cubic fourfold Y contains a plane but δ(T ) is even for any 2-cycle T on Y , so that the sheaf of Azumaya algebras B does not split. It is, however, still possible that Db (S, B) ∼ = Db (S ) for some other (or even for the same) K3 surface S . For this, however, the Picard group of S should be suﬃciently big, at least if Pic S ∼ = Z this is impossible. This follows from Proposition 4.8. Let Y be a cubic fourfold containing a plane P. If the group of codimension 2 algebraic cycles on Y modulo numerical equivalence is generated by P and H 2 , then AY ∼ Db (S ) for any surface S . = The proof will be given in the Appendix. Actually, we will show that K0 (S, B) ∼ K0 (S ) for any surface S as lattices with a bilinear form (the = Euler form χ), where S is the K3 surface corresponding to Y and B is the induced sheaf of Azumaya algebras on it. More precisely, we will show that K0 (S, B) does not contain a pair of vectors (v1 , v2 ) such that χ(S,B) (v1 , v2 ) = 1, χ(S,B) (v2 , v2 ) = 0 (while in K0 (S ) one can take v1 = [OS ], v2 = [Op ], where Op is a structure sheaf of a point).

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5 Singular cubic fourfolds Let Y ⊂ P(V ) = P5 be a singular cubic fourfold. Let P be its singular point. Let σ : Y# → Y be the blowup of P . The linear projection from P gives a regular map π : Y# → P4 . Lemma 5.1. The map π is the blowup of a K3 surface S ⊂ P4 which is an intersection of a quadric and a cubic hypersurfaces. Let D be the exceptional divisor of π, and Q the exceptional divisor of σ. Let i : D → Y# , j : S → P4 , and α : Q → Y# be the embeddings, and p : D → S the projection. α

Q σ

P

/Y

/ Y# o i D ?? ?? ?? p ?? π ?? ?? ?? ?? j 4 o S P

Then the map π ◦ α : Q → P4 identiﬁes Q with the quadric passing through S. Moreover, let H and h denote the pullbacks to Y# of the classes of hyperplanes in P(V ) and P4 , respectively. Then we have the following relations in Pic Y# : Q = 2h−D, H = 3h−D, h = H−Q, D = 2H−3Q, KYe = −5h+D = −3H+2Q. Proof. Let z0 , . . . , z5 be coordinates in V such that P = (1 : 0 : 0 : 0 : 0 : 0). Then (z1 : z2 : z3 : z4 : z5 ) are the homogeneous coordinates in P4 and the rational map π ◦ σ −1 takes (z0 : z1 : z2 : z3 : z4 : z5 ) to (z1 : z2 : z3 : z4 : z5 ). Since the point P is singular for Y , the equation of Y is given by z0 F2 (z1 , z2 , z3 , z4 , z5 ) + F3 (z1 , z2 , z3 , z4 , z5 ) = 0, where F2 and F3 are homogeneous forms of degree 2 and 3, respectively. It follows that π is the blowup of the surface S = {F2 (z1 , z2 , z3 , z4 , z5 ) = F3 (z1 , z2 , z3 , z4 , z5 ) = 0} ⊂ P4 , and that the rational map σ ◦ π −1 is given by the formula (z1 : z2 : z3 : z4 : z5 ) → (−F3 (z1 , z2 , z3 , z4 , z5 ) : z1 F2 (z1 , z2 , z3 , z4 , z5 ) : . . . : z5 F2 (z1 , z2 , z3 , z4 , z5 )). It follows that the proper preimage of the quadric {F2 (z1 , z2 , z3 , z4 , z5 ) = 0} ⊂ P4 in Y# is contracted by σ, whence Q = 2h−D. It also follows that H = 3h−D. Solving these with respect to h and D we get the other two relations. Finally, since Y# is the blowup of P4 in a surface S we have KYe = −5h+D. Substituting h = H − Q, D = 2H − 3Q we deduce the last equality.

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Theorem 5.2. The category Db (S) is a crepant categorical resolution of AY . In other words, there exists a pair of functors ρ∗ : Db (S) → AY ,

ρ∗ : Aperf → Db (S), Y

where Aperf = AY ∩ Dperf (Y ) and Dperf (Y ) ⊂ Db (Y ) is the category of perfect Y complexes on Y , such that ρ∗ is both left and right adjoint to ρ∗ and ρ∗ ◦ ρ∗ ∼ = id. The notion of a crepant categorical resolution of singularities is introduced and discussed in [K2]. To prove the theorem we start with consider# of Db (Y ). Following [K2], to construct ing a crepant categorical resolution D # such D one starts with a dual Lefschetz decomposition (with respect to the conormal bundle) of the derived category of the exceptional divisor of a usual resolution. We take the resolution σ : Y# → Y . Then the exceptional divisor is a three-dimensional quadric Q and the conormal bundle is isomorphic to OQ (−Q) ∼ = OQ (h−H) ∼ = OQ (h). We choose the dual Lefschetz decomposition Db (Q) = B2 ⊗ OQ (−2h), B1 ⊗ OQ (−h), B0 with B2 = B1 = OQ and B0 = OQ , SQ , where SQ is the spinor bundle on Q. Then by [K2] the triangulated category # = ⊥ α∗ (B2 ⊗ OQ (−2h)), α∗ (B1 ⊗ OQ (−h)) = ⊥ α∗ OQ (−2h), α∗ OQ (−h) D is a crepant categorical resolution of Db (Y ) and there is a semiorthogonal decomposition # Db (Y# ) = α∗ OQ (−2h), α∗ OQ (−h), D . # induced by the Further, we consider the semiorthogonal decomposition of D, b decomposition (3) of D (Y ): # = A˜Y , O e , O e (H), O e (2H) . D Y Y Y One can easily show that A˜Y is a crepant categorical resolution of AY (see Lemma 5.8 below). So, to prove the theorem, it suﬃces to check that A˜Y ∼ = Db (S). Now we describe the way we check this. Substituting the above decompo# into the above decomposition of Db (Y# ) we obtain sition of D Db (Y# ) = α∗ OQ (−2h), α∗ OQ (−h), A˜Y , OYe , OYe (H), OYe (2H) .

(16)

On the other hand, since π : Y# → P4 is the blowup of S we have by [Or] the following semiorthogonal decomposition: Db (Y# ) = Φ(Db (S)), π ∗ (Db (P4 )) ,

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where the functor Φ : Db (S) → Db (Y# ) is given by F → i∗ p∗ F (D). Using one of the standard exceptional collections Db (P4 ) = OP4 (−3), OP4 (−2), OP4 (−1), OP4 , OP4 (1) we obtain a semiorthogonal decomposition: Db (Y# ) = Φ(Db (S)), OYe (−3h), OYe (−2h), OYe (−h), OYe , OYe (h) .

(17)

Now we are going to make a series of mutations, transforming decomposition (17) into (16). This will give the required equivalence Db (S) ∼ = A˜Y . Now let us describe the series of mutations. Step 1. Left mutation of OYe (−3h), OYe (−2h), and OYe (−h) through Φ(Db (S)). Lemma 5.3. For any F ∈ Db (P4 ) we have LΦ(Db (S)) (π ∗ F ) = π ∗ F (D). Proof. Recall that by deﬁnition we have LΦ(Db (S)) (G) = Cone(Φ(Φ! (G)) → G)

(18)

for any G ∈ Db (Y# ). But Φ! (π ∗ F ) ∼ = p∗ i! π ∗ F (−D) ∼ = p∗ i∗ π ∗ F [−1] ∼ = p∗ p∗ j ∗ F [−1] ∼ = j ∗ F [−1], so

Φ(Φ! (π ∗ F )) ∼ = i∗ p∗ j ∗ F (D)[−1] ∼ = i∗ i∗ π ∗ F (D)[−1]

and it is clear that the triangle (18) boils down to i∗ i∗ π ∗ F (D)[−1] → π ∗ F → π ∗ F (D) obtained by tensoring exact sequence 0 → OYe → OYe (D) → i∗ OD (D) → 0 with π ∗ F and rotating to the left.

It follows that after this mutation we obtain the following decomposition: Db (Y# ) = OYe (−3h + D), OYe (−2h + D), OYe (−h + D), Φ(Db (S)), OYe , OYe (h) . (19) Step 2. Right mutation of Φ(Db (S)) through OYe and OYe (h). After this mutation we obtain the following decomposition: Db (Y# ) = OYe (−3h + D), OYe (−2h + D), OYe (−h + D), OYe , OYe (h), Φ (Db (S)) , (20) b b # where Φ = ROYe (h) ◦ ROYe ◦ Φ : D (S) → D (Y ). Lemma 5.4. We have Ext• (OYe (−h+D), OYe ) = 0, so that the pair (OYe (−h+ D), OYe ) is completely orthogonal.

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Proof. We have Ext• (OYe (−h+D), OYe ) = H • (Y# , OYe (h−D)) = H • (P4 , JS (1)), where JS is the sheaf of ideals of S. On the other hand, since S is the intersection of a quadric and a cubic, we have an exact sequence 0 → OP4 (−5) → OP4 (−3) ⊕ OP4 (−2) → JS → 0. Twisting by OP4 (1) we deduce the required vanishing.

By Lemma 2.10 the transposition of the pair (OYe (−h + D), OYe ) gives a semiorthogonal decomposition. Step 3. Transpose the pair (OYe (−h + D), OYe ). After the transposition we obtain the following decomposition: Db (Y# ) = OYe (−3h + D), OYe (−2h + D), OYe , OYe (−h + D), OYe (h), Φ (Db (S)) . (21) Step 4. Simultaneous right mutation of OYe (−2h + D) through OYe and of OYe (−h + D) through OYe (h). Lemma 5.5. We have ROYe (OYe (−2h + D)) ∼ = α∗ OQ [−1],

ROYe (h) (OYe (−h + D)) ∼ = α∗ OQ (h)[−1].

Proof. Note that Ext• (OYe (−2h + D), OYe ) = H • (Y# , OYe (2h − D)) = H • (P4 , JS (2)). Using the above resolution of JS we deduce that ! 1, for p = 0 p dim Ext (OYe (−2h + D), OYe ) = 0, otherwise. It follows that we have a distinguished triangle ROYe (OYe (−2h + D)) → OYe (−2h + D) → OYe . Since 2h−D = Q, the right map in the triangle is given by Q, so it follows that the ﬁrst vertex is α∗ OQ [−1]. The same argument proves the second claim.

We conclude that the following semiorthogonal decomposition is obtained: Db (Y# ) = OYe (−3h + D), OYe , α∗ OQ , OYe (h), α∗ OQ (h), Φ (Db (S)) . Step 5. Left mutation of OYe (h) through α∗ OQ . Lemma 5.6. We have Lα∗ OQ (OYe (h)) ∼ = OYe (3h − D).

(22)

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Proof. Note that Ext• (α∗ OQ , OYe (h)) = Ext• (OQ , α! OYe (h)) = Ext• (OQ , OQ (h + Q)[−1]). But the divisor Q is contracted by σ, and h + Q = H is a pullback by σ, hence OQ (h + Q) ∼ = OQ , so we conclude that ! 1, for p = 1 p dim Ext (α∗ OQ , OYe (h)) = 0, otherwise. It follows that we have a distinguished triangle α∗ OQ [−1] → OYe (h) → Lα∗ OQ (OYe (h)). Comparing it with the rotation of the exact sequence 0 → OYe (h) → OYe (3h − D) → α∗ OQ → 0, we deduce the required isomorphism.

As a result of this step we obtain the following semiorthogonal decomposition: Db (Y# ) = OYe (−3h+D), OYe , OYe (3h−D), α∗ OQ , α∗ OQ (h), Φ (Db (S)) . (23) Step 6. Left mutation of the subcategory α∗ OQ , α∗ OQ (h), Φ (Db (S)) through its orthogonal subcategory α∗ OQ , α∗ OQ (h), Φ (Db (S)) ⊥ = OYe (−3h + D), OYe , OYe (3h − D) and a twist by OYe (H). Applying Lemma 2.11 and taking into account equalities KYe = D − 5h and 3h − D = H, we obtain Db (Y# ) = α∗ OQ (−2h), α∗ OQ (−h), Φ (Db (S)), OYe , OYe (H), OYe (2H) , (24) where

Φ = TOYe (3h−D) ◦ TOYe (−5h+D) ◦ Φ = TOYe (−2h) ◦ Φ .

Comparing (24) with (16) we obtain the following: Corollary 5.7. The functor Φ = ROYe (−h) ◦ ROYe (−2h) ◦ TOYe (D−2h) ◦ i∗ ◦ p∗ : Db (S) → Db (Y# ) induces an equivalence of categories Db (S) ∼ = A˜Y . Now we can ﬁnish the proof of Theorem 5.2 by the following: Lemma 5.8. The category A˜Y is a crepant categorical resolution of the category AY .

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# is a crepant categorical resolution of Proof. Recall that by [K2] the category D b b # → D (Y ) and σ ∗ : Dperf (Y ) → D. # So, to prove D (Y ) via the functors σ∗ : D ∗ ˜ ˜ the lemma we only have to check that σ∗ (AY ) ⊂ AY and σ (Aperf Y ) ⊂ AY . ˜ But this is straightforward — if F ∈ AY , then Hom(OY (t), σ∗ (F )) = Hom(σ ∗ (OY (t)), F ) = Hom(OYe (tH), F ) = 0 for t = 0, 1, 2 by adjunction between σ∗ and σ ∗ and the deﬁnition of A˜Y . Similarly, if G ∈ Aperf Y , then Hom(OYe (tH), σ ∗ (G)) = Hom(σ ∗ (OY (t)), σ ∗ (G)) = Hom(OY (t), σ∗ σ ∗ (G)) ∼ = Hom(OY (t), G) = 0

again by adjunction and by the fact that σ∗ ◦ σ ∗ ∼ = id on Dperf (Y ).

Proof (of Theorem 5.2). By Corollary 5.7 we have Db (S) ∼ = A˜Y and by

Lemma 5.8 the category A˜Y is a crepant categorical resolution of AY . Remark 5.9. The resulting functors between Db (S) and AY take form ρ∗ = σ0∗ ◦ ROYe (−h) ◦ ROYe (−2h) ◦ TOYe (D−2h) ◦ i∗ ◦ p∗ : Db (S) → AY1, σ∗ i∗ (p∗ F (D − 2h)) → Hom• (F, OS [−1])∗ ⊗ OYe (−2H) → F → Hom(F, ΩP4 (h)|S [−1])∗ ⊗ OYe (−H) ρ∗ = p∗ ◦ i∗ ◦ TOYe (−3h)[2] ◦ LOYe (2h−D) ◦ LOYe (4h−D) ◦ σ ∗ : Aperf → Db (S), Y ∗ ∗ G → p∗ (i σ G(−3h))[2]. It will be interesting to describe the subcategory of Db (S) which is mapped to zero by ρ∗ : Db (S) → AY .

6 Appendix. The Grothendieck group of a twisted K3 surface associated with a cubic fourfold containing a plane For the computation of the Grothendieck group we will use the notion of a twisted Chern character introduced by Huybrechts and Stellari in [HS1], see also [HS2]. Let S be a polarized K3 surface of degree 2 with Pic S = Zh (and h2 = 2), and B a nonsplit Azumaya algebra of rank 4 on S. Let α ∈ H 2 (S, OS∗ ) be the class of B in the Brauer group Br(S) = H 2 (S, OS∗ ). It follows from the exponential sequence 0

/Z

/ OS

exp(2πi(−))

/ O∗ S

/0

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and from H 3 (S, Z) = 0 that there exists β ∈ H 2 (S, OS ) such that α = exp(2πiβ). Since the order of α is 2, it follows that 2β is the image of some integer class B0 ∈ H 2 (S, Z), hence β is the image of B = 12 B0 ∈ H 2 (S, 12 Z) ⊂ H 2 (S, Q). Let us ﬁx such B. Certainly, the class B ∈ H 2 (S, 12 Z) such that α = exp(2πiB) is not unique. It is deﬁned up to addition of an element in H 2 (S, Z) (ambiguity in a choice of β) and of an element in H 1,1 (S, 12 Z) (ambiguity in the lifting of β to B). However, the following invariant does not depend on the choices. Let {t} := t − !t" denote the fractional part of t ∈ Q. Lemma 6.1. The fractional part {Bh} of the product Bh does not depend on the choice of B. Moreover, if {Bh} = 12 , then {B 2 } does not depend on the choice of B. Proof. Take any u ∈ H 2 (S, Z). Then (B + u)h = Bh + uh,

(B + u)2 = B 2 + 2Bu + u2 .

It is clear that uh, (2B)u, and u2 are integral, so {Bh} and {B 2 } do not change. Further, note that H 1,1 (S, 12 Z) = 12 Zh, so it remains to look at 1 1 (B+ h)h = Bh+ h2 = Bh+1, 2 2

1 1 1 (B+ h)2 = B 2 +Bh+ h2 = B 2 +Bh+ . 2 4 2

We see that {Bh} does not change and that if {Bh} = 12 , then {B 2 } does not change as well.

One can compute {Bh} from the geometry of the P1 -bundle associated with the corresponding Brauer class α = exp(2πiB). It is an interesting question as to how to compute {B 2 } in a similar fashion. Lemma 6.2. Let M → S be a P1 -ﬁbration given by the Brauer class α = exp(2πiB) and C ⊂ S, a smooth curve in the linear system |h| on S. Then there exists a rank 2 vector bundle E on C such that M ×S C ∼ = PC (E). Moreover 1 0 1 deg det E , {Bh} = 2 for any such bundle E. Proof. A vector bundle E exists since the Brauer group of a smooth curve is trivial. The equality of fractional parts can be deduced as follows. Note that the class α ∈ H 2 (S, OS∗ ) is of order 2, hence it comes from a class α0 ∈ H 2 (S, μ2 ), where μ2 stands for the group of square roots of unity. Note also that we have a canonical isomorphism 1 H 2 (C, μ2 ) ∼ = μ2 ∼ = Z/Z. 2

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Let us ﬁrst check that {Bh} coincides with the image of the restriction α0|C ∈ H 2 (C, μ2 ) under this isomorphism. Indeed, this follows immediately from the commutative diagram: H 2 (S, 12 Z)

/ H 2 (C, 1 Z) 2

1 2Z

H 2 (S, μ2 )

/ H 2 (C, μ2 )

1 Z/Z 2

{−}

in which the vertical arrows are induced by the map exp(2πi(−)) : 12 Z → μ2 , and the horizontal arrows are given by the/restriction to C. So, it remains to check that α0|C equals 12 deg det E . For this we consider exact sequence of groups 1

/ GL2

/ μ2

(p,det)

/ PGL2 × Gm

/1,

where p : GL2 → PGL2 is the canonical projection. From this we obtain an exact sequence of cohomologies ∗ ) → H 2 (C, μC ) H 1 (C, GL2 (OC )) → H 1 (C, PGL2 (OC )) ⊕ H 1 (C, OC

which can be rewritten as the following commutative (up to a sign which in H 2 (C, μC ) = 12 Z/Z is insigniﬁcant) diagram H 1 (C, GL2 (OC )) p

H 1 (C, PGL2 (OC ))

det

/ H 1 (C, O∗ ) C { 12 deg(−)}

/ H 2 (C, μC )

By deﬁnition, α0|C comes from the class in H 1 (C, PGL2 (OC )) of the restriction of the Azumaya algebra B|C and the class of E in H 1 (C, GL2 (OC )) is its lift. On the other hand, the top horizontal and the right vertical arrows take E precisely to { 12 deg det E}.

Consider the bounded derived category Db (S, B) of coherent sheaves of B-modules on the surface S. Its Grothendieck group comes with the Euler bilinear form on it: χ(S,B) ([F ], [G]) = (−1)i dim Exti (F, G). We denote by K0 (S, B) the quotient of the Grothendieck group by the kernel of the Euler form, i.e., the numerical Grothendieck group. Lemma 6.3. Assume that {Bh} = {B 2 } = 12 . Then there is no such pair of vectors v1 , v2 ∈ K0 (S, B) for which χ(S,B) (v1 , v2 ) = 1, χ(S,B) (v2 , v2 ) = 0.

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Proof. Recall that h ∈ H 1,1 (S, Z) denotes the positive generator of Pic S and let p ∈ H 4 (S, Z) be the class of a point. By [HS1, Proposition 1.2], there exists a linear map (the B-twisted Chern character) chB : K0 (S, B) → H • (X, Q), such that: 1. Im chB = H ∗,∗ (S, B, Z) := exp(B)(H 0 (S, Q) ⊕ H 1,1 (S, Q) ⊕ H 4 (S, Q)) ∩ H • (S, Z) (Corollary 2.4 and Remark 1.3 (ii)); 2 2 2. χ(S,B) (F, G) = chB (F ) td(S), chB (G) td(S) , where −, − is the Mukai pairing r1 + d1 h + s1 p, r2 + d2 h + s2 p = r1 s2 − 2d1 d2 + s1 r2 ,

ri , di , si ∈ Z.

Let us describe H ∗,∗ (S, B, Z). We have H 0 (S, Q)⊕H 1,1 (S, Q)⊕H 4 (S, Q) = {r + dh + sp | r, d, s ∈ Q} since H 1,1 (S, Q) = Qh. Further exp(B)(r + dh + sp) = r + (rB + dh) + (rB 2 /2 + dBh + s)p. So, to obtain an element of H ∗,∗ (S, B, Z) we must have r ∈ Z,

rB + dh ∈ H 2 (S, Z),

rB 2 /2 + dBh + s ∈ Z.

Let us show that r is even. Indeed, for r odd rB + dh ∈ H 2 (S, Z) would imply B +dh ∈ H • (S, Z), hence the image of B in H 2 (S, OS )/H 2 (S, Z) = H 2 (S, OS∗ ) would be zero. So, r is even. Therefore rB ∈ H 2 (S, Z), hence dh ∈ H 2 (S, Z), so d ∈ Z. We conclude that H ∗,∗ (S, B, Z) is generated by elements 2 + 2B, h and p, and it is easy to see that the matrix of the bilinear form χ(S,B) in this basis is ⎛ ⎞ 8 − 4B 2 −2Bh 2 ⎝ −2Bh −2 0⎠ . 2 0 0 Note that the only potentially odd integer in the matrix is −2Bh, all the rest are deﬁnitely even. So, if we have χ(S,B) (v1 , v2 ) = 1, then the coeﬃcient of v2 at h (in the decomposition of v2 with respect to the above basis of K0 (S, B)) is odd. Thus, it suﬃces to check that for any v2 ∈ K0 (S, B) such that χ(S,B) (v2 , v2 ) = 0 the coeﬃcient of v2 at h is even. Indeed, assume that v2 = x(2 + 2B) + yh + zp. Then 0 = χ(S,B) (v2 , v2 ) = (8 − 4B 2 )x2 − 4Bhxy + 4xz − 2y 2 implies y 2 = 4x2 + 2xz − x(2B 2 x − 2Bhy). Taking into account that 2B 2 ≡ 2Bh ≡ 1 mod 2 we deduce that y 2 ≡ x(x + y) mod 2. It is clear that for y ≡ 1 mod 2 this has no solutions, so we conclude that y should be even.

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In what follows we check that for the Azumaya algebra B on a K3 surface S arising from a cubic fourfold Y containing a plane the conditions of Lemma 6.3 are satisﬁed. Then it will follow that Db (S, B) ∼ Db (S ) for any surface S . = We will use freely the notation introduced in Section 4. Lemma 6.4. We have {Bh} = {B 2 } = 12 . Proof. Let us start with a computation of {Bh}. Recall that we have a ﬁbration in two-dimensional quadrics π : Y# → P(B) and M is the Hilbert scheme of lines in ﬁbers of π. Let L be a generic line in P(B) and C = L ×P(B) S = f −1 (L). Restricting the ﬁbration π : Y# → P(B) to L we obtain a pencil of quadrics {Qλ }λ∈L . Then MC = M ×S C is the Hilbert scheme of lines in Qλ . By Lemma 6.2 it suﬃces to check that MC ∼ = PC (E) with deg det E being odd. Consider each Qλ as a quadric in P(V ). It intersects with P = P(A) in a conic qλ , so we have a pencil of conics {qλ }λ∈L in P. It is clear that for suﬃciently general L the following two conditions are satisﬁed: 1. the base locus of the pencil {qλ } on P is a quadruple of distinct points such that any triple of them is noncollinear; 2. the points λ1 , . . . , λ6 ∈ L corresponding to singular quadrics Qλ with λ ∈ L (and hence to the ramiﬁcation points of f : C → L) are pairwise distinct from the points λ1 , λ2 , λ3 ∈ L corresponding to reducible conics qλ . We will assume that for the chosen L both properties (1) and (2) are satisﬁed. Our main observation is that each line on a quadric Qλ intersects the conic qλ = Qλ ∩ P in a unique point, hence the Hilbert scheme of lines on Qλ identiﬁes either with qλ qλ , if qλ is irreducible and Qλ is smooth (i.e., λ ∈ {λ1 , λ2 , λ3 , λ1 , λ2 , λ3 , λ4 , λ5 , λ6 }); if qλ is irreducible and Qλ is singular qλ , (i.e., λ ∈ {λ1 , λ2 , λ3 , λ4 , λ5 , λ6 }); + − qλ qλ , if qλ = qλ+ ∪ qλ− is reducible and Qλ is smooth (i.e., λ ∈ {λ1 , λ2 , λ3 }). ˜ of P in the base An immediate consequence is the following. The blowup P locus of qλ has a natural structure of a conic bundle over L with three reducible ˜C = P ˜ ×L C. This is a conic bundle over C ﬁbers (over the points λi ). Let P with six reducible ﬁbers, over the preimages λ± i ∈ C of points λi ∈ L. Then ˜ C by contracting the components q + in the ﬁbers over MC is obtained from P λi − − λ+ i and qλi in the ﬁbers over λi . It follows that a vector bundle E on C such that MC = PC (E) can be obtained as follows. ˜ over First, consider the contraction of components qλ+i in the ﬁbers of P 1 λi ∈ L. We will obtain a P -ﬁbration over L which is isomorphic to PL (E0 )

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for some vector bundle E0 on L of rank 2. Then PC (f ∗ E0 ) is obtained from ˜ C by contracting the components q + both in the ﬁbers over λ+ and in P i λi ∗ the ﬁbers over λ− i , whereof it follows that MC is obtained from PC (f E0 ) − by simple Hecke transformations in the ﬁbers over the points λi (a simple Hecke transformation of a P1 -bundle is a blowup of a point followed by contraction of the proper preimage of the ﬁber containing this point). So, PC (E) diﬀers from PC (f ∗ E0 ) by three simple Hecke transformations. It remains to note that deg det f ∗ E0 = deg f ∗ det E0 is even and to use the well-known fact that whenever PC (E) and PC (E ) are related by a simple Hecke transformation the parity of deg det E and deg det E is diﬀerent, so after three Hecke transformations we obtain PC (E) with deg det E being odd. Now, to compute {B 2 } we are going to use Lemma 6.3. By this lemma for any α-twisted sheaf F on S of rank 2 we have 2 2 χ(S,α) (F, F ) = chB (F ) td(S), chB (F ) td(S) . On the other hand, for any α-twisted sheaf of rank 2 we have chB (F ) = 2 + 2B + dh + s for some d, s ∈ Z, so χ(S,α) (F, F ) = 8 − 4B 2 + 4s − 2d2 − 4dBh. Note that 8 + 4s − 2d2 − 4dBh = 4s − 2d(d + 2Bh) is divisible by 4. Indeed, 2d is divisible by 4 for even d, while for d odd 2(d + 2Bh) is divisible by 4. We conclude that χ(S,α) (F, F ) ≡ 4B 2 mod 4. On the other hand, under identiﬁcation of the category of sheaves of Bmodules on S with the category of α-twisted sheaves, the sheaf B corresponds to a certain rank 2 twisted sheaf. We conclude that χ(S,B) (B, B) ≡ 4B 2 mod 4. Now consider the covering f : S → P(B). We have the pushforward and the pullback functors f∗ : Db (S, B) → Db (P(B)), f ∗ : Db (P(B)) → Db (S, B). Note that f ∗ OP(B) ∼ = B, so by adjunction χ(S,B) (B, B) = χ(S,B) (f ∗ OP(B) , B) = χP(B) (OP(B) , f∗ B) = χP(B) (OP(B) , B0 ) = χP(B) (B0 ). It remains to note that B0 ∼ = OP(B) ⊕ (Λ2 A ⊗ OP(B) (−1) ⊕ A ⊗ OP(B) (−2)) ⊕ Λ3 A ⊗ OP(B) (−3), so χP(B) (B0 ) = 2, whence {B 2 } = 2/4 = 1/2.

Now we can give a proof of Proposition 4.8. Combining Lemma 6.4 with Lemma 6.3 we conclude that K0 (S, B) does not contain a pair of elements (v1 , v2 ) such that χ(S,B) (v1 , v2 ) = 1, χ(S,B) (v2 , v2 ) = 0. On the other hand, for any surface S in K0 (S ) there is such a pair. Indeed, just take v1 = [OS ], v2 = [Op ] where Op is a structure sheaf of a point. We conclude that K0 (S, B) ∼ = K0 (S ) which implies that Db (S, B) ∼ = Db (S ).

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References [B]

[BK]

[BO] [H]

[Ha1] [Ha2] [HS1] [HS2]

[K1] [K2] [K3] [K4] [N] [Or]

[Tr1] [Tr2] [Ve]

A. Bondal, Representations of associative algebras and coherent sheaves, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 25–44; translation in Math. USSR-Izv. 34 (1990), no. 1, 23–42. A. Bondal, M. Kapranov, Representable functors, Serre functors, and reconstructions, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1183–1205, 1337; translation in Math. USSR-Izv. 35 (1990), no. 3, 519– 541. A. Bondal, D. Orlov, Semiorthogonal decomposition for algebraic varieties, math.AG/9506012. R. Hartshorne, Residues and duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20 Springer-Verlag, Berlin, 1966. B. Hassett, Special Cubic Fourfolds, PhD thesis, University of Chicago, 1998. B. Hassett, Special cubic fourfolds, Compos. Math. 120 (2000), 1–23. D. Huybrechts, P. Stellari, Equivalences of twisted K3 surfaces, Math. Ann. 332 (2005), no. 4, 901–936. D. Huybrechts, P. Stellari, Proof of Căldăraru’s conjecture, Appendix: “Moduli spaces of twisted sheaves on a projective variety” by K. Yoshioka. Moduli spaces and arithmetic geometry, 31–42, Adv. Stud. Pure Math., 45, Math. Soc. Japan, Tokyo, 2006. A. Kuznetsov, Derived categories of quadric ﬁbrations and intersections of quadrics, Adv. in Math. 218 (2008), N. 5, 1340–1369. A. Kuznetsov, Lefschetz decompositions and categorical resolutions of singularities, Selecta Math. 13 (2008), N. 4, 661–696. A. Kuznetsov, Homological projective duality for Grassmannians of lines, preprint math.AG/0610957. A. Kuznetsov, Derived categories and rationality, in preparation. A. Neeman, The Grothendieck duality theorem via Bousﬁeld’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205–236. D. Orlov, Projective bundles, monoidal transformations, and derived categories of coherent sheaves, (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 4, 852–862; translation in Russian Acad. Sci. Izv. Math. 41 (1993), no. 1, 133–141. S. Tregub, Three constructions of rationality of a cubic fourfold, Moscow Univ. Math. Bull. 39 (1984), no. 3, 8–16. S. Tregub, Two remarks on four dimensional cubics, Russian Math. Surveys 48 (1993), no. 2, 206–208. J.-L. Verdier, Categories dérivées. Quelques résultats, Séminaire de Géométrie Algébrique du Bois-Marie (SGA 4 1/2), Lecture Notes in Mathematics 569, Springer-Verlag, Berlin, 1977, 262–311.

Fields of Invariants of Finite Linear Groups Yuri G. Prokhorov Department of Algebra, Faculty of Mathematics (and Mechanics) Moscow State University Moscow 117234, Russia [email protected] Summary. We study Noether’s rationality problem for actions of ﬁnite groups on projective three-space.

Key words: Noether’s problem, rationality problem 2000 Mathematics Subject Classiﬁcation codes: 14E08, 13A50

1 Introduction Let G be a ﬁnite group of order n and let k be a ﬁeld. Consider a rational (i.e., pure transcendental) extension K/k of transcendence degree n. We may assume that K = k({xg }), where g runs through all the elements of G. The group G naturally acts on K via h(xg ) = xhg . E. Noether [Noe13] asked whether the ﬁeld of invariants K G is rational over k or not. On the language of algebraic geometry, this is a question about the rationality of the quotient variety An /G. The most complete answer on this question is known for Abelian groups. Thus, if G is Abelian of exponent e, char k does not divide e, and k contains a primitive eth roots of unity, then An /G is rational [Fis15]. On the other hand, over an arbitrary ﬁeld k the rationality of An /G is related to some number theoretic questions. In this case An /G can be nonrational [Swa69] (see also [Vos73], [EM73], [Len74]). Noether’s question can be generalized as follows. Problem 1.1. Let G be a ﬁnite group, let V be a ﬁnite-dimensional vector space over an algebraically closed ﬁeld k, and let ρ : G → GL(V ) be a representation. Whether the quotient variety V /G is rational over k? The following fact is well known to specialists.

F. Bogomolov, Y. Tschinkel (eds.), Cohomological and Geometric Approaches to Rationality Problems, Progress in Mathematics 282, DOI 10.1007/978-0-8176-4934-0_10, © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

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Proposition 1.2. V /G is birationally equivalent to P(V )/G × P1 . Indeed, by blowing up the origin we get a generically C1 -ﬁbration f : V3 /G → P(V )/G which is locally trivial in the étale topology. By the geometric form of Hilbert’s theorem 90 the ﬁbration f is also locally trivial in the Zariski topology. We refer to [Miy71], [KV89], [CTS05] for detailed algebraic proofs. By Proposition 1.2 the rationality of P(V )/G implies the rationality of V /G. The inverse implication is not known (cf. [BCTSSD85]). The aﬃrmative answer to the Lüroth problem gives us the rationality of P(V )/G (and therefore V /G) for dim V ≤ 3 over any algebraically closed ﬁeld, cf. [Bur55, Ch. 17]. Thus dim V = 4 is the ﬁrst nontrivial case. Most of this survey is devoted to reviewing the current status of this problem in the special case where V is of dimension four. We refer to surveys [For84], [Dol87], [KV89], [Haj00], [CTS05] for other aspects of 1.1. If the representation G "→ GL(V ) is not irreducible, then the decompo) → sition V = V1 ⊕ V2 gives us a G-equivariant P1 -bundle structure P(V ) is the blowup of P(V ) along P(V1 ) ∪ P(V2 ). ObviP(V1 ) × P(V2 ), where P(V ously, this P1 -bundle has a G-invariant section, exceptional divisors over P(V1 ) and P(V2 ). Therefore, P(V )/G ≈ (P(V1 ) × P(V2 ))/G × P1 . If dim V = 4, this implies the rationality of P(V )/G. Thus, in the rationality question of P3 /G, we may assume that G ⊂ GL(V ) is irreducible. Definition 1.3. Let G ⊂ GL(V ) be a ﬁnite irreducible subgroup. We say that G is imprimitive of type (mk ) if V is a nontrivial direct sum V = V1 ⊕· · ·⊕Vk with dim Vi = m such that gVi = Vj for all g ∈ G. If such a direct splitting does not exist, G is called primitive. As a consequence of Jordan’s theorem [Jor70] one can see that modulo scalar matrices in any dimension n there is only a ﬁnite number of primitive ﬁnite subgroups G ⊂ GL(n, C). Lower-dimensional primitive groups have been completely classiﬁed, see references in [Fei71, §8.5]. 1.4. Let G ⊂ GL(4, C) be a ﬁnite subgroup. By 1.2 we may regard G modulo scalar matrices. Following [Bli17] we reproduce the list of all primitive subgroups G ⊂ GL(4, C) modulo scalar multiplications. In particular, we assume that G ⊂ SL(4, C). Notation is taken from [Fei71, §8.5] with small modiﬁ# cations. Here o is the order of the group and z is the order of its center, G denotes some central extension of G. (I) A×B/Z, where A, B are two-dimensional primitive subgroups in SL(2, C) and Z is the central subgroup of order 2 which is contained in neither A nor B, (V) (VI) (VII)

A5 , # 6, A # A7 ,

S5 , o = 60, 120, z = 1, # 6 , o = 360z, 720z, z = 2, S o = ( 12 7!)z, z = 2,

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(VIII) (IX) (X) (XI)

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# 5 o = 60z, 120z, z = 2, SL(2, F5 ), S SL(2, F7 ), o = 168z, z = 2, G25920 , o = 25920z, z = 2, G is such that N ⊂ G ⊂ M ⊂ SL(4, C), where N is a special imprimitive subgroup of order 32 and M is its extension by the automorphism group, |M | = 32 · 6!.

Note that (XI) is the biggest class. It includes also three groups from (I). We discuss the rationality problem of P3 /G for primitive and imprimitive subgroups in GL(4, C) in case-by-case manner. Notation Throughout this paper we will use the following notation unless otherwise speciﬁed. • • • •

• •

X ≈ Y denotes the birational equivalence of algebraic varieties X and Y . P(a1 , . . . , an ) denotes the weighted projective space. Z(G) usually denotes the center of a group G. Sn and An denote the symmetric alternating groups on n letters, respectively. Let Sn act on Cn by permuting the coordinates xi . The restriction of this representation to the invariant hyperplane xi = 0 we call the standard representation. T, O, I ⊂ SL(2, C) are binary tetrahedral, octahedral, and icosahedral groups, respectively. There are well-known isomorphisms T/Z(T) A4 , O/Z(O) S4 , I/Z(I) A5 , T SL(2, F3 ), and I SL(2, F5 ). If V is a vector space, then P(V ) denotes its projectivization and for an element x ∈ V , [x] denotes the corresponding point on P(V ).

For the sake of simplicity, we work over the complex number ﬁeld C. However, some results can be extended for an arbitrary algebraically closed ﬁeld (at least if the characteristic is suﬃciently large). Acknowledgment. I express my thanks to V.A. Iskovskikh for many stimulating discussions on the rationality problems. I am grateful to F.A. Bogomolov for helpful suggestions and Ming-chang Kang for very useful corrections and comments. The work was partially supported by grants CRDF-RUM, No. 1-2692-MO05 and RFBR, No. 05-01-00353-a, 06-01-72017.

2 Groups of type (I) In this section we follow [KMP92]. Regard V = C4 as the space of 2 × 2matrices. The group SL(2, C) × SL(2, C) acts on V from the left and the right. Therefore, for any two subgroups G1 , G2 ⊂ SL(2, C) there is a natural

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representation Ψ : G1 × G2 → SL(V ). Denote G := Ψ (G1 , G2 ) ⊂ SL(V ). For Ψ (T, T), Ψ (T, O), Ψ (T, I), Ψ (O, O), Ψ (O, I), and Ψ (I, I) we get primitive groups of type (I) (they are denoted by 1o , 3o , 4o , 5o , 6o , and 7o in Blichfeldt’s classiﬁcation [Bli17]). Theorem 2.1 ([KMP92]). The variety P3 /G is rational for G = Ψ (T, T), Ψ (T, O), Ψ (T, I), Ψ (O, O), Ψ (I, I). Another proof of the rationality of P3 /G for G = Ψ (T, T), Ψ (T, O), Ψ (O, O) will be given in §3. Proof. We have P(V )/G ≈ V /(G · C∗ ) ≈ SL(2, C)/G ≈ G1 \SL(2, C)/G2 . The aﬃne variety G1 \SL(2, C) is a homogeneous space under the action of SL(2, C). In all cases G1 = T, O, and I, there is a natural nonsingular projective compactiﬁcation W of G1 \SL(2, C) with Picard number one. We describe this construction below. Denote by Mn the space of binary forms of degree n. In the case G1 = T, the group T has two semi-invariants √ x4 , x4 = t41 ± 2 −3 t21 t22 + t42 ∈ M4 (see [Web96, vol. 2], [Spr77, §4.5]). Let W2 := SL(2, C) · [x4 ] ⊂ P(M4 ) = P4 be the closure of the SL(2, C)-orbit of [x4 ] ∈ P(M4 ). This set is SL(2, C)invariant and contains an open orbit isomorphic to T\SL(2, C). On the other hand, there is a nondegenerate symmetric bilinear form q( , ) on M4 (see [Web96, vol. 2], [Spr77, §3.1]). For an element a ∈ T of order 3 one has a · x4 = ωx4 , where ω is a primitive 3rd root of unity. Therefore, q(x4 , x4 ) = q(a · x4 , a · x4 ) = ω 2 q(x4 , x4 ), so that q(x4 , x4 ) = 0. Thus the variety W2 is deﬁned by the equation q(x, x) = 0. This shows that W2 ⊂ P4 is a smooth quadric. Similarly, in cases G1 = O and I by [Web96, vol. 2] or [Spr77, §4.5], the group O (resp. I) has a semi-invariant x6 = t1 t2 (t41 − t42 ) ∈ M6 5 5 10 (resp. invariant x12 = t1 t2 (t10 1 + 11t1 t2 − t2 ) ∈ M12 ).

By [MU83] the closure W5 := SL(2, C) · [x6 ] ⊂ P(M6 ) = P6 (resp. W12 := SL(2, C) · [x12 ] ⊂ P(M12 ) = P12 ) is a smooth compactiﬁcation of O\SL(2, C) (resp. I\SL(2, C)).

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Remark 2.2. Varieties W2 , W5 , and W22 are smooth Fano threefolds with ρ = 1. One has Pic W = Z · H and −KW ∼ rH, where H is the class of hyperplane section and r = 3, 2, and 1, respectively. The theory of SL(2, C)varieties with an open orbit was developed from the minimal model theoretic viewpoint in [MU83], [Ume88], [Nak89]. The main result is as follows: for any smooth SL(2, C)-variety X with an open orbit, there is a sequence of SL(2, C)equivariant birational morphisms X = X1 → · · · → Xn = Y , where each Xi is smooth, Xi → Xi+1 is the blowup of a smooth SL(2, C)-invariant curve, and Y is a so-called minimal SL(2, C)-variety. Minimal SL(2, C)-varieties are completely classiﬁed. In the event the stabilizer of a general point is I (resp. O, T), there is only one case Y = W22 (resp. W5 , W2 ). In the event the stabilizer is dihedral or cyclic group, the situation is more complicated: either Y P3 or Y has the form PS (O), where S is a smooth surface admitting an SL(2, C)-action or P1 and O is an equivariant vector bundle of rank 4−dim S. A complete description of pairs (S, O) is given. Now we consider the following cases. 1) G = Ψ (T, T). Then P(V )/Ψ (T, T) ≈ T\SL(2, C)/T ≈ W2 /T. The point [x4 ] ∈ W2 is T-invariant. Projection from this point gives a birational isomorphism W2 /T ≈ P3 /T. The last variety is obviously rational (because T has no irreducible four-dimensional representations). 2) G = Ψ (I, T) or Ψ (I, I). As above, P(V )/G ≈ W22 /G2 and the point P := [x12 ] ∈ W22 is G2 -invariant. Since the point P is contained in an open orbit, it does not lie on a line [MU83]. Triple projection from P (the rational map given by the linear system |−KW −3P |) gives a birational G2 -equivariant map W22 P3 [Tak89], see also [IP99, §4.5]. Therefore, P(V )/G ≈ W22 /G2 ≈ P3 /G2 . If G2 = T, the last variety is rational, as in case 1). For G2 = I, the rationality of P3 /G2 will be proved below. 3) G = Ψ (O, T) or Ψ (O, O). Then P(V )/G ≈ W6 /G2 and the point P := [x6 ] ∈ W5 is G2 -invariant. Double projection from P (i.e., the rational map deﬁned by the linear system |− 1/2KW5 − 2P |) gives a G2 -equivariant rational map W5 P2 [KMP92], [FN89]. There is the following diagram of G2 equivariant maps: χ W _ _ _/ W + σ

ϕ

ψ W5 _ _ _/ P2 where σ is the blowup of P , χ is a ﬂop, and ϕ is a P1 -bundle. Let S := σ −1 (P ) be the exceptional divisor and let S + be its proper transform on W + . Then S + is a G2 -invariant section. Therefore, the quotient W + /G2 → P2 /G2 has a birational structure of P1 -bundle. This implies that P3 /G ≈ W + /G2 is rational.

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The rationality of P3 /Ψ (O, I) is an open question. Theorem 2.3. Notation as above. (i) P3 /Ψ (O, I) ≈ W22 /O ≈ W5 /I. (ii) P3 /Ψ (O, I) is stably rational. More precisely, P3 /Ψ (O, I) × P2 is rational. Proof. (i) follows by the above arguments. We prove (ii). Put V := C3 . There is a faithful three-dimensional representation A5 → GL(V ) that induces an action of A5 on P2 = P(V ). We have the following ﬁbrations: (W5 × P2 )/A5 MMM p MMMg f pppp MMM p p p MM& xppp W5 /A5

P2 /A5

where f (resp. g) is a generically P2 (resp. W5 )-bundle in the étale topology. Since the action of A5 on P2 is faithful, the map f admits a section. Put X := (W5 × P2 )/A5 and K := C(P2 /A5 ). Then X ≈ W5 /A5 × P2 . On the other hand, XK := X ⊗ K is a smooth Fano threefold of index 2 and degree 5 (see [IP99]) deﬁned over a nonclosed ﬁeld K. A general pencil of hyperplane sections deﬁnes a structure of del Pezzo ﬁbration of degree 5 on XK . By [Man86, Ch. 4] the variety XK is K-rational (cf. Proof of Theorem 5.1, case 9 Γ12 ).

3 The primitive group of order 64 · 6! and the Segre cubic In this section we prove the rationality of quotients of P3 by primitive groups of type (XI) with two exceptions. We also give an alternative proof of the rationality P3 /G for groups of type (I) for (A, B) = (T, T), (T, O), (O, O). This section is a modiﬁed and simpliﬁed version of [KMP93] but basically follows the same idea. The primitive group of order 64 · 6! ([Bli17]) Let Q8 ⊂ SL(2, C) be the quaternion group of order 8. We may assume that Q8 = {±E, ±I, ±J, ±K}, where √ √ 0 0 −1 −1 √0 −1 √ I= , J= , K =I·J = . 1 0 −1 0 0 − −1 Regard V = C4 as the space of 2 × 2 matrices. The group Q8 × Q8 naturally acts on V by multiplications from the left and the right. This induces a representation ρ : Q8 × Q8 → SL(V ).

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The image of ρ is an imprimitive group of order 32 isomorphic to Q8 × Q8 /{±1}. Indeed, Q8 ×Q8 interchanges the one-dimensional subspaces Vi ⊂ V generated by e1 := E, e2 := I, e3 := J, and e4 := K. Denote by N the subgroup √ of order 64 in SL(V ) generated by ρ(Q8 ×Q8 ) and scalar multiplications by −1. Let M ⊂ SL(V ) be the normalizer of N . We study the rationality question for diﬀerent subgroups of M . The group M naturally acts on ∧2 V C6 as an imprimitive group (see [Bli17]). This easily follows from the fact that the image of N in GL(Λ2 V ) is an Abelian group. The eigenvectors + + + := e1 ∧ e2 + e3 ∧ e4 , w13 := e1 ∧ e3 + e2 ∧ e4 , w14 := e1 ∧ e4 + e2 ∧ e3 , w12 − − − w12 := e1 ∧ e2 − e3 ∧ e4 , w13 := e1 ∧ e3 − e2 ∧ e4 , w14 := e1 ∧ e4 − e2 ∧ e3

form a basis of ∧2 V . This gives a decomposition 42

V =

6

± Wi = C · wj,k

Wi ,

(1)

i=1

such that N · Wi = Wi . The group M/N permutes subspaces Wi and is isomorphic to the symmetric group S6 . Thus we have the exact sequence π

1 −→ N −→ M −→ S6 −→ 1

(2)

and any ﬁnite group containing N as a normal subgroup is uniquely determined by its image under π : M → S6 . √ √ √ Example 3.1. It is easy to see that the matrix S := 1+√2−1 Diag( −1, −1, 1, 1) is contained in M . The action on ∧2 V is as follows: √ + − + + + + w12 → − −1 w12 , w13 → −w13 , w14 → −w14 , √ − + − − − − w12 → − −1 w12 , w13 → −w13 , w14 → −w14 . Therefore, S is a transposition (1, 2) in M/N = S6 . Similarly, the matrix B := √ 1+√ −1 Diag(1, 1, 1, −1) also normalizes N . The corresponding substitution in 2 S6 is (1, 2)(3, 4)(5, 6) (odd element of order 2). Remark 3.2. The group N/Z(N ) = Q8 × Q8 /{±1, ±1} is Abelian of order 16 isomorphic to (F2 )4 . The exact sequence (2) induces an embedding S6 "→ Aut F42 SL(4, F2 ). The results Theorem 3.3 ([KMP93]). The quotient P3 /N is isomorphic to the Igusa quartic I4 ⊂ P4 (see below ).

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Theorem 3.4 ([KMP93]). Let G ⊂ SL(4, C) be a ﬁnite group having a normal imprimitive subgroup N of order 64. Then P3 /G is birationally isomorphic to the quotient of the Segre cubic S3 ⊂ P4 by G/N , where the action of G/N on P4 is given by the composition of the canonical embedding π : G/N "→ S6 , an outer automorphism λ : S6 → S6 (see, e.g., [Mil58]), and the standard action of S6 on S3 . Corollary 3.5 ([KMP93]). In the above notation, the variety P3 /G is rational except possibly for the following two groups (we use Blichfeldt’s notation [Bli17] ): 20o |G| = 64 · 360 and G/N A6 , 17o |G| = 64 · 60, G/N A5 , and π(G/N ) ⊂ S6 as a transitive subgroup. Corollary 3.6. In cases 20o and 17o above, the variety P3 /G is birationally isomorphic to S3 /λ ◦ π(G/N ). Note that in case 17o one of the xi is λ ◦ π(G/N )-invariant. Segre cubic Regard the variety S3 given by the equations x1 + x2 + x3 + x4 + x5 + x6 = 0, x31 + x32 + x33 + x34 + x35 + x36 = 0

(3)

as a cubic hypersurface in P4 . This cubic satisﬁes many remarkable properties (see [Seg63], [SR49, Ch. 8], [DO88]) and is called the Segre cubic. For example, any cubic hypersurface in P4 has at most ten isolated singular points, this bound is sharp and achieved exactly for the Segre cubic (up to projective isomorphism). The symmetric group S6 acts in S3 in the standard way. Moreover, it is easy to show that Aut(S3 ) = S6 (see, e.g., [Fin87]). We refer to [DO88] and [Koi03] for further interesting properties of S3 . The singular locus of S3 consists of ten nodes given by xi1 = xi2 = xi3 = −xi4 = −xi5 = −xi6 ,

(4)

where {i1 , i2 , i3 , i4 , i5 , i6 } = {1, 2, 3, 4, 5, 6}. We denote such a point by 123 i1 ,i2 ,i3 P i4 ,i5 ,i6 . For example, P 345 = (1, 1, 1, −1, −1, −1). It is easy to see that i1 ,i2 ,i3 j1 ,j2 ,j3 P i ,i ,i = P j ,j ,j if and only if corresponding matrices are obtained from 4 5 6 4 5 6 each other by permutations of rows and elements in each row. Hence there is an S6 -equivariant 1–1 correspondence Sing(S3 ) ←→ {Sylow 3-groups in S6 }. i1 ,i2 ,i3 Since the action of S6 on Sing(S3 ) is transitive, the stabilizer St(P i4 ,i5 ,i6 ) i1 ,i2 ,i3 123 of P i ,i ,i is a group of order 72. For example, St(P 345 ) is generated by 4 5 6 S3 × S3 and (1, 3)(2, 4)(3, 5).

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Further, there are 15 planes on S3 (see e.g. [Fin87]). Each of them is given by equations xi1 + xi4 = xi2 + xi5 = xi3 + xi6 = 0, (5) i1 ,i2 ,i3 where {i1 , i2 , i3 , i4 , i5 , i6 } = {1, 2, 3, 4, 5, 6}. Denote such a plane by Π i4 ,i5 ,i6 i1 ,i2 ,i3 and the set of all the planes on S3 by Ω. It is easy to see that Π i4 ,i5 ,i6 = j1 ,j2 ,j3 Π j4 ,j5 ,j6 if and only if corresponding matrices are obtained from each other by permutations of columns and elements in each column. Thus there is an S6 -equivariant 1-1-correspondence Ω ←→ {(i1 , i4 )(i2 , i5 )(i3 , i6 )} ⊂ S6 . For σ = (i1 , i4 )(i2 , i5 )(i3 , i6 ), we often will write Π(σ) instead of Π

i1 ,i2 ,i3 i4 ,i5 ,i6

.

Corollary 3.7. St(Π(σ)) = Z(σ) and |Z(σ)| = 48. Comparing (5) and (4) one can see the following facts. a) Every plane Π(σ) contains exactly four singular points and there are six planes passing through 5 every6singular point. i1 ,i2 ,i3 b) Π(σ1 ) ∩ Π(σ2 ) = P i4 ,i5 ,i6 ⇐⇒ σ1 ◦ σ2 = (i1 , i2 , i3 )(i4 , i5 , i6 ). c) Π(σ1 ) ∩ Π(σ2 ) is a line ⇐⇒ σ1 and σ2 contain a common transposition.

Recall that there exists an outer automorphism λ : S6 → S6 (see, e.g., [Mil58]). For any standard embedding S5 ⊂ S6 , the subgroup λ(S5 ) is a “nonstandard” transitive subgroup isomorphic to S5 . We need the following simple lemmas. Lemma 3.8 (cf. [SR49, pp. 169–170]). Let G ⊂ S6 be a subgroup of order 120 (isomorphic to S5 ). One of the following holds: (i) G is not transitive, then G ∩ St(Π(σ)) is of order 8 and G acts on Ω transitively; (ii) G is transitive, then the order of G ∩ St(Π(σ)) is either 12 or 24 and Ω splits into two G-orbits Ω and Ω consisting of 10 and 5 planes, respectively. A plane Π(σ) is contained in Ω if and only if σ ∈ G. Moreover, every two planes from Ω intersect each other at a (singular ) point. Proof. (ii) Assume that there are two planes Π(σ1 ), Π(σ2 ) ∈ Ω such that Π(σ1 ) ∩ Π(σ2 ) is a line. Then σ1 and σ2 contain a common transposition and σ1 , σ2 ∈ / G. This is equivalent to that σ1 ◦ σ2 is an element of order 2. But then λ(σ1 ), λ(σ2 ) are two transpositions such that λ(σ1 ) ◦ λ(σ2 ) is an element of order 2 and both λ(σ1 ), λ(σ2 ) are not contained in a standard subgroup S5 ⊂ S6 . Clearly, this is impossible.

Lemma 3.9. Let Π = Π(σ) be a plane on S3 and let H be the pencil of hyperplane sections through Π. Let Q be the pencil of residue quadrics to Π (i.e., H = Π + Q). Then the base locus of Q consists of four singular points of S3 contained in Π and a general member of Q is smooth.

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Proof. It is suﬃcient to check the statement only for one plane. In this case it can be done explicitly.

Igusa quartic Consider the dual map Ψ : S3 P4∗ sending a smooth point P ∈ S3 to the tangent space TP,S3 . Let S3∗ = Ψ (S3 ) be the dual variety. Lemma 3.10 ([Koi03], [DO88]). The map Ψ : S3 S3∗ is birational and deg S3∗ = 4. Proof. Assume that Ψ is not birational. Then for a general P ∈ S3 there is at least one point P = P such that Ψ (P ) = Ψ (P ). This means that tangent spaces TP,S3 and TP ,S3 coincide. We may assume that P, P are not contained in any plane Π(σ). The line L passing through P and P is contained in S3 . This line meets some plane Π = Π(σ) ⊂ S3 . Thus the linear span L, Π is a hyperplane in P4 . It is easy to see that L, Π ∩ S3 = Π ∪ Q, where Q is a two-dimensional quadric. Clearly, TP,Q = TP ,Q . This is possible only if Q is a quadratic cone. By Bertini’s theorem its vertex is contained in Π. This contradicts Lemma 3.9. Therefore, Ψ is birational and a general tangent hyperplane section TP,S3 ∩ S3 has exactly one node. This implies that a general pencil H of hyperplane sections of S3 is a Lefschetz pencil. Let H1 , . . . , Hr be singular members of H. Ten of them pass through singular points of S3 . Hence the degree of S3∗ ⊂ P4∗ is equal to r − 10. Finally, it is easy to compute that the topological Euler number of a cubic hypersurface in P4 having only nodes as singularities is equal to m − 6, where m is the number of singular points. Using this fact one can see that χtop (S3 ) = 4 = 9(2 − r) + 8r, so r = 14 and deg S3∗ = 4.

Therefore, S3∗ is a hypersurface of degree 4 in P4 . This famous quartic is called the Igusa quartic (cf. [DO88], [Koi03]) and denoted by I4 . An interesting fact is that I4 is the compact moduli space of Abelian surfaces with the level two structure. By the above, there is an S6 -equivariant birational map Ψ : S3 I4 . We need the following characterization of I4 in terms of the action of S6 . Lemma 3.11. The image of every plane Π = Π(σ) is a line on S3∗ . Therefore, S3∗ contains an S6 -invariant conﬁguration of 15 lines such that the action on these lines is transitive. Moreover, the above 15 lines form the singular locus of S3∗ and the stabilizer of such a line is Z(σ), where σ = (i1 , i2 )(i3 , i4 )(i5 , i6 ). Proof. The tangent space TP,S3 at P ∈ Π contains Π. Hence TP,S3 is contained in the pencil of hyperplane sections passing through Π. The fact that Ψ (Π) ⊂ Sing(S3∗ ) can be checked by direct computations, see [Koi03].

Lemma 3.12. Let X ⊂ P4 be an S6 -invariant quartic under the standard action of S6 on P4 . Assume that there is an odd element σ ∈ S6 of order 2 such that

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(i) the ﬁxed point locus of σ on P4 is a disjointed union of a line L and a plane Π, (ii) L is contained in X. Then X is the Igusa quartic (up to projective equivalence) and L ⊂ Sing(X). Proof. It is clear that X must be deﬁned by an invariant ψ of degree 4. On the other hand, modulo xi there are exactly two linearly independent invariants of degree 4: s4 and s22 , where sd := xdi . Hence, ψ = αs4 + βs22 , α, β ∈ C. Applying a suitable automorphism of S6 we may assume that σ = (12)(34)(56). Therefore, L is given by the following equations: x1 = x2 ,

x3 = x4 ,

x5 = x6 ,

x1 + x3 + x5 = 0.

On the other hand, in the pencil ψ = αs4 + βs22 there is exactly one quartic containing such an L.

Invariants of N Lemma 3.13. The ring of invariants C[x, y, z, u]N is generated by the following elements of degree 4: f1 = x2 u2 + y 2 z 2 ,

f2 = x2 z 2 + y 2 u2 ,

f4 = x4 + y 4 + z 4 + u4 ,

f5 = xyzu.

f3 = x2 y 2 + z 2 u2 ,

The only relation is . (f4 + 2f1 + 2f2 + 2f3 ) f1 f2 f3 − f4 f52 + 2f52 (f1 + f2 + f3 ) 2 − f1 f2 + f2 f3 + f3 f1 + 4f52 = 0. (6) Proof. Indeed, it is easy to see that all the fi are N -invariants. Hence, C[f1 , . . . , f5 ] ⊂ C[x, y, z, t]N . Now it is suﬃcient to show that [C(x, y, z, u) : C(f1 , f2 , f3 , f4 , f5 )] = |N | = 64. Using the relations C(f1 , f2 , f3 , f4 , f5 ) : C(f1 , f2 , f3 , f4 , f52 ) = 2, C(f1 , f2 , f3 , f4 , f52 ) : C(xu + yz, xz + yu, xy + zu, x2 + y 2 + z 2 + u2 , f5 ) = 16, [C(xu + yz, xz + yu, xy + zu, x2 + y 2 + z 2 + u2 , f5 ) : : C(xu + yz, xz + yu, xy + zu, x + y + z + u, f5 )] = 2 we get the desired equality. Relation (6) can be checked directly. It is easy to see that the quartic given by (6) is smooth in codimension one. Hence it is

irreducible and (6) is the only relation between the fi .

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Lemma 3.14. The quartic X := P3 /N given by (6) is singular along 15 lines L1 , . . . , L15 . Moreover, S6 acts on the Li transitively. Proof. This can be obtained immediately from equation (6) but we prefer to use the quotient structure on X. The group N/Z(N ) is an Abelian group of order 16 isomorphic to (μ2 )4 . Every nontrivial element a ∈ N/Z(N ) ﬁxes ¯ i ⊂ P3 , i = 1, 2. Thus in P3 there are 30 lines whose points on two lines L a ¯ ia are contained general points have stabilizer of order 2. The images of the L in the singular locus of X.

Proof (of Theorem 3.3). Put X := P3 /N and ﬁx the embedding X "→ P4 by the fi ’s. Clearly, the group S6 M/N naturally acts on X. Since Pic X Z · (−KX ), the action is induced by a linear action on P4 . The transposition S from Example 3.1 acts on the fi by the diagonal matrix Diag(1, 1, −1, −1, 1). In particular, it is not a reﬂection. This shows that up to scalar multiplication the action of S6 on X ⊂ P4 is given by the composition of the standard action of S6 on P4 and the outer automorphism λ. Now it is suﬃcient to show that Sing(X) contains a line L such as in Lemma 3.12. Indeed, the ﬁxed point locus of S on P4 consists of disjointed union of the plane f3 = f4 = 0 and the line f1 = f2 = f5 = 0 contained in X. Moreover, X is singular along f3 = f4 = 0. Thus Lemma 3.12 can be applied and X S3 .

Proof (of Theorem 3.4 and Corollary 3.5). Consider any subgroup G such that N ⊂ G ⊂ M . By Theorem 3.3 the quotient P3 /N is the Igusa quartic I4 ⊂ P4 , where the action of G/N = S6 on P4 is given by the composition of the canonical embedding π : G/N "→ M/N = S6 , an outer automorphism λ : S6 → S6 , and the standard action of S6 on P4 . Thus P3 /G ≈ I4 /(G/N ). By Lemma 3.10 there is an S6 -equivariant birational map I4 S3 . Hence P3 /G ≈ S3 /(G/N ). This proves Theorem 3.4. To prove Corollary 3.5 we consider two cases. 3.15. G/N has a fixed point P ∈ Sing(S3 ). Projection from this point is G/N -equivariant, and therefore S3 /(G/N ) is birationally equivalent to P3 /(G/N ), where G/N is a group of order ≤ 72. Corollary 3.5 in this case follows from the rationality of P3 /(G/N ). 3.16. π(G) is a subgroup of a S5 ⊂ S6 , a standard (nontransitive) permutation group. This exactly means that one of the Wi (see (1)) is an eigenspace for G. Then P3 /G ≈ S3 /(G/N ), where G/N is embedded into a transitive S5 . By Lemma 3.8 there is a G-invariant set Ω of 5 planes Πi = Π(σi ) and Πi ∩Πj ⊂ Sing(S3 ) for i = j. We claim that there are exactly two planes Πi ∈ Ω passing through every singular point. Indeed, this follows by the fact that S5 transitively acts on the set of Sylow 3-subgroups in S6 . Further, for every singular point P ∈ S3 , there is exactly two small (possibly nonprojective) resolutions μP : S¯3 → S3 and μP : S¯3 → S3 . Recall that a birational contraction is said to be small if it does not contract any divisors.

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¯ i of planes Πi ∈ Ω do For one of them, say for μP , the proper transforms Π −1 not meet each other over μP (P ). Thus there is exactly one small resolution ¯ i of the μ : S¯3 → S3 of all singular points such that the proper transforms Π planes Πi ∈ Ω are disjointed. This resolution must be S5 -equivariant (cf. ¯ i P2 . It is easy to check that the Π ¯i [Fin87, §5]). By our construction, Π ¯ satisfy the contractibility criterion, i.e., Πi |Π¯ i OP2 (−1). So there is an S5 equivariant contraction ϕ : S¯3 → W of Moishezon complex manifolds. Since Pic S¯3 Z⊕6 (see [Fin87]), we have Pic W Z. Obviously, the anticanonical divisor −KW is eﬀective and divisible by four in Pic W . By [Nak87] the variety W is projective and W P3 . Therefore, we have P3 /G ≈ S3 /(G/N ) ≈ P3 /(G/N ), where (G/N ) is a subgroup of S5 . The latter reduces the question of rationality of P3 /G to a smaller group G of order ≤ 120. This will be discussed below. In fact, it will be shown in §4 that there is another (diﬀerent from μ : S¯3 → S3 ) small projective G/N -equivariant resolution μ+ : S¯3+ → S3 and there is a G/N -equivariant P1 -bundle structure on S¯3+ (see Proof of Proposition 4.6). To ﬁnish the proof of Corollary 3.5 we note that, except for 20o and 17o , there are only two groups which do not satisfy conditions 3.15 or 3.16 above: π(G) = S6 and π(G) ⊂ S6 is a transitive S5 . The ring of invariants C[S3 ]S6 is 6 generated by symmetric functions s2 , s4 , s5 , s6 , where sk = i=1 xki . Therefore, S3 /S6 P(2, 4, 5, 6) is rational. Consider the case when G/N S5 and π(G) ⊂ S6 is a transitive S5 . Then G/N = S5 ﬁxes some of x1 , . . . , x6 on S3 . Assume that S5 · x6 = x6 . The ring of invariants C[S3 ]S6 is generated by 5 x6 and s1 , . . . , s5 , where sk = i=1 xki . Thus, C[S3 ]S6 C[s2 , s4 , s5 , x6 ] and S3 /S5 P(2, 4, 5, 1) is rational. This proves Corollary 3.5.

Remark 3.17. Another approach to the treatment of case 3.16 is to note that there is a G/N -equivariant P1 -bundle structure on S3 . Indeed, consider the family of lines L = L(S3 ) on S3 . Let L1 , . . . , Lr be all covering irreducible components (i.e., components Li such that there is a line from Li passing through a general point of S3 ). Since there are at most 6 lines passing through a general point, r ≤ 6. On the other hand, there is a hyperplane section of the form Π1 + Π2 + Π3 , where Πi are planes. A general line from a covering family meets exactly one of the planes Π1 , Π2 , Π3 . This shows that r ≥ 3. Using the action of S6 of S3 one can see that r = 6. Therefore, each family Li gives us (birationally) a P1 -bundle structure on S3 . Now it remains to note that if G is such as in 3.16, then one of families Li is G/N -invariant.

4 Groups of types (V)–(X) Case (X) Recall that an element g ∈ GL(n, C) of ﬁnite order is said to be (complex) reﬂection if exactly n − 1 eigenvalues are equal to 1. For convenience of the

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reader we recall the following well-known theorem of Chevalley and ShephardTodd: Theorem 4.1 (see [ST54], [Che55], see also [Spr77]). Let V = Cn and let G ⊂ GL(V ) be a ﬁnite subgroup. The following are equivalent: (i) G is generated by reﬂections, (ii) V /G Cn . Proof (Outline of Proof ). Let W := V /G and f : V → W be the quotient morphism. (ii) =⇒ (i). Assume the converse. Let G0 ⊂ G be the maximal subgroup generated by reﬂections. Then the morphism V /G0 → W is étale over W \ Z, where Z is of codimension of least two. On the other hand, π1 (V \ Z) = {1}, a contradiction. (i) =⇒ (ii). Put R := C[x1 , . . . , xn ]. First we claim that R is a free RG module. Let I ⊂ R be the ideal generated by homogeneous invariants of positive degree. Lemma 4.2. Assume that for some homogeneous elements yi ∈ R and zi ∈ RG the following relation holds: z1 y1 + · · · + zm ym = 0.

(7)

If z1 ∈ / RG z2 + · · · + RG zm , then y1 ∈ I. Proof. We can If deg y1 = 0, then applying take yi so that deg y1 is minimal. 1 G the map |G| g to (7) we obtain z ∈ R z + · · · + RG zm . Assume that 1 2 g∈G deg y1 > 0. Let s ∈ G be a reﬂection and let {ls = 0} be its ﬁxed hyperplane. For any homogeneous f ∈ R, the polynomial s · f − f is divisible by ls . Deﬁne a map Δs (f ) = (s · f − f )/ls . Δs : Rd → Rd−1 , It is easy to check that Δs (f1 f2 ) = (Δs f1 )f2 ,

∀f1 ∈ R,

∀f2 ∈ RG .

This gives us z1 Δs (y1 ) + · · · + zm Δs (ym ) = 0. Since deg Δs (yi ) < deg yi , we may assume that Δs (y1 ) ∈ I. Therefore, s · y1 − y1 ∈ I for all reﬂections s ∈ G. This implies that g · y1 − y1 ∈ I for all g ∈ G,

so y1 ∈ I. Take homogeneous elements yi ∈ R so that the images y¯i in R/I form a basis over C. It is clear that the yi generate R as an RG -module. By the above lemma these yi are linearly independent over RG . Indeed, if z1 y1 +· · ·+zm ym = 0 for some zi ∈ RG , then z1 = z2 u2 + · · · + zm um , ui ∈ RG . So,

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z2 (y2 + y1 u2 ) + · · · + zm (ym + y1 um ) = 0 and we can apply the induction by m. Since R is integral over RG , the basis yi is ﬁnite. This proves our claim. Further, let J be the ideal of RG generated by homogeneous elements of positive degree. Since RG is a Noetherian algebra, J is ﬁnitely generated. Take a minimal system of generators f1 , . . . , fr . It is clear that f1 , . . . , fr generate RG as C-algebra. On the other hand, one can check that they are algebraically independent. This proves the theorem.

Note that the statement of Theorem 4.1 fails if the characteristic of the base ﬁeld divides |G|. However the ﬁeld of invariants is rational in this case under the additional assumption that G is irreducible [KM99]. Corollary 4.3. Let G ⊂ GL(n, C) be a ﬁnite group generated by reﬂections. Then C[x1 , . . . , xn ]G C[f1 , . . . , fn ], where f1 , . . . , fn are homogeneous polynomials. Numbers di := deg fi are called the degrees of G. They are uniquely determined by G. Thus for any group generated by reﬂections we have Cn /G Cn and Pn−1 /G is a weighted projective space P(d1 , . . . , dn ). The list of all complex reﬂection groups can be found in [ST54], [Coh76]. According to this list there is a ﬁnite group No. 32 of order 25920 · 6 generated by complex reﬂections of order 3. The degrees of this group are 12, 18, 24, 30 and the intersection with SL(4, C) is exactly our ﬁrst group. Therefore, the quotient P3 /G25920 P(12, 18, 24, 30) P(2, 3, 4, 5) is rational. The group SL(2, F5 ) Let δ : SL(2, F5) "→ SL(2, C) be a faithful representation whose image is the icosahedron group I. For short, we identify SL(2, F5 ) with I. Then S 3 δ : I → SL(4, C) is an irreducible faithful representation as in (VIII). This gives the action of A5 = I/{±E} on P3 which leaves a rational cubic curve C invariant. Since I has a faithful two-dimensional representation, P3 /I is stably rational. We prove more: Theorem 4.4 ([KMP87], [KMP92]). P3 /I is rational. Proof. Let σ : X → P3 be the blowup of C. Then X is a Fano threefold and there is a K-negative extremal contraction [Mor82] diﬀerent from σ. This contraction is given by the birational transform of the linear system of quadrics passing through C and the ﬁbers are birational transforms of 2-secant lines of C. This deﬁnes a P1 -bundle structure ϕ : X → P2 . We have the following I-equivariant diagram: X } AAA ϕ σ }} AA }} AA } ~} _ _ _ _ _ _ _ / P2 P3

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Thus P3 /I ≈ X/I and there is a rational curve ﬁbration f : X/I → P2 /I. The action I on P2 is induced by an irreducible representation β : I/{±E} = A5 → SL(3, C). Note that the group β(A5 )·{±E} is generated by reﬂections and has degrees 2, 6, 10 (see [Bur55, Ch. 17, §266]). Therefore, P2 /A5 is the weighted projective plane P(2, 6, 10) P(1, 3, 5). Let Δ ⊂ P2 /A5 be the minimal curve such that F is smooth over (P2 /A5 )\Δ. It is easy to see that Δ is the image of the reﬂection lines in P2 . Since there are exactly 15 such lines (corresponding to order 2 elements in A5 ), the degree of Δ on P(1, 3, 5) is equal to 15 and Δ is irreducible. Let P(1, 3, 5) = Proj C[x0 , y0 , z0 ], where deg x0 = 1, deg y0 = 3, deg z0 = 5. Using only the equality deg Δ = 15 one can see that Δ is given by the equation c1 z03 + c2 y05 + c3 x0 y03 z0 + c4 x20 y0 z02 + c5 x30 y04 + c6 x40 y02 z0 + c7 x50 z02 + c8 x60 y03 12 15 + c9 x70 y0 z0 + c10 x90 y02 + c11 x10 0 z0 + c12 x0 y0 + c13 x0 = 0,

where the ci are some constants. Consider the open set U = P(1, 3, 5) ∩ {x0 = 0}. Then in coordinates y = y0 /x30 , z = z0 /x50 on U A2 the curve Δ is deﬁned by c2 y 5 + c5 y 4 + (c3 z + c8 ) y 3 + (c6 z + c10 ) y 2 + c4 z 2 + c9 z + c12 y + c1 z 3 + c7 z 2 + c11 z + c13 = 0. Put S0 := U \ Δ and V := f −1 (S0 ). Then f |V : V → S0 is a smooth morphism whose geometric ﬁbers are isomorphic to P1 . Thus, V → S0 is a Severi–Brauer scheme. Lemma 4.5. Let S be a smooth projective rational surface and let D ⊂ S be a reduced curve. Let S0 := S \ D and let V /S0 be a Severi-Brauer scheme. Assume that there is an irreducible component D1 ⊂ D which is a smooth rational curve and such that D1 meets the closure D − D1 at a single point. Then the Severi-Brauer scheme V /S0 can be extended to S \ D − D1 . Proof. According to general theory, there is 1–1 correspondence between isomorphism classes of Severi–Brauer S0 -schemes of relative dimension n − 1 and isomorphism classes of Azumaya OS0 -algebras of rank n2 . Let A be the corresponding Azumaya algebra over S0 . Denote by [A] its class in the Brauer group of the function ﬁeld Br C(S). Taking into account that S is rational we consider the Artin–Mumford exact sequence [AM72]: a

0 −→ Br C(S) −→

r

H 1 (C(C), Q/Z) −→

C⊂S

−→

μ−1 −→ μ−1 −→ 0, s

P ∈S

where μ−1 := ∪n Hom(μn , Q/Z), the ﬁrst sum runs through all irreducible curves C ⊂ S while the second runs through all closed points P ∈ S (for details

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we refer to [AM72], [Tan81]). Note that the map r “measures” ramiﬁcation and the map s is just the sum. Now let P := D1 ∩D − D1 . Since there are no cyclic coverings of D1 P1 ramiﬁed only at P , the H 1 (C(D1 ), Q/Z)-component of a([A]) is zero, i.e., the algebra A is unramiﬁed over D1 . In this situation, A can be extended over D1 , see, e.g., [Tan81, Prop. 6.2].

¯ . Let (x, y, z) Now we consider the natural embedding U = A2 "→ P2 = U 2 2 be homogeneous coordinates on P so that Px,y,z ∩ {z = 0} = A2x,y = U . Let Δ¯ ⊂ P2 be the closure of Δ ∩ U . Then Δ¯ intersects the inﬁnite line N := {x = 0} at a single point P := (0, 0, 1) which is cuspidal. By the above lemma the Severi–Brauer scheme V can be extended to N . Let Lt be the ¯ at P pencil of lines on P2 through P . Then a general member of Lt meets Δ and three more points. According to [Sar82] there is a standard conic bundle #2 and a commutative diagram: g: Y → P X/I o_ _ _ Y g ψ #2 P2 o P # 2 is a smooth surface, ψ is a birational morphism, and Y X/I is Here P # of g is contained a birational map. By the above the discriminant curve Δ ¯ Moreover, again by Lemma 4.5 the Severi–Brauer scheme V can in ψ −1 (Δ). be extended to all exceptional divisors over P (because P ∈ Δ¯ is a cuspidal # t is a base point free pencil such that point). Thus we may assume that L # # Lt · Δ = 3. In this situation, Y is rational (see [Isk87]), so are both X/I and

P3 /I. Groups of type (V) Since the ring of invariants of the standard representation of S5 on C4 is generated by symmetric polynomials s2 , . . . , s5 , the quotient P3 /S5 P(2, 3, 4, 5) is rational. Note also that A5 has a faithful three-dimensional representation, so P3 /A5 is stably rational (more precisely, P3 /A5 × P3 is rational). T. Maeda [Mae89] (see also [KV89]) proved the rationality of A5 /A5 ≈ P3 /A5 × P2 (over an arbitrary ﬁeld). The rationality of P3 /A5 over C was proved in [KV89] by an algebraic method. Here we propose an alternative, geometric approach. Proposition 4.6. There is the S5 -equivariant diagram

262

Y. G. Prokhorov χ X A_ _ _ _ _ _ _/ X + AA {{ ϕ+ AAϕ0 0 { AA {{ { A }{{ ϕ σ S3

P3 _ _ _ _ _ _ _ / W where χ is a ﬂop, ϕ0 and ϕ+ 0 are small contractions to the Segre cubic S3 , W is a smooth del Pezzo surface of degree 5, and ϕ is a P1 -bundle. As in the proof of Theorem 4.4 the rationality of P3 /A5 can be proved by a detailed analysis of the discriminant curve of the rational curve ﬁbration X + /A5 → W/A5 . Proof. Let G = S5 . Consider the standard representation G = S5 "→ GL(4, C) and the corresponding linear action on P3 . Then G permutes ﬁve points P1 , . . . , P5 ∈ P3 . Let σ : X → P3 be the blowup of P1 , . . . , P5 and let Si = σ −1 (Pi ) be the exceptional divisors. Let H be the class of hyperplane Pi is an intersection of quadric section on P3 and let H ∗ := σ ∗ H. Since (scheme-theoretically), the linear system |2H ∗ − Si | = |− 12 KX | is base point free. In particular, −KX is nef and big, i.e., X is a weak Fano threefold. It is easy to check that dim | − 12 KX | = 4 and the morphism ϕ0 : X → X0 ⊂ P4 given by | − 12 KX | is birational onto its image, a three-dimensional cubic. Moreover, ϕ0 is small and contracts proper transforms Li,j of lines passing through Pi and Pj . Thus X0 has ten singular points ϕ0 (Li,j ). Therefore, X0 is the Segre cubic S3 and our construction is inverse to the construction in §3, Proof of Corollary 3.5. Since (Pic X)G is of rank two, there is G-equivariant ﬂop χ : X X + . Here X + is a small resolution of X0 = S3 obtained from X by “changing all the signs” [Fin87]. There is a unique K-negative G-extremal ray on X + [Mor82]. Let ϕ : X + → W be its contraction. Assume that ρ(X + /W ) > 1. Then ϕ passes through a (non-G-equivariant) extremal contraction ϕ1 : X + → W1 . If ϕ1 is birational, then taking into account that −KX + is divisible by 2 and the classiﬁcation [Mor82] we get that W1 is smooth and ϕ1 is the blowup of a point. Let S1 be the corresponding exceptional divisor and let S1 , . . . , Sr be the G-orbit. By the extremal property the Si are disjointed and give us extremal rays on X + /W . Since ρ(X + /W ) ≤ 5, r ≤ 5. On the other hand, the image of Si on X0 = S3 is a plane. By Lemma 3.8 we have r = 5 and the Si are proper transforms of σ-exceptional divisors. Hence, χ = id and W = P3 . Clearly, this is impossible. Thus, ϕ1 is not birational. We claim that W is a surface. Indeed, W cannot be a point because −KX + is not ample. Assume that W is a curve and let F be a general ﬁber. Since KF = KX + |F is divisible by 2, F P1 × P1 . On the other hand, the Mori cone N E(X + /W ) has at least 5 (nonbirational) extremal rays. Each

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of them gives a contraction to a surface over W . This is impossible because the ﬁbers of these contractions are contained in the ﬁbers of ϕ. Thus W is a surface. By construction, the linear system | − 12 KX + | is the pullback of the system of hyperplane sections of S3 . In particular, | − 12 KX + | is base point free and a general element D ∈ | − 12 KX + | is a smooth cubic surface. Since the restriction ϕ|D : D → W is birational, the surface also is smooth and −KW is ample, i.e., W is a del Pezzo surface. The group G faithfully acts on W . 2 Clearly, this is not possible if ρ(W ) ≤ 5. Therefore, ρ(W ) = 5 and KW = 5. Finally, −KX + is divisible by 2, so the ﬁbration ϕ has no degenerate ﬁbers. This proves the statement.

5 Monomial groups An action of a group Γ on a ﬁeld C(x1 , . . . , xn ) is said to be monomial (with respect to x1 , . . . , xn ) if for every g ∈ Γ one has mi,1

g(xi ) = λi (g)x1

i,n · · · xm , n

λi (g) ∈ C∗ ,

mi,j ∈ Z.

(8)

Rationality of the ﬁelds of invariants of such actions was studied in the series of works [Hae71], [Haj83], [HK94], [Haj00]. Any monomial action (8) deﬁnes an integer representation π : Γ → GL(n, Z),

g → (mi,j ).

Now let V := Cn+1 and let G ⊂ GL(V ) be an imprimitive group of type (1 ). Then G permutes the one-dimensional subspaces Vi from Deﬁnition 1.3 and contains a normal Abelian subgroup A which acts diagonally in the corresponding coordinates x1 , . . . , xn+1 . Let Γ := G/A. By the above, there is a natural embedding Γ "→ Sn . Since we are assuming that the representation G "→ GL(V ) is irreducible, the group Γ ⊂ Sn+1 is transitive. Put n+1

y1 =

x1 xn , . . . , yn = ∈ C(P(V )). xn+1 xn+1

The action of A on y1 , . . . , yn is diagonal in these coordinates. If f = I aI yI ∈ C[y1 , . . . , yn ] is an A-invariant, then so are all the monomials aI yI . Hence the ring C[y1 , . . . , yn ]A and its fraction ﬁeld C(y1 , . . . , yn )A are generated by invariant monomials. Let z1 , . . . , zn be a basis of the free Z-module (A-invariant monomials in yi )∗ /C∗ . Then the ﬁeld of invariants C(P(V ))A is generated by these zi = zi (y1 , . . . , yn ) and they are algebraically independent. Further, the action of G on C(P(V ))A is monomial with respect to z1 , . . . , zn . Thus the rationality question of Pn /G is reduced to the rationality question of the invariant ﬁeld C(z1 , . . . , zn )Γ of a monomial (but in general nonlinear) action. We have two representations π0 , π : Γ → GL(n, Z), corresponding monomial actions of Γ on C(y1 , . . . , yn ) and C(z1 , . . . , zn ), respectively. Moreover,

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π is a restriction of π0 to an invariant sublattice of ﬁnite index. In particular, π ⊗ Q π0 ⊗ Q. In some cases the rationality of the ﬁeld C(z1 , . . . , zn )Γ can be proved by purely algebraic methods. For example, C(z1 , . . . , zn )Γ is rational in the following cases: (i) Γ is cyclic of order m, where the class number of mth cyclotomic ﬁeld is 1 [Hae71], [Haj83], (ii) dim V ≤ 4 and G is meta-Abelian [Haj83] (in fact, the author proved the rationality of V /G), (iii) n ≤ 3 and the action of Γ on C(z1 , . . . , zn ) is purely monomial (i.e., all constants λi (g) in (8) are equal to 1) with one exception1 [HK94]. From now on we consider the four-dimensional case, i.e., the case n = 4. Theorem 5.1. Let G ⊂ GL(4, C) be an imprimitive group of type (14 ). Then P3 /G is rational. There are the following possibilities for Γ ⊂ S4 : 1) Γ is a cyclic group of order 4, 2) Klein group V4 , 3) Γ is a dihedral group D4 of order 8, 4) Γ = A4 , and 5) Γ = S4 . By the above it is suﬃcient to prove the rationality of the ﬁeld C(z1 , z2 , z3 )Γ . The theorem was proved in the unpublished manuscript of I. Kolpakov-Miroshnichenko and the author (1986) by case-by-case consideration of the action of Γ on C(z1 , z2 , z3 ). We consider here only the case Γ = A4 . Case Γ = S4 can be treated in a similar way. Cases Γ = Z4 , V4 , and D4 are easier. Moreover, in these three cases G is also imprimitive of type (22 ) and then the rationality can be proved also by another method, see §6. The group A4 is generated by two elements: δ = (1, 2)(3, 4) and θ = (1, 2, 3). One has ⎛ ⎛ ⎞ ⎞ 010 0 1 −1 π0 (θ) = ⎝ 0 0 1 ⎠ π0 (δ) = ⎝ 1 0 −1 ⎠ . 100 0 0 −1 Using the classiﬁcation of ﬁnite subgroups in GL(3, Z) [Tah71] we get the following possibilities: ⎛ ⎞ ⎛ ⎞ 010 −1 0 0 Γ912 : π(θ) = ⎝ 0 0 1 ⎠ π(δ) = ⎝ 0 1 0 ⎠ ⎛1 0 0⎞ ⎛ 0 0 −1 ⎞ 010 0 −1 1 12 Γ10 : π(θ) = ⎝ 0 0 1 ⎠ π(δ) = ⎝ 0 −1 0 ⎠ 100 1 −1 0 1

Professor Ming-Chang Kang has informed me that the last unsettled case has been solved in the paper of A. Hoshi and Y. Rikuna (to appear in Math. of Comput.)

Fields of Invariants of Finite Linear Groups

⎛

⎞ 010 12 Γ11 : π(θ) = ⎝ 0 0 1 ⎠ 100

265

⎛

⎞ −1 −1 −1 π(δ) = ⎝ 0 0 1 ⎠ . 0 1 0

Case Γ912 By (8) the action of Γ = A4 on C(z1 , z2 , z3 ) has the following form (a1 z2 , a2 z3 , a3 z1 ) ←− (z1 , z2 , z3 ) −→ (b/z1 , cz2 , b /z3 ), θ

δ

ai , b, b , c ∈ C∗ .

After coordinate change of the form zi → λi zi , λi ∈ C∗ we may assume that a1 = a2 = a3 = b = 1. Since δ 2 = 1, c = ±1. From other relations between θ and δ we get b = c = ±1. Therefore, θ

δ

(z2 , z3 , z1 ) ←− (z1 , z2 , z3 ) −→ (b/z1 , bz2 , 1/z3 ),

b = ±1.

Assume that b = c = 1. Then after the coordinate change of the form z1 = (z1 + 1)/(z1 − 1),

z2 = (z2 + 1)/(z2 − 1),

z3 = (z3 + 1)/(z3 − 1)

the action will be linear: (z2 , z3 , z1 ) ←− (z1 , z2 , z3 ) −→ (−z1 , z2 , −z3 ). θ

δ

By Proposition 1.2 the ﬁeld C(z1 , z2 , z3 )Γ is rational. Now assume that b = c = −1. Regard z1 , z2 , z3 as nonhomogeneous coordinates on P1 × P1 × P1 . We get an action of Γ = A4 on P1 × P1 × P1 by regular automorphisms. Consider the Segre embedding P1 × P1 × P1 "→ P7 (z1 , z2 , z3 ) −→ (z1 z2 z3 , z1 , z2 , z3 , z2 z3 , z1 z3 , z1 z2 , 1) and let t1 , . . . , t8 be the corresponding coordinates in P7 . This induces the following representation of the tetrahedron group T into GL(8, C): θe

e δ

(t1 , t3 , t4 , t2 , t6 , t7 , t5 , t8 ) ←− (t1 , . . . , t8 ) −→ (t3 , −t4 , −t1 , t2 , −t7 , −t8 , t5 , t6 ). It is easy to check that this representation is the direct sum of four twodimensional faithful representations. Therefore, there are four Γ -invariant lines Li in P7 . Clearly P1 × P1 × P1 contains no A4 -invariant lines. On the other hand, the intersection P1 × P1 × P1 ∩ Li consists of at most two points (because the image of the Segre embedding is an intersection of quadrics). This immediately implies that all the Li are disjointed from P1 × P1 × P1 . Let V P5 be the linear span of L2 , L3 , L4 and let H be the pencil of hyperplane sections of P1 × P1 × P1 passing through V . We claim that the general member of H ∈ H is smooth. Let B = V ∩ P1 × P1 × P1 be the base locus of H. By Bertini’s theorem Sing(H) ⊂ B. If

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H is not normal, then by the adjunction formula H is singular along a line. On the other hand, P1 × P1 × P1 does not contain any A4 -invariant lines. This immediately implies that H is normal. Since −KH is ample, H is either a cone over an elliptic curve or a del Pezzo surface with Du Val singularities. In both cases the number of singular points is at most two. On the other hand, Sing(H) coincides with the set of points where the dimension of Zariski tangent space TP,B jumps. Therefore, Sing(H) is A4 -invariant. This immediately implies the existence of A4 -invariant point P ∈ P1 × P1 × P1 , a contradiction. Thus, a general member H ∈ H is a smooth del Pezzo surface of degree 6. Now consider the projection of P1 × P1 × P1 from V to L1 . By blowing up B we obtain a A4 -equivariant ﬁbration f : Y → P1 whose general ﬁber is a smooth del Pezzo surface of degree 6. This induces a ﬁbration Y /A4 → P1 /A4 with the same type of general ﬁber. Such a ﬁbration is rational over C(P1 /A4 ) (see [Man86, Ch. 4]), so (P1 × P1 × P1 )/A4 ≈ Y /A4 is also rational. 12 Case Γ10

Regard z1 , z2 , z3 as nonhomogeneous coordinates on P3 . We get a linear action of Γ = A4 on P3 . This action is either reducible or imprimitive. In the ﬁrst case, C(P(V ))G is rational. In the second one, we can repeat inductive procedure to reduce the problem to the smaller order of Γ . 12 Case Γ11

Then after coordinate change the action of Γ = A4 on C(z1 , z2 , z3 ) has the following form: θ

δ

(z2 , z3 , z1 ) ←− (z1 , z2 , z3 ) −→ (1/(z1 z2 z3 ), z3 , z2 ), i.e., it is purely monomial. The ﬁeld of invariants is rational by [HK94].

6 Imprimitive case (22 ) Theorem 6.1. Let G ⊂ GL(4, C) be an imprimitive group of type (22 ). Then P3 /G is rational. Proof (Outline of the proof ). Let G ⊂ SL(V ), dim V = 4 be an imprimitive group of type (22 ). Then there is a decomposition V = V1 ⊕ V2 , dim Vi = 2 and a subgroup N ⊂ G of index 2 such that N · Vi = Vi . We may assume that N is not Abelian (otherwise G is reducible). The decomposition V = V1 ⊕V2 deﬁnes two skew lines L1 , L2 in P3 = P(V ). Let H be the class of hyperplane in P3 . The image of the map P3 P3 given by the linear system 2H − L1 − L2 is the two-dimensional quadric P1 × P1 and

Fields of Invariants of Finite Linear Groups

267

ﬁbers of this map are lines meeting both L1 and L2 . We have the following G-equivariant diagram: #3 G P GG ~ GGϕ σ ~~ ~ GG GG ~~ ~ # ~ P3 _ _ _ _ _ _ _/ P1 × P1 # 3 → P3 is the blowup of L1 ∪ L2 and ϕ is a P1 -bundle. Let Si where σ : P be the corresponding exceptional divisors. There is a rational curve ﬁbration #3 /G → S = (P1 × P1 )/G. The rationality of P3 /G follows from detailed f: P analysis of the action of G on P1 × P1 (cf. Proof of Theorem 4.4). Consider for example the case when restrictions N → GL(Vi ) are injective. Let π : P1 × P1 → (P1 × P1 )/G be the quotient map and let Δ ⊂ S be the discriminant curve. Then, B := π −1 (Δ) is contained in the ramiﬁcation divisor, the union of one-dimensional components of the locus of points with nontrivial stabilizer. Take a point P = (x, y) ∈ B. If g ∈ St(P ), then g(x) = x, g(y) = y. If g ∈ N or g 2 = λE, there are only a ﬁnite number of such points. This implies that the ramiﬁcation index over each component of Δ is equal to 2. Consider the pencil Ct of (1, 0)-curves of P1 × P1 . Let Dt := π(Ct ). Then Dt is a base point free pencil. Hence, KS · Dt = 2pa (Dt ) − 2 − Dt2 = −2. Further, by the Hurwitz formula, KP1 ×P1

1 = π KS + Δ . 2 ∗

This yields 2 2 1 ∗ ∗ 0> KP1 ×P1 · π Dt = π K S + Δ · π ∗ Dt deg π deg π 2 1 = 2 KS + Δ · Dt = Δ · Dt − 4. 2 Therefore, Δ · Dt ≤ 3. Finally, as in the proof of Theorem 4.4 we deduce the # 3 /G using [Sar82] and [Isk87].

rationality of P

7 Final remarks and open questions As a consequence of the above results we have the following: Theorem 7.1. Let G ⊂ GL(4, C) a ﬁnite subgroup. Assume that G is solvable. Then P3 /G is rational.

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Remaining cases The rationality question for P3 /G and C4 /G is still open for the following groups (up to scalar multiplication). For convenience of the reader we give a short description of the group action. Type (I) The rationality of P3 /G is unknown only for G = Ψ (O, I). Note that P3 /Ψ (O, I) is stably rational, see Theorem 2.3. Types (VI), (VII), and (VIII) # 6, S # 6 , and A # 7 . The corresponding actions on P3 are # 5, A Unsolved cases are S given by projective representations of S5 , A6 , S6 , and A7 into P GL(4, C). Recall that the Schur multiplier H 2 (Sn , C∗ ) of the symmetric group is isomorphic to μ2 for n ≥ 4. Therefore, any projective representation of Sn , n ≥ 4 is ˜ n by μ . induced by a linear representation of a central extension S 2 Following Schur [Sch11], [Sch01] we give an explicit matrix representation ˜ 6 → GL(4, C). Consider the following matrices: S √ 10 0 0 −1 1 0 −1 √ E := A := B := C := . 01 1 0 0 −1 −1 0 Next take the following Kronecker products: M1 := C ⊗ A, M2 := C ⊗ B, M3 := A ⊗ E, M4 := B ⊗ E, M5 :=

√ −1C ⊗2 .

One can easily check the relations Mj2 = −E4 ,

Mj Mk = −Mk Mj ,

1 ≤ j = k ≤ 5,

where E4 is the identity 4 × 4 matrix. Now put . √ 1 - √ Tk := √ − k − 1Mk−1 + k + 1Mk , 2k

k = 1, . . . , 5.

(9)

(10)

From (9) we have Tk2 = −E4 , Tj Tk = −Tk Tj

(Tk Tk+1 )3 = −E4 , (11) for k > j + 1.

˜ 6 that is the Generators Tj and relations (11) determine an abstract group S central extension of S6 by μ2 . Here Tj corresponds to the transposition inter˜ 6 → GL(4, C) changing j and j + 1. In fact, the constructed representation S is obtained from the standard action of S6 on the Cliﬀord algebra A(C5 )

Fields of Invariants of Finite Linear Groups

269

(cf., e.g., [Che54]). Taking compositions with embeddings into S6 we get also projective representations S5 "→ P GL(4, C) and A6 "→ P GL(4, C). ˜ 7 , similarly put For A M1 := C ⊗2 ⊗ A, M3 := C ⊗ A ⊗ E, M5 := A ⊗ E ⊗2 , M2 := C ⊗2 ⊗ B, M4 := C ⊗ B ⊗ E, M6 := B ⊗ E ⊗2 , M7 :=

√ −1C ⊗3

and deﬁne Tk by formula (10). As above, we get an irreducible linear rep˜ 7 splits as a direct sum resentation S˜7 → GL(8, C). The restriction to A of two four-dimensional faithful representations. The explicit generators of A7 ⊂ P GL(4, C) can be taken as follows: ⎛ ⎛ 2 ⎞ ⎞ 10 0 0 p 1 1 1 ⎜0 β 0 0 ⎟ ⎟ 1 ⎜ ⎜ 1 −p −q −p⎟ ⎟ S=⎜ ⎝0 0 β 4 0 ⎠ W = √−7 ⎝ 1 −p −p −q ⎠ 1 −q −p −p 0 0 0 β2 where β is a primitive 7-th root of unity, p := β + β 2 + β 4 , and q := β 3 + β 5 + β 6 , see [Bli17]. Then the isomorphism between the subgroup in P GL(4, C) generated by S, W , and A7 is given by S → (1, 2, 3, 4, 5, 6, 7),

W → (2, 3, 5)(4, 6, 7).

ˆ 6 by μ which Remark 7.2. The group A6 has also a central extension A 3 ˆ ˆ 6 ) is admits a 3-dimensional representation δ : A6 → SL(3, C). The group δ(A projectively equivalent to the complex reﬂection group of order 360 · 6 [ST54], ˜ 6. [Coh76]. Possibly this can be used to prove the stable rationality of P3 /A Type (IX), G SL(2, F7 ) The representation G "→ SL(4, C) can be described as follows. Fix a character χ : F∗7 → C∗ and consider the following C-vector space of C-valued functions on F27 \ {0}: / Vχ := f : F27 \ {0} → C | f (λx) = χ(λ)f (x), ∀λ ∈ F∗7 . It is easy to see that dimC Vχ = 8 and SL(2, F7 ) naturally acts on Vχ . Thus we have a representation ρχ : SL(2, F7 ) → SL(Vχ ) = SL(8, C). If χ2 = 1, χ = 1, then ρχ splits as a direct sum of two four-dimensional faithful representations. This gives as a subgroup in SL(4, C) isomorphic to SL(2, F7 ). Explicit matrices can be found, e.g., in [Bli17]. The ring of invariants is completely described, see [MS73]. Note that SL(2, F7) has a three-dimensional nonfaithful representation δ : SL(2, F7 ) → SL(3, C). The group G := G × {±E} is generated by complex reﬂections and the degrees are 4, 6, and 14 (see [Web96, §139], [Coh76]).

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Type (XI) The rationality of P3 /G is unknown only for two groups of order 64 · 360 and 64 · 60, see Corollaries 3.5 and 3.6. Using Theorem 3.4 one can easily get equations for birational models of these quotients in terms of discriminants. p-Groups It is known that the answer to Noether’s problem is negative in higher dimensions: there are examples of p-groups such that k({xg }g∈G )G is not rational (and even not stably rational) [Sal84], [Sha90]. Moreover, for any prime p there is a group G of order p6 such that k({xg }g∈G )G is not stably rational [Bog88]. On the other hand, it is known that the ﬁeld of invariant of any linear action of a p-group of order ≤ p4 on k(x1 , . . . , xN ) is rational whenever k contains a primitive pe -th root of unity, where pe is the exponent of G [CK01], see also [Ben03]. The following question is open: Is it true that for any group G of order p5 and any linear action of G on Cn the quotient Cn /G is stably rational? It seems that the answer is positive (see [CHcKP08] for p = 2). Note that in characteristic p > 0 any linear action of p-group is rational [Kun55].

References [AM72]

M. Artin and D. Mumford, Some elementary examples of unirational varieties which are not rational. Proc. London Math. Soc. (3), 25:75–95, 1972. [BCTSSD85] A. Beauville, J.-L. Colliot-Thélène, J.-J. Sansuc, and P. Swinnerton-Dyer, Variétés stablement rationnelles non rationnelles. Ann. Math. (2), 121(2):283–318, 1985. [Ben03] E. Beneish, Stable rationality of certain invariant ﬁelds. J. Algebra, 269(2):373–380, 2003. [Bli17] H. F. Blichfeldt, Finite collineation groups. University of Chicago Press, Chicago, 1917. [Bog88] F. A. Bogomolov, The Brauer group of quotient spaces of linear representations. Math. USSR-Izv., 30(3):455–485, 1988. [Bur55] W. Burnside, Theory of groups of ﬁnite order. Dover Publications, New York, 1955. 2d ed. [Che54] C. C. Chevalley, The algebraic theory of spinors. Columbia University Press, New York, 1954. [Che55] C. Chevalley, Invariants of ﬁnite groups generated by reﬂections. Amer. J. Math., 77:778–782, 1955. [CHcKP08] H. Chu, Sh.-J. Hu, M.-c. Kang, and Yu. Prokhorov, Noether’s problem for groups of order 32. J. Algebra, 320(7):3022–3035, 2008. [CK01] H. Chu and M.-c. Kang, Rationality of p-group actions. J. Algebra, 237(2):673–690, 2001.

Fields of Invariants of Finite Linear Groups [Coh76] [CTS05]

[DO88] [Dol87]

[EM73] [Fei71]

[Fin87] [Fis15]

[FN89] [For84] [Hae71]

[Haj83] [Haj00]

[HK94] [IP99]

[Isk87] [Jor70] [KM99] [KMP87]

271

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I. Ya. Kolpakov-Miroshnichenko and Yu. G. Prokhorov, Rationality construction of ﬁelds of invariants of some ﬁnite fourdimensional linear groups associated with Fano threefolds. Math. Notes, 51:74–76, 1992. I. Ya. Kolpakov-Miroshnichenko and Yu. G. Prokhorov, Rationality of ﬁelds of invariants of some four-dimensional linear groups, and an equivariant construction related to the Segre cubic. Math. USSR Sb., 74(1):169–183, 1993. K. Koike, Remarks on the Segre cubic. Arch. Math. (Basel), 81(2):155–160, 2003. H. Kuniyoshi, On a problem of Chevalley. Nagoya Math. J., 8:65–67, 1955. M. Kervaire and Th. Vust, Fractions rationnelles invariantes par un groupe ﬁni: quelques exemples. In Algebraische Transformationsgruppen und Invariantentheorie, volume 13 of DMV Sem., pages 157–179. Birkhäuser, Basel, 1989. H. W. Lenstra, Jr., Rational functions invariant under a ﬁnite abelian group. Invent. Math., 25:299–325, 1974. T. Maeda, Noether’s problem for A5 . J. Algebra, 125(2):418–430, 1989. Yu. I. Manin, Cubic forms, Algebra, geometry, arithmetic, volume 4 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, second edition, 1986. Translated from the Russian by M. Hazewinkel. D. W. Miller, On a theorem of Hölder. Amer. Math. Monthly, 65:252–254, 1958. T. Miyata, Invariants of certain groups. I. Nagoya Math. J., 41:69–73, 1971. S. Mori, Threefolds whose canonical bundles are not numerically eﬀective. Ann. Math., 115:133–176, 1982. C. L. Mallows and N. J. A. Sloane, On the invariants of a linear group of order 336. Proc. Camb. Phil. Soc., 74:435–440, 1973. Sh. Mukai and H. Umemura, Minimal rational threefolds. In Algebraic geometry (Tokyo/Kyoto, 1982), volume 1016 of Lecture Notes in Math., pages 490–518. Springer, Berlin, 1983. I. Nakamura, Moishezon threefolds homeomorphic to P3 . J. Math. Soc. Japan, 39(3):521–535, 1987. T. Nakano, On equivariant completions of 3-dimensional homogeneous spaces of SL(2, C). Japan. J. Math. (N.S.), 15(2):221–273, 1989. E. Noether, Rationale Functionenkörper. Jahresber. Deutsch. Math. Verein., 22:316–319, 1913. D. J. Saltman, Noether’s problem over an algebraically closed ﬁeld. Invent. Math., 77(1):71–84, 1984. V. G. Sarkisov, On the structure of conic bundles. Math. USSR Izv., 120:355–390, 1982. I. Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen. J. Reine Angew. Math., 139:155–250, 1911.

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The Rationality Problem and Birational Rigidity Aleksandr V. Pukhlikov Department of Mathematical Sciences M& O Building, Peach Street The University of Liverpool Liverpool L69 7ZL, UK [email protected], [email protected] Summary. In this survey paper birational geometry of higher-dimensional rationally connected varieties is discussed. In higher dimensions the classical rationality problem generalizes to the problem of description of the structures of a rationally connected ﬁber space on a given variety. We discuss the key concept of birational rigidity and present examples of Fano ﬁber spaces with ﬁnitely many rationally connected structures.

Key words: Rationality problems, birational rigidity 2000 Mathematics Subject Classiﬁcation codes: 14E08, 14E05

Introduction 0.1. The Lüroth problem. The modern age in birational geometry started with the negative solution of the Lüroth problem: does unirationality imply rationality? In [3,12] negative answers were given for dimension three, in [2] for arbitrary dimension ≥ 3. The unirationality of the produced examples was proved by direct (sometimes almost obvious) constructions and the hardest part was to prove their nonrationality. The paper of Iskovskikh and Manin on the three-dimensional quartics [12] started a whole new ﬁeld of research in the framework of which new methods of proving nonrationality were developed, the methods that work eﬀectively for a large class of higher-dimensional algebraic varieties. The aim of this survey is to describe and explain by examples some of the main ideas in this ﬁeld. In [12] the following fact was shown.

F. Bogomolov, Y. Tschinkel (eds.), Cohomological and Geometric Approaches to Rationality Problems, Progress in Mathematics 282, DOI 10.1007/978-0-8176-4934-0_11, © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

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Theorem 0.1. Let χ : V V be a birational map between smooth threedimensional quartics V, V ⊂ P4 . Then χ is a biregular (projective) isomorphism. In particular, the group of birational self-maps Bir V = Aut V is ﬁnite (for a generic quartic V it is trivial). Corollary 0.1. The smooth three-dimensional quartic V ⊂ P4 is nonrational. Proof of the corollary. The group of birational self-maps of an algebraic variety X is a birational invariant. However, by Theorem 0.1 the group Bir V is ﬁnite, whereas the Cremona group Bir P3 is inﬁnite. Therefore, V cannot be birational to P3 , which is what we need. Q.E.D. Remark 0.1. The argument above is obvious. For a long time (for more than 20 years after the paper [12] was published) quite a few people believed that this was the only way to deduce nonrationality of the three-dimensional quartic. However, with all simplicity and brevity of this argument, there is a disadvantage, namely, if the group Bir X is “of the same size” as the Cremona group Bir P3 in the sense of cardinality, one cannot prove nonrationality of the variety X in this way. In particular, this method does not work for the complete intersection V2·3 ⊂ P5 of a quadric and a cubic (a description of the group Bir V2·3 is given below following [13]). It is almost certain that the groups Bir V2·3 and Bir P3 are nonisomorphic, but today we cannot even approach this problem. However, there are two (very close to each other) ways to derive Corollary 0.1 from the constructions of the paper [12], although not directly from Theorem 0.1. Their advantage is in the fact that they work for other Fano varieties, in particular, for V2·3 . Let us describe these arguments. A second proof of Corollary 0.1. In [12] the following fact was actually shown. Proposition 0.1. Let χ : V X be a birational map of a smooth threedimensional quartic V onto a smooth projective variety X, |R| a movable complete linear system on X, Σ ⊂ |nH| = | − nKV | its strict transform on V with respect to χ, where H ∈ Pic V is the class of a hyperplane section of V ⊂ P4 . Then, if for some positive integers a, b ∈ Z+ the linear system |aR + bKX | is empty, then the linear system |anH + bKV | is empty, either, that is, b > an. Corollary 0.2. Let α : X → S be a morphism. Assume that one of the following two cases holds: • •

S = P1 , the general ﬁber α−1 (s), s ∈ S, is a rational surface, dim S = 2, and the general ﬁber α−1 (s), s ∈ S, is an irreducible rational curve.

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Then there is no birational map χ : V X, where V ⊂ P4 is a smooth quartic. Since a linear projection P3 P2 or P3 P1 realizes P3 as a P1 or P2 -bundle, respectively, Corollary 0.2 implies nonrationality of the threedimensional quartic. Proof of Corollary 0.2. Assume the converse: there is a birational map χ : V X. Let Λ be a complete very ample linear system on S. Let |R| = α∗ Λ be its pullback on X. Obviously, the class R is trivial on the ﬁbers of α, so that for any a, b > 0 we get |aR + bKX | = ∅, since the ﬁber α−1 (s) has the negative Kodaira dimension. Let Σ ⊂ |nH| be the strict transform of the system |R| on V with respect to χ. By Proposition 0.1, we get b > an. Since a, b are arbitrary, we get n = 0. But Σ is a movable linear system, so that n ≥ 1. A contradiction. Q.E.D. for Corollary 0.2. Remark 0.2. We have just obtained a much stronger fact than nonrationality of V . Corollary 0.2 asserts that there is no rational map γ : V S onto a variety S of positive dimension, the generic ﬁber of which is a rational surface or a rational curve. In the modern terminology, on V there are no structures of a ﬁber space into rational curves or rational surfaces. Since on P3 there are inﬁnitely many such structures, the quartic V is nonrational. Although the argument above is much less obvious than the ﬁrst proof of Corollary 0.1, its potential is much greater: it shows in which direction one should generalize the rationality problem and what class of algebraic varieties should be involved in consideration. These generalizations will be considered below. Completing our discussion of the three-dimensional quartic, let us give A third proof of Corollary 0.1. The argument given below is also based on Proposition 0.1, but it is more direct than the previous one. Assume that χ : V P3 is a birational map and |R| is the complete linear system of planes in P3 . The linear system |aR + bKP3 | = |(a − 4b)R| is empty if and only if a < 4b. Let Σ ⊂ |nH| be the strict transform of the system |R| on V . By Proposition 0.1, for any positive integers a, b, satisfying the inequality a < 4b, we get b > an. Thus n ≤ 14 , that is, n = 0, which is impossible. A contradiction. Q.E.D. for nonrationality of the three-dimensional quartic. Keeping in mind the three proofs of nonrationality of the three-dimensional quartic, we will show in this paper what class of varieties it is natural to consider in general, what questions it is natural to ask, and what answers it is natural to expect. 0.2. Rationally connected varieties. Recall [14,15] that an algebraic variety X is said to be rationally connected, if any two (generic) points x, y ∈ X

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can be joined by an irreducible rational curve, that is, there exists a morphism f : P1 → X such that x, y ∈ f (P1 ). The projective space PM and smooth Fano varieties are rationally connected. In [5] the following fundamental fact was proved. Theorem 0.2. Let π : X → S be a ﬁber space (that is, a surjective morphism of projective varieties with connected ﬁbers), the base S and generic ﬁber π −1 (s), s ∈ S, of which are rationally connected. Then the variety X itself is rationally connected. The ﬁber spaces π : X → S described in the theorem above are called rationally connected ﬁber spaces. From the viewpoint of classiﬁcation of algebraic varieties, rationally connected varieties are the most natural generalization of rational varieties in dimension three and higher. Obviously, the rationality problem makes sense for rationally connected varieties only. Definition 0.1. A structure of a rationally connected ﬁber space on a rationally connected variety X is an arbitrary rational dominant map ϕ : X S, the ﬁber of general position of which ϕ−1 (s), s ∈ S, is irreducible and rationally connected. If the base S is a point, then the structure is said to be trivial. An alternative deﬁnition: a structure of a rationally connected ﬁber space on a variety X is a birational map χ : X X onto a variety X equipped with a surjective morphism π : X → S realizing X as a rationally connected ﬁber space. We identify the structures of a rationally connected ﬁber space ϕ1 : X S1 and ϕ2 : X S2 , if there exists a birational map α : S1 S2 such that the following diagram commutes: id

X ↔ X ϕ1 ↓ ↓ ϕ2 α S1 S2 ,

(1)

that is, ϕ2 = α ◦ ϕ1 . In other words, ϕ1 and ϕ2 have the same ﬁbers. The set of nontrivial structures of a rationally connected ﬁber space on the variety X (modulo the identiﬁcation above) is denoted by RC(X). On the set RC(X) there is a natural relation of partial order: for ϕ1 , ϕ2 ∈ RC(X) we have ϕ1 ≤ ϕ2 , if there is a rational dominant map α : S1 S2 such that the diagram (1) commutes. In other words, the ﬁbers of ϕ1 are contained in the ﬁbers of ϕ2 . For a general point s ∈ S2 we have α−1 (s) = ϕ1 (ϕ−1 2 (s)), therefore α ∈ RC(S1 ) is a structure of a rationally connected ﬁber space on S1 . It is easy to see that the correspondence ϕ2 → α determines a bijection of the sets {ψ ∈ RC(X)|ψ ≥ ϕ1 } and RC(S1 ). Therefore from the geometric viewpoint of primary interest are the minimal elements of the ordered set RC(X). Denote the set of minimal elements by RCmin (X). Set also RCd (X) ⊂

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RC(X) to be the set of structures, the generic ﬁber of which is of dimension d. Obviously, if d = min{e ∈ Z+ |RCe = ∅}, then RCd ⊂ RCmin . For each d ∈ {1, . . . , dim X − 1} on the set RCd (X) there is a natural relation of ﬁber-wise birational equivalence: ϕ1 ∼ ϕ2 if there exists a birational transformation χ ∈ Bir X and a birational map α : S1 S2 such that the diagram χ X X ϕ1 ↓ ↓ ϕ2 α S1 S2 commutes, that is, ϕ2 ◦ χ = α ◦ ϕ1 . In other words, the birational self-map χ transforms the ﬁbers of ϕ1 into the ﬁbers of ϕ2 . The quotient set RCd (X)/ ∼ we denote by the symbol RC d (X). For instance, any two linear projections ϕ1 , ϕ2 : PM PM−d are ﬁberwise birationally equivalent and realize the same element in RC d (PM ). On the other hand, let V ⊂ PM be a smooth Fano hypersurface of index two, that is, a hypersurface of degree M − 1. Proposition 0.2. Any two distinct generic linear projections ϕ1 , ϕ2 : PM P1 determine the structures of a rationally connected ﬁber space on V , ϕi | V : V P1 , which are not ﬁber-wise birationally equivalent. For the proof, see Section 3. The ﬁbers of the structures ϕi | V are Fano hypersurfaces of index 1, that is, hypersurfaces of degree M − 1 in PM−1 . Since for a general hypersurface V , a general projection ϕ : PM P1 , and a general point p ∈ P1 for M ≥ 5 we have RC(ϕ| −1 V (p)) = ∅ (see [18] and Section 1 of the present paper), the structures ϕ| V are minimal elements of the set RC(V ). Conjecture 0.1. For e ≤ M −2 and a general hypersurface V ⊂ PM of degree M − 1 we have RCe (V ) = ∅. For a general four-dimensional quartic V = V4 ⊂ P5 Conjecture 0.1 asserts that V has no structures of a rationally connected ﬁber space with the base of dimension two or three. The assumption of genericity is essential: if V ⊃ P , where P ⊂ P5 is a two-dimensional plane, then the projection from that plane πP : P5 P2 ﬁbers V into cubic surfaces, that is, πP | V ∈ RC2 (V ). Proposition 0.2 shows that the set RC d (X) can be quite big and possess a natural structure of an algebraic variety. The second proof of Corollary 0.1 now can be formulated in the following way: Proposition 0.1 implies that for a smooth three-dimensional quartic V ⊂ P4 we have RC(V ) = ∅. Since RC(P3 ) = ∅, the quartic V is nonrational. The arguments of Section 0.1 show that the rationality problem generalizes to the following questions concerning birational geometry of a rationally connected variety X: • •

compute the sets RC(X), RCmin (X), RCd (X), and RC d (X), compute the group of birational self-maps Bir X.

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We single out computing the group of birational self-maps as a separate problem, since it is of independent interest. In fact, it is necessary to compute this group to describe the quotient set RC d (X); moreover, one should know the action of the group Bir X on the set RCd (X). Besides, the interest in the problem of computing the group Bir X (like the special interest in the rationality problem) comes from tradition. 0.3. The structure of the paper. The aim of this paper is to explain the main ideas connected with the problems that were set up above, for certain natural classes of rationally connected varieties. Section 1 is devoted to discussing the key concept of birational rigidity. We give the necessary deﬁnitions and describe the main steps in proving birational rigidity (that is, excluding and “untwisting” maximal singularities). As an example of description of a group of birational self-maps we give (following [13]) a proof of the theorem on generators and relations in the group Bir V2·3 for the three-dimensional complete intersection of a quadric and a cubic in P5 . Here we follow [13], giving all details of the proof, since the paper [13] is not easily accessible. This group by its “size” is comparable with the Cremona group Bir P3 , so that the cardinality argument is insuﬃcient to prove nonrationality of the variety V2·3 (which at the same time automatically follows from birational rigidity: RC(V2·3 ) = ∅). Description of the group Bir V2·3 presents an exceptionally visual example of “untwisting” maximal singularities. In Section 2 we consider examples of rationally connected varieties, the set of rationally connected structures of which is nonempty but ﬁnite: the direct products of divisorially canonical Fano varieties (Section 2.1), Fano ﬁber spaces V /P1 with a nontrivial group of birational self-maps Z/2Z×Z/2Z, permuting the two elements in RC(V ), so that &RC(V ) = 1 (Section 2.2) and Fano ﬁber spaces V /P1 with no nontrivial birational self-maps, Bir V = Aut V and &RC(V ) = &RC(V ) = 2 (Section 2.3). The varieties, considered in Sections 2.2 and 2.3, present examples of ﬂops in higher dimensions. These are the ﬁrst examples of nontrivial untwisting of maximal singularities in dimensions higher than three; the varieties of the type of Section 2.3 are the ﬁrst examples of nontrivial links in higher dimensions (in the terminology of Sarkisov program [4,33]). In Section 3, following [21], we prove Proposition 0.2. Computation of the group of birational self-maps of a rationally connected variety V , which is the total space of a rationally connected ﬁber space π : V → S, dim S ≥ 1, naturally breaks into two separate problems: that of comparison of the group Bir V with the group of ﬁber-wise (with respect to π) birational self-maps Bir(V /S) and that of computation of the group Bir(V /S). In Section 3 we consider the problem of computing the group Bir(V /S), where C is a curve, for an essentially bigger class of ﬁber spaces. Proposition 0.2 follows from the main theorem of Section 3 in a straightforward way. A birational correspondence

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between two rationally connected structures described in Proposition 0.2 turns out to be a biregular map, for a generic V it is identical. If a rationally connected ﬁber space V /S determines a unique nontrivial rationally connected structure on V , then the exact sequence 1 → Bir(V /S) → Bir V → Bir S reduces computation of the group of birational self-maps to computation of the group of the proper birational self-maps, preserving the ﬁbers of π: χ

V V π ↓ ↓ π S ←→ S, or, equivalently, the group Bir Fη of birational self-maps of the ﬁber Fη over the generic (nonclosed) point of the base S. We can also look at χ as a continuous family of birational self-maps of ﬁbers S * s → χs ∈ Bir Fs . If V /S is a Fano ﬁber space, the general ﬁber of which is birationally superrigid, then the results of Section 3 make it possible to give a complete description of the group of birational self-maps of the variety V , as it is done below in Sections 2.2 and 2.3 for Fano ﬁber spaces over P1 .

1 Birational rigidity 1.1. Termination of canonical adjunction. A rationally connected variety X satisﬁes the classical condition of termination of canonical adjunction: for any eﬀective divisor D the linear system |D + nKX | is empty for n 0, since KX is negative on some family of rational curves sweeping out X. The classical proof of the Castelnuovo rationality criterion [1] makes use of this condition, ﬁxing the precise step n∗ of canonical adjunction when |D + n∗ KX | is still nonempty, but |D +(n∗ +1)KX | = ∅: it turns out that the linear system |D + n∗ KX | has very useful properties. To formalize this idea, for a smooth rationally connected variety X consider the Chow group Ai X of algebraic cycles of codimension i modulo rational equivalence, A1 X ∼ = Pic X, and set AiR X = Ai X ⊗ R. Let Ai+ X ⊂ AiR X be the closed cone generated by eﬀective classes, that is, the cone of pseudoeﬀective classes. Set also A1mov X ⊂ A1R X to be the closed cone generated by the classes of movable divisors (that is, divisors in movable linear systems). Definition 1.1. The threshold of canonical adjunction of a divisor D on the variety X is the number c(D, X) = sup{ε ∈ Q+ |D + εKX ∈ A1+ X}. If Σ is a

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nonempty linear system on X, then we set c(Σ, X) = c(D, X), where D ∈ Σ is an arbitrary divisor. Example 1.1. (i) Let X be a primitive Fano variety, that is, a smooth projective variety with the ample anticanonical class and Pic X = ZKX . For any eﬀective divisor D we have D ∈ |− nKX | for some n ≥ 1, so that c(D, X) = n. If we replace the condition Pic X = ZKX by the weaker one rk PicX = 1, that is, KX = −rH, where Pic X = ZH, r ≥ 2 is the index of the variety X, then for D ∈ |nH| we get c(D, X) = nr . (ii) Let π : V → S be a rationally connected ﬁber space with dim V > dim S ≥ 1, Δ an eﬀective divisor on the base S. Obviously, c(π ∗ Δ, V ) = 0. If Pic V = ZKV ⊕ π ∗ Pic S, that is, V /S is a primitive Fano ﬁber space, and D is an eﬀective divisor on V , which is not a pullback of a divisor on the base S, then D ∈ | − nKV + π ∗ R| for some divisor R on S, where n ≥ 1. Obviously, c(D, V ) ≤ n, and moreover, if the divisor R is eﬀective, then c(D, V ) = n. (iii) Let F1 , . . . , FK be primitive Fano varieties, V = F1 × · · · × FK their direct product. Let Hi = −KFi be the positive generator of the group Pic Fi . Set Fi , Si = j=i

so that V ∼ = Fi × Si . Let ρi : V → Fi and πi : V → Si be the projections onto the factors. Abusing our notations, we write Hi instead of ρ∗i Hi , so that Pic V =

K

ZHi

i=1

and KV = −H1 − · · · − HK . For any eﬀective divisor D on V we get D ∈ |n1 H1 + · · · + nK HK | for some nonnegative n1 , . . . , nK ∈ Z+ , and obviously c(D, V ) = min{n1 , . . . , nK }. This example can be reduced to the previous one: assume that c(D, V ) = n1 and set n = n1 , π = π1 , F = F1 , S = S1 . We get Σ ⊂ | − nKV + π ∗ Y |, where Y =

K i=2

(ni − n)Hi is an eﬀective class on the base S of the ﬁber space

π : V → S. This is the case of Example 1.1 (ii) above. The threshold of canonical adjunction is easy to compute, but the main disadvantage of this concept is that it is not a birational invariant.

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Example 1.2. Let π : PM Pm be a linear projection from an (M − m − 1)-dimensional plane P ⊂ PM . Consider a movable linear system Λ of hypersurfaces of degree n in Pm and let Σ be its pullback via π. Obviously, n c(Σ, PM ) = M+1 . However, let us blow up the plane P , say σ : P+ → PM , so that the composite map π ◦ σ : P+ → Pm is a PM−m -bundle. Let Σ + be the strict transform of Σ on P+ . Since π ◦ σ is a morphism with rationally connected ﬁbers, we get c(Σ + , P+ ) = 0. This example can be easily generalized to linear projections of Fano complete intersections V ⊂ PM of index 2 or higher, similar to the case considered in Proposition 0.2. 1.2. Birationally rigid varieties. In order to overcome birational noninvariance of the threshold of canonical adjunction, we give Definition 1.2. For a movable linear system Σ on the variety X deﬁne the virtual threshold of canonical adjunction by the formula cvirt (Σ) = inf {c(Σ , X )}, X →X

where the inﬁmum is taken over all birational morphisms X → X, X is a smooth projective model of C(X), Σ the strict transform of the system Σ on X . The virtual threshold is obviously a birational invariant of the pair (X, Σ): if χ : X X + is a birational map, Σ + = χ∗ Σ is the strict transform of the system Σ with respect to χ−1 , we get cvirt (Σ) = cvirt (Σ + ). Proposition 1.1. (i) Assume that on the variety V there are no movable linear systems with the zero virtual threshold of canonical adjunction. Then on V there are no structures of a nontrivial ﬁbration into varieties of negative Kodaira dimension, that is, there is no rational dominant map ρ : V S, dim S ≥ 1, the generic ﬁber of which has negative Kodaira dimension. (ii) Let π : V → S be a rationally connected ﬁber space. Assume that every movable linear system Σ on V with the zero virtual threshold of canonical adjunction, cvirt (Σ) = 0, is the pullback of a system on the base: Σ = π ∗ Λ, where Λ is some movable linear system on S. Then any birational map χ

V V π ↓ ↓ π S S,

(2)

where π : V → S is a ﬁbration into varieties of negative Kodaira dimension, is ﬁber-wise, that is, there exists a rational dominant map ρ : S S , making the diagram (2) commutative, π ◦ χ = ρ ◦ π. In other words, π ≥ π in the sense of the order on the set of rationally connected structures: π is the least element of RC(V ).

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Thus for certain rationally connected varieties the virtual threshold of canonical adjunction reduces the problem of describing the set RC(V ) to the same problem for the base S. This is a crucial step that in many cases leads to an exhaustive description of the set RC(V ). But the main disadvantage of the virtual thresholds is that they are extremely hard to compute. To be precise, the only known way to compute them is by reduction to the ordinary thresholds. Definition 1.3. (i) The variety V is said to be birationally superrigid, if for any movable linear system Σ on V the equality cvirt (Σ) = c(Σ, V ) holds. (ii) The variety V (respectively, the Fano ﬁber space V /S) is said to be birationally rigid, if for any movable linear system Σ on V there exists a birational self-map χ ∈ Bir V (respectively, a ﬁber-wise birational self-map χ ∈ Bir(V /S)), providing the equality cvirt (Σ) = c(χ∗ Σ, V ). In the following examples the main classes of Fano varieties and Fano ﬁber spaces, for which birational rigidity or superrigidity is known today, are listed. Example 1.3. Smooth three-dimensional quartics V = V4 ⊂ P4 are birationally superrigid: this follows immediately from the arguments of [12]. Generic smooth complete intersections V2·3 ⊂ P5 of a cubic and a quadric hypersurface are birationally rigid, but not superrigid. For description of their groups of birational self-maps (which also demonstrates how the thresholds of canonical adjunction are decreased by means of birational automorphisms), see Section 1.3 below. Example 1.4. Generic hypersurfaces of index one VM ⊂ PM are birationally superrigid [18]. The same is true for generic complete intersections V ⊂ PM+k of index one and codimension k, provided that M ≥ 2k + 1 [22]. Example 1.5. Let σ : V → Q ⊂ PM+1 be a double cover, where Q = Qm ⊂ PM+1 is a smooth hypersurface of degree m, and the branch divisor W ⊂ Q ∗ is cut out on Q by a hypersurface W2l ⊂ PM+1 , where m + l = M + 1. The Fano variety V is birationally superrigid for general Q, W ∗ [19]. Instead of a double cover an arbitrary cyclic cover could be considered, instead of a hypersurface Q ⊂ PM+1 a smooth complete intersection Q ⊂ PM+k of appropriate index and codimension k < 12 M . A general variety in each of

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these classes is birationally superrigid [23,27]. Another example is given by iterated double covers [24]. All varieties mentioned in Examples 1.4 and 1.5 can be realized as Fano complete intersections in weighted projective spaces. Conjecture 1.1. A smooth Fano complete intersection of index one and dimension ≥ 4 in a weighted projective space is birationally rigid, of dimension ≥ 5 birationally superrigid. Now let us consider the known examples of ﬁber spaces. Example 1.6. (V. G. Sarkisov [31,32]) Let π : V → S be a conic bundle with a suﬃciently positive discriminant divisor D, satisfying the Sarkisov condition |4KS + D| = ∅. Then &RC1 (V ) = 1, that is, there is exactly one structure of a conic bundle on V , namely, the projection π. Example 1.7. Let F be any of the classes of Fano varieties listed in Examples 1.3–1.5. Let π : V → P1 be a smooth Fano ﬁber space, such that every ﬁber Ft = π −1 (t), t ∈ P1 , is in F . Assume furthermore that the strong K 2 condition is satisﬁed: KV2 ∈ Int A2+ V . In a certain natural sense almost all ﬁber spaces V /P1 satisfy the strong K 2 -condition, which can be considered as a characteristic of “twistedness” over the base. In these assumptions, a general ﬁber space V /P1 is birationally superrigid [17,20,25,29]. Example 1.8. Three-dimensional del Pezzo ﬁbrations, satisfying strong K 2 condition, are birationally rigid [17]. In fact, the strong K 2 -condition can be considerably relaxed [7,8,35]. 1.3. The method of maximal singularities. In order to prove birational (super)rigidity of a smooth projective rationally connected variety V , ﬁx a movable linear system Σ on V and set n = c(Σ) ∈ Z+ . Assume that the inequality cvirt (Σ) < n holds (otherwise no work is required). In particular, n > 0. By deﬁnition, there exists a birational morphism σ : V# → V of smooth varieties such that # V# ) < n, c(Σ, # is the strict transform of Σ on V# . where Σ Definition 1.4. An exceptional divisor E ⊂ V# is called a maximal singularity of the system Σ, if the Noether-Fano inequality νE (ϕ∗ Σ) > na(E)

(3)

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holds, where νE (·) is the multiplicity of the pullback of Σ on V# along E and a(E) is the discrepancy of E. Proposition 1.2. In the assumptions above, a maximal singularity of Σ does exist. For a (very simple) proof, see [12,18,20]. It turns out that maximal singularities of movable linear systems are a very special phenomenon. For many classes of Fano varieties and Fano ﬁber spaces a movable linear system cannot have a maximal singularity which in view of Proposition 1.2 implies superrigidity. In this section we present one of the most sophisticated examples of a birationally rigid, but not superrigid, Fano three-fold, known today, namely, the complete intersection of a quadric and a cubic in P5 . The proof was started in [11] and completed in [16]. A detailed exposition can be found in [13]. Here we concentrate on the “untwisting” procedure. Let us ﬁx notations. We study the complete intersection V = Q ∩ F ⊂ P5 , where Q is a quadric and F is a cubic hypersurface. The variety V is assumed to be smooth and, moreover, generic in the sense described below, in particular, Pic V = ZH, where H = −KV is the class of a hyperplane section of V ⊂ P5 . 1.3.1. Lines on the complete intersection V . Let L ⊂ V be a line in P5 . Proposition 1.3. For the normal sheaf NL/V there are two possible cases: • •

∼ OL (−1)⊕OL ; in this case the line L is said to be of general either NL/V = type, or NL/V ∼ = OL (−2) ⊕ OL (1); in this case the line L is said to be of nongeneral type.

Moreover, the line L is of nongeneral type if and only if any of the following two equivalent conditions holds: • •

there exists a plane P ⊂ P5 such that L ⊂ P and the scheme-theoretic intersection V ∩ P is not reduced everywhere along L, let σ : V# → V be the blow up of L, E = σ −1 (L) the exceptional divisor. Then restricting to E the strict transform on V# of a generic hyperplane section containing L, we get a nonample divisor on E.

The proof is straightforward and left to the reader. We will consider the general complete intersections V = Q ∩ F , satisfying the following conditions:

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V does not contain lines of nongeneral type (it is easy to check by the usual dimension count that this condition is justiﬁed, that is, a general complete intersection satisﬁes it), there are no three lines on V lying in one plane and having a common point, the quadric Q is nondegenerate.

Let L ⊂ V be a line. The projection P5 P3 from L deﬁnes a rational map πL : V P3 of degree two. Set αL ∈ Bir V to be the corresponding Galois involution. More formally, let σ : V# → V be the blow up of L, E = σ −1 (L) ⊂ V# the exceptional divisor. The map πL extends to a morphism p = πL ◦ σ : V# → P3 . Lemma 1.1. The morphism p is a ﬁnite morphism of degree 2 outside a closed subset W ⊂ V# of codimension two, and p (W ) ⊂ P3 is a ﬁnite set of points. The involution αL extends to a biregular involution of V# \ W . Its action on Pic V# = ZH ⊕ ZE is given by the formulas α∗L (H) = 4H − 5E,

α∗L (E) = 3H − 4E.

Proof. The projection p : V# → P3 is a ﬁnite morphism outside the set W ⊂ V# that consists of curves that are contracted by the morphism p. We will show there are ﬁnitely many of them. Set H = nH − νE and E = mH − μE to be the classes in Pic V# of the strict transform of a general hyperplane section and the divisor E with respect to αL . The linear system |H − E| is clearly invariant under αL . Take a general surface S ∈ |H − E|. Since KS = 0, the birational involution αL | S extends to a biregular involution of this surface. Denote it by αS , and the restrictions of H and E to S by HS and ES , respectively. We get α∗S HS = nHS − νES ,

α∗S ES = mHS − μES

and the class HS − ES is α∗S -invariant, whence we get n = m + 1, ν = μ + 1. Since αS is an automorphism, (α∗S HS · (HS − ES )) = (HS · (HS − ES )) = 5 and (α∗S HS )2 = (HS )2 = 6, whence by the obvious equalities (HS · ES ) = 1, (ES2 ) = −2 we get the following two possibilities for n, m, ν, μ: • •

either H = 4H − 5E, E = 3H − 4E, or H = H, E = E,

the latter being clearly impossible because αL cannot be extended to a biregular automorphism of V . By construction, the system |4H − 5E| is movable. However, if a curve C is contracted by the morphism p, then (C · (H − E)) = 0 and therefore

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(C · H ) < 0. We conclude that there can be only ﬁnitely many such curves. Q.E.D. Now let P ⊂ P5 be a 2-plane such that P ∩ V is a union of three lines, P ∩ V = L ∪ L1 ∪ L2 . This is possible only if P ⊂ Q. Let σ : V# → V be the composition of three blowups: ﬁrst, we blow up L, then the strict transform of L1 , then the strict transform of L2 . We denote the exceptional divisors on V# , corresponding to the lines L, L1 , L2 , by the symbols E, E1 , E2 , respectively. Lemma 1.2. The involution αL extends to a biregular involution on V# \ W , where W is a closed subset of codimension two. The action of αL on Pic V# = ZH ⊕ ZE ⊕ ZE1 ⊕ ZE2 is given by the formulas: α∗L = 4H − 5E − 2E1 − 2E2 , α∗L E = 3H − 4E − 2E1 − 2E2 , α∗L Ei = Ej , where {i, j} = {1, 2}. Proof is obtained in the same way as for the previous lemma; one has to consider, along with the projection πL , the projection πP : P5 P2 from the plane P . The considerations are more subtle but essentially similar. 1.3.2. Conics on the complete intersection V . It is easy to see that there is a one-dimensional family of irreducible conics Y ⊂ V such that the plane P (Y ) =< Y > is contained entirely in the quadric Q. Obviously, P (Y ) ∩ V = Y ∪ L(Y ), where L(Y ) is the residual line. We will call the conics described above the special conics. Every special conic Y generates the following construction. Set P = P (Y ). Consider the projection πP : P5 P2 from the plane P . The ﬁbers of πP are 3-planes S ⊃ P , so that S ∩ Q = P ∪ P (S), where P (S) is the residual plane. Therefore, πP ﬁbers V over P2 into elliptic curves CS = P (S) ∩ F , that is, plane cubics. A general curve CS intersects the residual line L(Y ) an one point, which is L(Y ) ∩ P (S). We deﬁne the involution βY ∈ Bir V as a ﬁber-wise map, setting βY |CS to be the elliptic reﬂection, where the group law on CS is deﬁned by the point L(Y ) ∩ P (S) as the zero. Let σ : V# → V be the composition of the blowup of the conic Y and the blowup of the strict transform of the line L(Y ), E and E + be the corresponding exceptional divisors. Obviously, πP ◦σ : V → P2 is a morphism, the general ﬁber of which is an elliptic curve Ct , t ∈ P2 . The divisor E + is a section of this elliptic ﬁbration, (E + · Ct ) = 1.

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Lemma 1.3. The birational involution βY extends to a biregular involution on the complement V# \ W , where W is a closed subset of codimension two, and moreover, πP ◦ σ(W ) ⊂ P2 is a ﬁnite set. The action of βY on Pic V# = ZH ⊕ ZE ⊕ ZE + is given by the formulas βY∗ H = 13H − 14E − 8E + , βY∗ E = 12H − 13E − 8E + , βY∗ E + = E + .

The Proof is quite similar to the proof of Lemma 1.1. Let H , E , E ∈ Pic V# be the classes of the strict transforms of a general hyperplane section and the divisors E and E + , respectively. On the general curve Ct , t ∈ P2 , the involution βY maps a point x ∈ Ct to the point βY (x) ∈ Ct satisfying the relation βY (x) + x ∼ 2(Ct ∩ E + ) as divisors on Ct . The kernel of the restriction of Pic V# onto a general ﬁber Ct is Z(H − E − E + ) = (πP ◦ σ)∗ Pic P2 , so that H + H = 6E + + m(H − E − E + ),

E + E = 4E + + l(H − E − E + )

and E = E + + k(H − E − E + ). Now we proceed exactly as in the proof of Lemma 1.1: we restrict βY and all the classes involved onto a general surface S ∈ |H − E − E + | (that is, S is the inverse image of a general line in P2 via πP ◦σ). Since KS = 0, βY |S extends to a biregular involution of S. Comparing intersection indices, we get m = 14, l = 12. Now βY is well deﬁned on irreducible ﬁbers, and it is easy to see that any reducible ﬁber Ct contains a component which intersects H negatively. Therefore, there are only ﬁnitely many of them. Now k = 0 and the proof is complete. Q.E.D. 1.3.3. Relations between the involutions αL . Let P ⊂ P5 be a plane such that P ⊂ Q and P ∩ F = L1 ∪ L2 ∪ L3 is a union of three lines. Lemma 1.4. The following relation holds: (αL1 ◦ αL2 ◦ αL3 )2 = idV . Proof. Obviously, each of the three involutions αLi preserves the ﬁbers of the projection πP : V P2 from the plane P . Recall that a general ﬁber πP−1 (t)

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is a cubic curve Ct , where Ct ∩ P = {x1 , x2 , x3 }, xi = Ct ∩ Li . Take a point x ∈ Ct ; obviously, αLi (x) + x + xi ∼ x1 + x2 + x3 on Ct . Therefore we compute: αL3 (x) αL2 ◦ αL3 (x) αL1 ◦ αL2 ◦ αL3 (x) (αL1 ◦ αL2 ◦ αL3 )2 (x)

∼ ∼ ∼ ∼

x1 + x2 − x, x3 − x2 + x, 2x2 − x, x,

which is what we need. Q.E.D. 1.3.4. Copresentation of the group Bir V . After this preparatory work we can formulate the main theorem describing birational geometry of V . Set L and C to be the sets of lines and special conics on V , respectively. Let G+ be the free group generated by symbols AL and BY for all L ∈ L and Y ∈ C, respectively. Let R+ ⊂ G+ be the normal subgroup, generated by the words A2L for all L ∈ L, BY2 for all Y ∈ C and, ﬁnally, (AL1 AL2 AL3 )2 for all triples of distinct lines L1 , L2 , L3 ∈ L such that < L1 ∪ L2 ∪ L3 >= P2 . Set G = G+ /R+ to be the quotient group. We construct a semidirect product G Aut V using the obvious action of Aut V on G: for ρ ∈ Aut V set ρAL ρ−1 = Aρ(L) ,

ρ BY ρ−1 = Bρ(Y ) .

Let ε : Aut V → Bir V be the homomorphism, sending AL to αL , BY to βY and identical on Aut V . Theorem 1.1. V is birationally rigid and ε is an isomorphism of groups. Proof. Set B = L ∪ C. Take any movable linear system Σ ⊂ |nH| on V . Obviously, c(Σ, V ) = n. In order to prove that ε is a bijection, we take Σ to be the strict transform of the linear system |H| of hyperplane sections with respect to a ﬁxed birational self-map χ ∈ Bir V . Clearly, in that case n = 1 if and only if χ ∈ Aut V (and by construction biregular automorphisms are in the image of ε). We will prove birational rigidity and surjectivity of ε simultaneously, using the following crucial technical fact. Proposition 1.4. Assume that cvirt (Σ) < n. Then there exists a subvariety B ∈ B (that is, a line or a special conic) such that multB Σ > n. Moreover, there are at most two subvarieties in B with that property, and if there are two, say B1 , B2 ∈ B, then they are lines, B1 , B2 ∈ L, their span < B1 , B2 > is a plane P = P2 , and P ⊂ Q.

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The Proof is very technical and represents the main step in the study of birational geometry of V . A subvariety B ∈ B satisfying the inequality multB Σ > n is called a maximal subvariety of the linear system Σ. By Proposition 1.2, we know that a maximal singularity exists. Now the hard part is to show that this implies existence of a maximal curve and this curve is necessarily a line or a special conic. For the details, see [13]. 1.3.5. The untwisting procedure. Now we derive Theorem 1.1 from Proposition 1.4. Lemma 1.5. (i) Let L ⊂ V be a line, Σ + ⊂ |n+ H| the strict transform of the linear system Σ with respect to αL . The following equalities hold: n+ = 4n − 3 multL Σ,

multL Σ + = 5n − 4 multL Σ.

(ii) Let Y ∈ C be a special conic, L = L(Y ) ∈ L the residual line, Σ + ⊂ |n+ H| the strict transform of the linear system Σ with respect to βY . The following equalities hold: n+ = 13n − 12 multY Σ,

multY Σ + = 14n − 13 multY Σ,

multL Σ + = 8n − 8 multY Σ + multL Σ. (iii) Let P ⊂ P5 be a 2-plane such that P ∩ V = L ∪ L1 ∪ L2 , Σ + as in (i) above. Then for {i, j} = {1, 2} we have multLi Σ + = 2n − 2 multL Σ + multLj Σ. Proof is a straightforward application of Lemmas 1.1–1.3. Q.E.D. Corollary 1.1. An involution τ = αL or βY satisﬁes the inequality n+ < n if and only if L or Y is a maximal curve of the linear system Σ, respectively, where Σ + ⊂ |n+ H| is the strict transform of Σ with respect to τ . Corollary 1.2. In the notations of the previous corollary assume that n+ = n. Then τ = αL for line L ∈ L and there exist lines L1 , L2 ∈ L, such that L ∪ L1 ∪ L2 = P ∩ V , where P ⊂ Q is a plane. Now let us prove birational rigidity of V and surjectivity of ε. Assume that cvirt (Σ) < n for a movable linear system Σ. By Proposition 1.4, there exists a curve B ∈ B such that multB Σ > n. Let τ ∈ ε(G) be the corresponding involution (that is, τ = αL if B = L ∈ L and τ = βY if B = Y ∈ C). By Corollary 1.1, Σ + ⊂ |n+ H| with n+ < n, where Σ + is the strict transform of Σ with respect to τ . Iterating this procedure, we construct a sequence of involutions τi ∈ ε(G) such that the strict transforms Σ (i) ⊂ |ni H| of the system Σ with respect to the compositions τi . . . τ1 satisfy the inequalities

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ni < ni−1 . Since ni ∈ Z+ , at some step we cannot decrease the threshold c(Σ (i) , V ) any longer. Therefore, for some k ≥ 1 we get c(Σ (k) , V ) = cvirt (Σ (k) , V ) = cvirt (Σ, V ), which is birational rigidity. Moreover, if we ﬁx a birational self-map χ ∈ Bir V and take Σ to be the strict transform of the system |H| via χ, then the procedure described above gives nk = 1 for some k, that is, Σ (k) ⊂ |H|. Comparing dimensions, we get Σ (k) = |H|, which implies that τk . . . τ1 χ ∈ Aut V is a biregular map. This proves surjectivity of ε. The last step in the proof of Theorem 1.1 is to show that ε has the trivial kernel. 1.3.6. The set of relations is complete. For convenience of notations, we write down words in AL , BY , using capital letters and corresponding birational self-maps using small letters, say t = ε(T ), etc. For a self-map t ∈ Bir V we deﬁne the integer n(t) ∈ Z+ by the formula Σ ⊂ |n(t)H|, where Σ is the strict transform of the system |H| via t; obviously, n(t) = 1 if and only if t ∈ Aut V . Theorem 1.1 immediately follows from Proposition 1.5. Let W = T1 . . . Tl be an arbitrary word in the alphabet {AL , BY | L ∈ L, Y ∈ C}. If w ∈ Aut V , then using the relations in R+ one can transform the word W into the empty word. Proof. Denote by Wi , i ≤ l(W ) = l, the left segment of the word W of length i, that is, Wi = T1 . . . Ti . Set n∗ (W ) = max{n(wi ) | 1 ≤ i ≤ l(W )}, ω(W ) = &{i | n(wi ) = n∗ (W ), 1 ≤ i ≤ l(W )}. Now we associate with every word W the ordered triple (n∗ (W ), ω(W ), l(W )). We order the set of words, setting W > W , if either n∗ (W ) > n∗ (W ), or n∗ (W ) = n∗ (W ) and ω(W ) > ω(W ), or n∗ (W ) = n∗ (W ), ω(W ) = ω(W ) and l(W ) > l(W ). It is easy to see that every decreasing chain of words W (1) > W (2) > . . . breaks. Therefore, it is suﬃcient to show that if w ∈ Aut V , then the word W can be transformed into a word W such that W > W , w = w . If the word W contains the subword AL AL or BY BY , then, eliminating this subword, we get a smaller word W (because the image of each left segment of the word W coincides with the image of some left segment of the word W and the map of the set of left segments of W into the set of left segments of W is injective).

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So we can assume that W does not contain subwords AL AL or BY BY . Since n(w) = 1, we can assume that n∗ (W ) ≥ 2 (otherwise there is nothing to prove). Let s = min{i | n(wi ) = n∗ (W )} ≤ l(W ) − 1. Let us consider the two cases Ts = AL and Ts = BY separately. Case 1. Ts = BY . In this case n(ws−1 ) = n(ws βY ) < n(ws ), by the choice of s. By Corollary 1.1, multY Σs > n(ws ), where Σs is the strict transform of |H| via ws . Since by construction n(ws+1 ) ≤ n(ws ), we get Ts+1 = Ts = BY . A contradiction to our assumption that W does not contain subwords AL AL and BY BY . Case 2. Let Ts = AL . By the choice of s we get multL Σs > n(ws ). By assumption, Ts+1 = Ts and n(ws+1 ) ≤ n(ws ). By Corollary 1.2, Ts+1 = AL , where L ⊂ V is a line such that there exists a third line Z ⊂ V , L ∪ L ∪ Z = P ∩ V for some plane P ⊂ Q. Lemma 1.6. (i) Z is a maximal line of the map ws−1 , that is, multZ Σs−1 > n(ws−1 ). Therefore, n(ws−1 αZ ) < n(ws−1 ). (ii) The equality n(ws−1 αZ ) − multL Σ = n(ws ) − multL Σs ≤ 0 holds, where Σ is the strict transform of |H| with respect to ws−1 αZ . Therefore, n(ws−1 αZ αL ) ≤ n(ws−1 αZ ). Proof. Straightforward computations based on Lemma 1.5. We will consider the claim (i) only, leaving (ii) to the reader. Since ws = ws−1 αL , we get ws−1 = ws αL and by Lemma 1.5, n(ws−1 ) = n(ws αL ) = 4n(ws ) − 3 multL Σs , multZ Σs−1 = 2n(ws ) − 2 multL Σs + multL Σs . Therefore, n(ws−1 )−multZ Σs−1 = 2n(ws )−multL Σs −multL Σs < 0, which is what we need. For the claim (ii), the arguments are similar. Q.E.D. Now let us complete the proof of Theorem 1.1. Consider ﬁrst the case when multL Σs > n(ws ). Using the relations A2Z = e and AZ AL AL = AL AL AZ , we can replace the subword AL AL by the subword AZ AL AL AZ . This operation increases the length. Denote the new word by W + . Obviously, Wi+ = Wi for i ≤ s − 1. Furthermore,

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ws+ = ws−1 αZ ,

+ ws+1 = ws−1 αZ αL

+ and ws+2 = ws−1 αZ αL αL = ws+1 αZ , whereas + ws+i = ws+i−2

for i ≥ 3. By the lemma above, n(wi+ ) < n(ws ) = n∗ (W ) for i = s, s + 1, s + 2 (and by construction this is true for the smaller values i < s, as well). Therefore, if ω(W ) ≥ 2, then n∗ (W + ) = n∗ (W ) and ω(W + ) = ω(W ) − 1. If ω(W ) = 1, then n∗ (W + ) < n∗ (W ). In any case, W + < W . It remains to consider the case multL Σs = n(ws ). In this case n(ws+1 ) = n(ws ), multL Σs+1 = n(ws+1 ). Since by assumption there are no subwords AL AL , we must have Ts+2 = AZ . Now let us replace the subword Ts Ts+1 Ts+2 = AL AL AZ by the subword AZ AL AL . Denote the new word by W + . Now the length is the same, and by Lemma 1.5 we obtain the inequalities n(wi+ ) < n∗ (W ) for i = s, s + 1, s + 2. Arguing as in the previous case, we complete the proof.

2 Varieties with ﬁnitely many structures In this section, we discuss three types of rationally connected varieties with ﬁnitely many (but more than just one) structures of a rationally connected ﬁber space: Fano direct products and two classes of varieties with a pencil of Fano double covers. Our considerations are based on [26,28,30]. For other examples, see [6,9,10,34,35]. 2.1. Fano direct products. Recall that a smooth projective variety F is a primitive Fano variety, if Pic F = ZKF , the anticanonical class is ample and dim F ≥ 3. Definition 2.1. We say that a primitive Fano variety F is divisorially canonical, or satisﬁes the condition (C) (respectively, is divisorially log canonical, or satisﬁes the condition (L)), if for any eﬀective divisor D ∈ | − nKF |, n ≥ 1, the pair 1 (4) (F, D) n has canonical (respectively, log canonical) singularities. If the pair (4) has canonical singularities for a general divisor D ∈ Σ ⊂ | − nKF | of any movable linear system Σ, then we say that F satisﬁes the condition of movable canonicity, or the condition (M ). Explicitly, the condition (C) is formulated in the following way: for any birational morphism ϕ : F# → F and any exceptional divisor E ⊂ F# , the inequality

Birational Rigidity

νE (D) ≤ na(E)

295

(5)

holds. The inequality (5) is opposite to the Noether–Fano inequality (3). The condition (L) is weaker: the inequality νE (D) ≤ n(a(E) + 1)

(6)

is required. It is well known (essentially starting from the classical paper of V. A. Iskovskikh and Yu. I. Manin [12]) that the condition (M ) ensures birational superrigidity. This condition is proved for many classes of primitive Fano varieties, see [12,18,22,24]. Note also that the condition (C) is stronger than both (L) and (M ). The following fact was proved in [28]. Theorem 2.1. Assume that primitive Fano varieties F1 , . . . , FK , K ≥ 2, satisfy the conditions (L) and (M ). Then their direct product V = F1 × · · · × FK is birationally superrigid. Now let us show how birational superrigidity makes it possible to describe rationally connected structures on V . Corollary 2.1. (i) Every structure of a rationally connected ﬁber space on the variety V is given by a projection onto a direct factor. More precisely, let β : V → S be a rationally connected ﬁber space and χ : V − − → V a birational map. Then there exists a subset of indices I = {i1 , . . . , ik } ⊂ {1, . . . , K} * Fi S , such that the diagram and a birational map α : FI = i∈I χ

V V πI ↓ ↓ β α FI S commutes, that is, β ◦ χ = α ◦ πI , where πI :

K * i=1

Fi →

*

Fi is the natural

i∈I

projection onto a direct factor. In particular, the variety V admits no structures of a ﬁbration into rationally connected varieties of dimension smaller than min{dim Fi }. In particular, V admits no structures of a conic bundle or a ﬁbration into rational surfaces. (ii) The groups of birational and biregular self-maps of the variety V coincide: Bir V = Aut V . In particular, the group Bir V is ﬁnite. (iii) The variety V is nonrational.

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Proof. Let us prove the claim (i). Let β : V → S be a rationally connected ﬁber space, χ : V V a birational map. Take a very ample linear system ΣS on the base S and let Σ = β ∗ ΣS be a movable linear system on V . As we have mentioned above (Example 1.1, (ii)), c(Σ ) = 0. Let Σ be the strict transform of the system Σ on V . By our remark, cvirt (Σ) = 0, so that by Theorem 2.1 we conclude that c(Σ) = 0. Therefore, in the presentation Σ ⊂ | − n 1 H1 − · · · − n K HK | some coeﬃcient ne = 0. We may assume that e = 1. Setting S = F2 × · · ·× FK and π : V → S to be the projection, we get Σ ⊂ |π ∗ Y | for a nonnegative class Y on S. But this means that the birational map χ of the ﬁber space V /S onto the ﬁber space V /S is ﬁber-wise: there exists a rational dominant map γ : S S , making the diagram χ

V V π ↓ ↓ β γ S S commutative. For a point z ∈ S of general position let Fz = β −1 (z) be the corresponding ﬁber, Fzχ ⊂ V its strict transform with respect to χ. By assumption, the variety Fzχ is rationally connected. On the other hand, Fzχ = π −1 (γ −1 (z)) = F × γ −1 (z), where F = F1 is the ﬁber of π. Therefore, the ﬁber γ −1 (z) is also rationally connected. Thus we have reduced the problem of description of rationally connected structures on V to the same problem for S. Now the claim (i) of Corollary 2.1 is easy to obtain by induction on the number of direct factors K. For K = 1 it is obvious that there are no nontrivial rationally connected structures (see Proposition 1.1, (i)). The second part of the claim (i) (about the structures of conic bundles and ﬁbrations into rational surfaces) is obvious since dim Fi ≥ 3 for all i = 1, . . . , K. Nonrationality of V is now obvious. Let us prove the claim (ii) of Corollary 2.1. Set RC(V ) to be the set of all structures of a rationally connected ﬁber space on V with a nontrivial base. By the part (i) we have Fi | ∅ = I ⊂ {1, . . . , K}}. RC(V ) = {πI : V → FI = i∈I

Now recall (Section 0.2) that the set RC(V ) has a natural structure of an ordered set: α ≤ β if β factors through α. Obviously, πI ≤ πJ if and only if J ⊂ I. For I = {1, . . . , K} \ {e} set πI = πe , FI = Se . It is obvious that π1 , . . . , πK are the minimal elements of RC(V ).

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Let χ ∈ Bir V be a birational self-map. The map χ∗ : RC(V ) → RC(V ), χ∗ : α −→ α ◦ χ, is a bijection preserving the relation ≤. From here it is easy to conclude that χ∗ is of the form χ∗ : πI −→ πI σ , where σ ∈ SK is a permutation of K elements and for I = {i1 , . . . , ik } we deﬁne I σ = {σ(i1 ), . . . , σ(ik )}. Furthermore, for each I ⊂ {1, . . . , K} we get the diagram χ V V ↓ πI σ πI ↓ χI FI FI σ , where χI is a birational map. In particular, χ induces birational isomorphisms χe : Fe Fσ(e) , e = 1, . . . , K. However, all the varieties Fe are birationally superrigid, so that all the maps χe are biregular isomorphisms. Thus χ = (χ1 , . . . , χK ) ∈ Bir V is a biregular isomorphism, too: χ ∈ Aut V . Q.E.D. for Corollary 2.1. Remark 2.1. The group of biregular automorphisms Aut V is easy to compute. Let us break the set F1 , . . . , FK into subsets of pair-wise isomorphic varieties: l 7 I = {1, . . . , K} = Ik , k=1

where Fi ∼ = Fj if and only if {i, j} ⊂ Ik for some k ∈ {1, . . . , l}. It is easy to see that l Aut V = Aut( Fi ). j=1

i∈Ij

In particular, if the varieties F1 , . . . , FK are pair-wise nonisomorphic, we get Aut V =

K

Aut Fi

i=1

(and this group acts on V component-wise). In the opposite case, if F1 ∼ = F2 ∼ = ... ∼ = FK ∼ = F,

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we obtain the exact sequence 1 → (Aut F )×K → Aut V → SK → 1, where SK is the symmetric group of permutations of K elements. In fact, in this case Aut V contains a subgroup isomorphic to SK which permutes direct factors of V , so that Aut V is a semidirect product of the normal subgroup (Aut F )×K and the symmetric group SK . It seems that the following generalization of Theorem 2.1 is true. Conjecture 2.1. Assume that F1 , . . . , FK are birationally (super)rigid primitive Fano varieties. Then their direct product V = F1 × · · ·× FK is birationally (super)rigid. Of course, Theorem 2.1 is meaningful only provided that we are able to prove the condition (C) for some particular Fano varieties. Certain examples were shown in [28]: generic Fano hypersurfaces F = FM ⊂ PM for M ≥ 6 and generic Fano double spaces of index 1. More examples (Fano complete intersections) were given in [29]. 2.2. Varieties with an involution. Following [26], let us construct a series of rationally connected varieties with exactly two nontrivial structures of a rationally connected ﬁber space. Fix positive integers m, l, satisfying the equality m + l = M + 1, M ≥ 4. Set P = PM+1 and take a hypersurface WP ⊂ P of degree 2l. Let σY : Y → P the double cover branched over the divisor WP . Consider the variety Y = P1 × Y, which is realized as the double cover σY : Y → X = P1 × P branched over the divisor W = P1 × WP . Set V = σY−1 (Q), where Q ⊂ X = P1 × P is a smooth divisor of the type (2, m), that is, it is given by the equation A(x∗ )u2 + 2B(x∗ )uv + C(x∗ )v 2 = 0, where A(·), B(·), C(·) are homogeneous of degree m. Here (u : v) and (x∗ ) = (x0 : . . . : xM+1 ) are homogeneous coordinates on P1 and P, respectively. Furthermore, let HP be the class of a hyperplane in P, LX = p∗X HP the tautological class on X, where pX : X → P is the projection onto the second factor, LV = σY∗ LX |V . It is easy to see that KV = −LV , so that the anticanonical linear system | − KV | is free and determines the projection pV = pX ◦ σ : V → P. On the other hand, the projection π : V → P1 , which is the composition of σY |V and the projection of P1 × P onto the ﬁrst factor, realizes V as a primitive Fano ﬁber space, the ﬁber of which is a Fano double hypersurface of index 1 [19]: Pic V = ZLV ⊕ ZF , where F is the class of a ﬁber of π. Lemma 2.1. The projection pV factors through the double cover σY : Y → P. More precisely, there is a morphism p : V → Y such that

Birational Rigidity

299

pV = σY ◦ p. The degree of the morphism p at a general point is equal to 2. Proof. Consider a point x ∈ P \ WP of general position. Set {y + , y − } = σY−1 (x) ⊂ Y. Set also Lx = P1 × {x} ⊂ X,

1 ± L± x = P × {y } ⊂ Y.

It is obvious that the inverse image σY−1 (Lx ) is the disjoint union of the lines − L+ x and Lx , whereas ± pY (L± x)= y , where pY : Y → Y is the projection onto the second factor. The divisor Q intersects Lx at two distinct (for a general point x) points q1 , q2 . Set − σ −1 (qi ) = {o+ i , oi } ⊂ V,

± o± i ∈ Lx .

The morphism p is the restriction pY |V . Obviously, ± p−1 (y ± ) = {o± 1 , o2 },

where the sign + or − is the same in the right-hand and left-hand side. This proves the lemma. Let Δ ⊂ V be a subvariety of codimension 2, given by the system of equations A = B = C = 0. The subvariety Δ is swept out by the lines Ly = P1 × {y} which are contracted by the morphism p. Set ΔY = p(Δ). Obviously, p : V \ Δ → Y \ ΔY is a ﬁnite morphism of degree 2. Let τ ∈ Bir V be the corresponding Galois involution. It is easy to see that τ commutes with the Galois involution α ∈ Aut V of the double cover σ : V → Q, so that τ and α generate a group of four elements. Since the involution τ is biregular outside the invariant closed subset Δ of codimension 2, that is, τ ∈ Aut(V \ Δ), the action of τ on the Picard group Pic V is well deﬁned. Let Σ ⊂ | − nKV + lF | be a movable linear system. Lemma 2.2. (i) The involution τ transforms the pencil |F | of ﬁbers of the morphism π into the pencil |mLV − F |. (ii) If l < 0, then the involution τ transforms the linear system Σ into the linear system Σ + ⊂ |n+ LV + l+ F |, where n+ = n + lm ≥ 0, l+ = −l > 0. Proof. Obviously, τ ∗ LV = LV . Let Ft = π −1 (t) be a ﬁber. We get p−1 (p(Ft )) = Ft ∪ τ (Ft ).

300

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However, p(Ft ) ∼ mHY = mσY∗ HP by the construction of the variety V . Since p∗ HY = LV , we obtain the claim (i). Thus τ ∗ F = mLV − F . This directly implies the second claim of the lemma. Now let us formulate the main result on birational geometry of the variety V. Theorem 2.2. The variety V is birationally superrigid. The group Bir V of birational self-maps is isomorphic to Z/2Z×Z/2Z with α and τ as generators. On the variety V there are exactly two nontrivial structures of a rationally connected ﬁber space, the projection π : V → P1 and the map πτ : V P1 . For the proof, see [26]. Let us just recall the scheme of the arguments modulo the hardest technical part. Let Σ ⊂ | − nKV + lF | be a movable linear system. If l ∈ Z+ , then the general constructions of [26, Theorem 2] imply that cvirt (Σ) = c(Σ), which is what we need. If l < 0, then consider the system Σ + = τ∗ Σ. Since τ is an isomorphism in codimension one, we have c(Σ + ) = c(Σ). Since the virtual threshold is a birational invariant, cvirt (Σ + ) = cvirt (Σ). However, Σ + ⊂ |−n+ KV +l+ F |, where by Lemma 2.2 n+ = n+lm, l+ = −l ≥ 1. Applying to Σ + the general theory ([26, Theorem 2]), we get cvirt (Σ + ) = c(Σ + ), which implies birational rigidity by what has been said above. The very same arguments prove that there are exactly two nontrivial structures of a ﬁber space into varieties of negative Kodaira dimension on V , that is, the projection π and πτ . Finally, if χ ∈ Bir V , then twisting by τ if necessary, one may assume that χ preserves the structure π, that is, transforms the ﬁbers of Ft into the ﬁbers Fγ(t) for some isomorphism γ : P1 → P1 . However, for a generic variety V a general ﬁber Ft has the trivial group of birational (= biregular) self-maps and moreover, a general ﬁber Ft is not isomorphic to any other ﬁber Fs , s = t, which implies that χ ∈ Aut V is either the identity map, or the Galois involution α. Therefore, Bir V = Z/2Z × Z/2Z = {idV , τ, α, ατ }. Q.E.D. for the theorem. 2.3. Varieties with two nonequivalent structures. Following [30], let us construct a family of rationally connected varieties with exactly two nontrivial structures of a rationally connected ﬁber space and this time the trivial group of birational self-maps. Let X be a projective bundle, X = P(E), where the ⊕OP1 (1)⊕2 . Thus X is a P = PM+1 locally free sheaf E is of the form E = OP⊕M 1 1 bundle over P . Let LX ∈ Pic X = ZLX ⊕ ZR be the class of the tautological sheaf, R the class of a ﬁber of the ﬁber space X/P1 . Let Q ∼ mLX be a smooth divisor, σ : V → Q the double cover branched over a smooth hypersurface W ∩ Q, where W ∼ 2lLX , m + l = M + 1. Obviously, π : V → P1 is a Fano ﬁber space, the ﬁber of which is a Fano double hypersurface of index 1. We get Pic V = ZLV ⊕ ZF , where LV = σ ∗ (LX |Q ) and F is the class of a ﬁber of π. It is easy to see that −KV = LV and thus the linear system

Birational Rigidity

301

$ $ | − KV − F | = σ ∗ (|LX − R| $ ) Q

is movable. Let ϕ : V P1 be the rational map, given by the pencil | − KV − F |. Birational geometry of the variety V is completely described by Theorem 2.3. (i) The variety V is birationally superrigid: for any movable linear system Σ on V its virtual and actual thresholds of canonical adjunction coincide, cvirt (Σ) = c(Σ). (ii) On the variety V there are exactly two nontrivial structures of a rationally connected ﬁber space, namely, π : V → P1 and ϕ : V P1 . These structures are birationally distinct, that is, there is no birational self-map χ ∈ Bir V , transforming the ﬁbers of π into the ﬁbers of ϕ. The groups of birational and biregular self-maps of the variety V coincide: Bir V = Aut V . (iii) There is a unique, up to a ﬁber-wise isomorphism, Fano ﬁber space π + : V + → P1 of the same type ((1, 1), (0, 0)), such that the following diagram commutes: χ V V + ϕ ↓ ↓ π+ P1 = P1 , where χ is a birational map. The construction V → V + is involutive, that is, (V + )+ = V . Proof. The space H 0 (X, LX ⊗ π ∗ OP1 (−1)) is two-dimensional and deﬁnes a pencil of divisors |LX − R|. Its base set ΔX = Bs |LX − R| is of codimension 2: it is easy to see that ΔX = P(OP⊕M )∼ = PM−1 × P1 . 1 Set ΔQ = ΔX ∩Q, Δ = σ −1 (ΔQ ) ⊂ V . Obviously, ΔQ is a smooth divisor of bidegree (m, 0) on ΔX = PM−1 ×P1 , Δ ⊂ V is a smooth irreducible subvariety of codimension 2. Lemma 2.3. The base set of the movable linear system | − KV − F | is equal to Bs | − KV − F | = Δ. Furthermore, −KV − F ∈ ∂A1mov V . More precisely, | − nKV + lF | = ∅ for l < −n. The Proof is straightforward (see [30]). Now let us study the rational map ϕ : V P1 . In order to do this, we need an explicit coordinate presentation of the varieties X, Q, and W , participating in the construction of the Fano ﬁber space V /P1 . Consider the locally free subsheaves E0 = OP⊕M "→ E 1

and E1 = OP1 (1)⊕2 "→ E.

302

A. V. Pukhlikov

Obviously, E = E0 ⊕ E1 . Let Π0 ⊂ H 0 (X, LX ) be the subspace, corresponding to the space of sections of the sheaf H 0 (P1 , E0 ) "→ H 0 (P1 , E). Set also Π1 = H 0 (X, LX ⊗ π ∗ OP1 (−1)) = H 0 (P1 , E1 (−1)). Let x0 , . . . , xM−1 be a basis of the space Π0 , y0 , y1 a basis of the space Π1 . Then the sections x0 , . . . , xM−1 , y0 t0 , y0 t1 , y1 t0 , y1 t1 ,

(7)

where t0 , t1 is a system of homogeneous coordinates on P1 , make a basis of the space H 0 (X, LX ). It is easy to see that the complete linear system (7) deﬁnes a morphism ¯ ⊂ PM+3 , ξ: X → X the image X of which is a quadratic cone with the vertex space PM−1 = ξ(ΔX ) and a smooth quadric in P3 , isomorphic to P1 × P1 , as a base. The morphism ¯ \ ξ(ΔX ) is an isomorphism ξ is birational, more precisely, ξ : X \ ΔX → X M−1 1 × P onto the vertex space of the cone. Let and ξ contracts ΔX = P u0 , . . . , uM−1 , u00 , u01 , u10 , u11 be the homogeneous coordinates on PM+3 , corresponding to the ordered set ¯ is given by the equation of sections (7). The cone X u00 u11 = u01 u10 . ¯ there are two pencils of (M + 1)-planes, corresponding to On the cone X the two pencils of lines on a smooth quadric in P3 . Let τ ∈ Aut PM+3 be the automorphism permuting the coordinates u01 and u10 and not changing ¯ is an automorphism of the cone the other coordinates. Obviously, τ ∈ Aut X ¯ X, permuting the above-mentioned pencils of (M + 1)-planes. One of these pencils is the image of the pencil of ﬁbers of the projection π, that is, the pencil ξ(|R|). For the other pencil we get the equality τ ξ(|R|) = ξ(|LX − R|). The automorphism τ induces an involutive birational self-map τ + ∈ Bir X. More precisely, τ + is a biregular automorphism outside a closed subset ΔX ˜ → X be the blowup of the smooth subvariety of codimension 2. Let ε : X ˜ is isomorphic to the blowup of the cone X ¯ at ΔX . Obviously, the variety X + its vertex space ξ(ΔX ). It is easy to check that τ extends to a biregular ˜ automorphism of the smooth variety X. + + + + Set Q = τ (Q) ⊂ X, W = τ (W ) ⊂ X. The divisors Q+ and W + are well deﬁned because τ + is an isomorphism in codimension 1. Lemma 2.4. The divisors Q+ and W + are divisors of general position in the linear systems |mLX | and |2lLX |, respectively. In particular, Q+ , W + , and Q+ ∩ W + are smooth varieties.

Birational Rigidity

303

Proof. The claim follows immediately from the fact that the linear systems |kLX |, k ∈ Z+ , are invariant under τ + , whereas Q and W are suﬃciently general divisors of the corresponding linear systems. Note that if a divisor D ∈ |kLX | is given by a polynomial h(u0 , . . . , uM−1 , u00 , u01 , u10 , u11 ), of degree k, then its image τ + (D) is given by the polynomial h+ (u∗ ) = h(u0 , . . . , uM−1 , u00 , u10 , u01 , u11 ) with permuted coordinates u01 and u10 . Q.E.D. for the lemma. Let σ + : V + → Q+ be the double cover, branched over a smooth divisor + Q ∩W + . Obviously, V + /P1 is a general Fano ﬁber space of type ((1, 1), (0, 0)). Lemma 2.5. (i) The map τ + lifts to a birational map χ : V V + , biregular in codimension 1. (ii) The action of χ on the Picard group is given by the formulas χ∗ K V + = K V ,

χ∗ F + = −KV − F,

where F + is the class of the ﬁber of the projection V + → P1 , so that Pic V + = ZKV + ⊕ ZF + . (iii) The construction of the variety V + is involutive: (V + )+ ∼ =V. Proof. The claims (i)–(iii) are obvious. Just note that the following presentation holds: χ = q + ◦ q −1 , where q : V˜ → V and q + : V˜ → V + are blowups of the smooth subvarieties of codimension two Δ ⊂ V and Δ+ ⊂ V + , respectively. Furthermore, E = q −1 (Δ) is the exceptional divisor of both blowups, E = Δ × P1 = ΔF × P1 × P1 , whereas the projections q | E and q + | E are projections with respect to the second and third direct factors, respectively. Finally, let us prove Theorem 2.3. Let Σ ⊂ | − nKV + + lF | be a movable linear system. If l ∈ Z+ , then by Theorem 2 of the paper [26] we get the desired coincidence of the thresholds: cvirt (Σ) = c(Σ). Assume that l < 0. Consider the linear system Σ + = τ + (Σ) on V + . By Lemma 2.5, Σ + ⊂ | − n+ KV + + l+ F + |, where l+ = −l ≥ 1. Since τ + is an isomorphism in codimension 1, we get c(Σ) = c(Σ + ). Again applying Theorem 2 of the paper [26], we obtain the desired coincidence of thresholds cvirt (Σ + ) = cvirt (Σ) = c(Σ + ) = c(Σ) = n+ = n + l. This proves birational superrigidity. Let us prove the claim (ii). Arguing as in Section 2.2, we show that on V there are exactly two nontrivial structures of a rationally connected ﬁber space (the arguments above imply that if a movable linear system Σ satisﬁes the equality cvirt (Σ) = 0, then either Σ is composed from the pencil |F |, or Σ is

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composed from the pencil |−KV −F |, which gives a description of the existing structures). For a general variety V these structures cannot be birationally equivalent. Indeed, by birational superrigidity of Fano double hypersurfaces of index 1, any birational map χ+ ∈ Bir V , which transforms the pencil |F | into the pencil | − KV − F |, induces a biregular isomorphism of the ﬁbers of general position in the pencils |F | and |F + | (the latter is taken on the variety V + ). Therefore, χ+ induces a biregular isomorphism of the ﬁbers of general position of the ﬁber spaces Q/P1 and Q+ /P1 . Now by Theorem 3.1 below for m ≥ 3 we get that these ﬁber spaces are globally ﬁber-wise isomorphic. It checks easily that for a suﬃciently general divisor Q ⊂ X this is impossible. For m = 2 we argue in a similar way, using the branch divisor W . Finally, the claim (iii) follows from the arguments above. Q.E.D. for Theorem 2.3.

3 Fiber-wise birational correspondences In this section, following [21], we study ﬁber-wise birational correspondences of ﬁber spaces, the ﬁber of which is a hypersurface. 3.1. Fibrations into complete intersections. Let C be a smooth algebraic curve with a marked point p ∈ C, and C ∗ = C \ {p} a “punctured” curve. In what follows our arguments remain correct if we replace C by a smooth germ of a curve p ∈ C, or a small disk Δε = {|z| < ε} ⊂ C. The symbol P stands for the complex projective space PM , M ≥ 3. Let V(d) be the set of smooth divisors V ⊂ X = C × P, each ﬁber of which Fx = V ∩ {x} × P, x ∈ C, is a hypersurface of degree d ≥ 2. Set X ∗ = C ∗ × P,

V ∗ = V ∩ X ∗,

so that V ∗ is obtained from V by throwing away the ﬁber Fp over the marked point. Theorem 3.1. Assume that d ≥ 3. Take V1 , V2 ∈ V(d) and let χ∗ : V1∗ → V2∗ be a ﬁber-wise isomorphism. Then χ∗ extends to a ﬁber-wise isomorphism χ : V1 → V2 . In other words, within the limits of the class V(d) these varieties do not permit nontrivial birational transforms of the ﬁbers. Let Z≥2 be the set of integers m ≥ 2. Conjecture 3.1. For a given k ≥ 2 there exist an integer M∗ ≥ k + 2 and a ﬁnite set S ⊂ Zk≥2 (which may occur to be empty) such that for each M ≥ M∗ and each set (d1 , . . . , dk ) ∈ Zk≥2 \S the statement of Theorem 3.1 is true for the

Birational Rigidity

305

class V(d1 , . . . , dk ) of smooth complete intersections of the type (d1 , . . . , dk ) in C × PM . Theorem 3.1 implies the following global fact. Corollary 3.1. Let V /P1 and V /P1 be smooth ﬁbrations into Fano hypersurfaces of index 1. Assume that V /P1 is suﬃciently twisted over the base [20,26]. Then any birational map χ : V V is a ﬁber-wise biregular isomorphism. Let us start with the following question: which singularities can acquire a special ﬁber if the total space is smooth? Let (d1 , . . . , dk ) ∈ Zk≥2 be a ﬁxed type of complete intersection. Consider the class of subvarieties in C × P, which can be represented locally over C as f1 = · · · = fk = 0, where the equations fi with respect to a system (x0 : · · · : xM ) of homogeneous coordinates on P are of the form aI xI , fi = |I|=di

I = (j0 , . . . , jM ) are multi-indices of degree j0 + · · · + jM = di , and the coeﬃcients aI are regular functions on C, whereas for each point y ∈ C the set of equations {f∗ }, restricted on the ﬁber Xy = {y} × P ∼ = P, deﬁnes a complete intersection of codimension k in P. Let us denote the class of these varieties by Z(d1 , . . . , dk ). Take V ∈ Z(d1 , . . . , dk ). Let F = V ∩ Xp be the ﬁber over the marked point. Fix a system of equations {f∗ } for V near the point p ∈ C and a local parameter t on the curve C at the point p. Now the equations fi can be expanded into their Taylor series (0)

fi = fi

(1)

+ tfi

(j)

+ · · · + tj f i

+ ··· ,

(j)

where fi are homogeneous polynomials of degree di in (x∗ ). The ﬁber F ⊂ P (0) is given by the system of equations {f∗ = 0}. Lemma 3.1. The following estimate holds dim(Xp ∩ Sing V ) ≥ dim Sing F − 1. The Proof is similar to the proof of Lemma 3.4.2 in [13]. The set Sing F is given on F by the condition 8 8 8 ∂f (0) 8 8 i 8 rk 8 8 ≤ k − 1. 8 ∂xj 8

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A. V. Pukhlikov

If dim Sing F ≤ 0, then there is nothing to prove. Otherwise, let Y ⊂ Sing F be a component of maximal dimension, dim Y ≥ 1. The set Xp ∩ Sing V is given on F by the condition 8 8 8 ∂f (0) 8 8 i (1) 8 (8) rk 8 | fi 8 ≤ k − 1. 8 ∂xj 8 (0)

If the set D = {x ∈ Y | rk +∂fi /∂xj + ≤ k − 2} is of codimension 1 in Y , then the lemma is proved, since D ⊂ Xp ∩ Sing V . Assume the converse: codimY D ≥ 2. Take a general curve Γ ⊂ Y disjoint from D. At each point (0) of the curve Γ the rank of the matrix +∂fi /∂xj + is equal to k − 1. Consider the morphisms of sheaves μj :

k

OΓ (1 − di ) → OΓ

i=1

that are deﬁned locally on the sets of sections (s1 , . . . , sk ) by the formula μj : (s1 , . . . , sk ) →

k

(i)

si

i=1

∂f0 ∂xj

with respect to a ﬁxed isomorphism O(−a) ⊗ O(a) ∼ = O. By assumption the subsheaf M k Ker(μ∗ ) = Ker μj ⊂ OΓ (1 − di ) j=0

i=1

is of constant rank 1. Now consider the morphism of sheaves λ : Ker(μ∗ ) → OΓ (1), (1) λ : (s1 , . . . , sk ) → ki=1 si fi . Assume that the condition (8) is not true at each point of the curve Γ . Then λ is an isomorphism of invertible sheaves, which means that OΓ (1) "→

k

OΓ (1 − di ).

i=1

But this is impossible. Q.E.D. for Lemma 3.1. Let us consider ﬁbrations into hypersurfaces. In accordance with Lemma 3.1, a variety V ∈ Z(d) with a local equation f = f (0) + tf (1) + · · · is smooth, that is, V ∈ V(d), if and only if the following two conditions hold: (i) the hypersurface F = {f (0) = 0} has at most zero-dimensional singularities; (ii) for each point x ∈ Sing F we have f (1) (x) = 0.

Birational Rigidity

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3.2. The diagonal presentation. Take V1 , V2 ∈ V(d), d ≥ 2, and let χ∗ : V1∗ → V2∗ be a ﬁber-wise isomorphism outside the marked point p ∈ C. Since the ﬁbers over generic points y ∈ C are smooth hypersurfaces of degree d ≥ 2, the isomoprhisms χ∗y over the points y ∈ C ∗ are induced by automorphisms of the ambient projective space ξy ∈ Aut P. Thus χ∗ = ξ ∗ |V1 , where ξy∗ = ξy is an algebraic curve ξ ∗ : C ∗ → Aut P of projective automorphisms. Let P = P(L) be the projectivization of a linear space L ∼ = CM+1 . The curve ξ ∗ can be lifted to a curve ξ : C → End L, where ∗ ξ(C ) ⊂ Aut L. If ξ(p) ∈ Aut L, then χ∗ extends to the ﬁber-wise (biregular) isomorphism χ = ξ|V1 , and the varieties V1 and V2 are ﬁber-wise isomorphic. Assume the converse: det ξ(p) = 0. Fix a local parameter t on the curve C at the point p, and let ∞

ti ξ (i)

i=0

be the Taylor series of the curve ξ. We may assume that ξ (0) = 0. Lemma 3.2. There exist curves of linear self-maps β, γ : C → End L, β(p), γ(p) ∈ Aut L, and a basis (e0 , . . . , eM ) of the space L such that with respect to this basis the curve βξγ −1 : C → End L has a diagonal form: βξγ −1 : ei → tw(ei ) ei ,

(9)

where w(ei ) ∈ Z+ . Proof. This is a well-known fact of elementary linear algebra. Now replace V1 by γ(V1 ), V2 by β(V2 ). We may simply assume that the ﬁber-wise birational correspondence ξ has the form (9) from the beginning. We claim that if m = max{w(ei )} ≥ 1, then this is impossible. Let {a0 = 0 < a1 < · · · < ak } = {w(ei ), i = 0, . . . , M } ⊂ Z+ be the set of weights of the diagonal transform (9), k ≤ M , m = ak the maximal weight. Take the system of homogeneous coordinates (x0 : · · · : xM ), dual to the basis (e∗ ). We deﬁne the weight of monomials in x∗ , setting w(xn0 0 xn1 1 . . . xnMM ) =

M

ni w(ei ).

i=0

Set Ai = {xj |w(ej ) = ai } ⊂ A = {x0 , . . . , xM } to be the collection of coordinates of the weight ai . The distinguished sets of coordinates of the maximal and minimal weight we denote by A∗ = A0 and A∗ = Ak .

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3.3. Birational = biregular. Let f = f (0) (x)+tf (1) +. . . be a local (over the base C) equation of the hypersurface V2 ⊂ C × P, where f (i) are homogeneous polynomials of degree d ≥ 3 in the coordinates x∗ . The series fξ =

∞ l=0

(l)

tl fξ (x) =

∞

tl f (l) (tw(x0 ) x0 , . . . , tw(xM ) xM )

l=0

vanishes on V1 , and outside the marked ﬁber F1 , that is, for t = 0, gives an equation of V1 . Let b ∈ Z+ be the maximal degree of the parameter t, dividing fξ . Then ∞ t−b fξ = g = tl g (l) (x0 , . . . , xM ) l=0

gives an equation of the hypersurface V1 at the marked ﬁber Xp , too. Lemma 3.3. For each l ∈ Z+ the polynomial f (l) belongs to the linear span of monomials of weight ≥ b − l, whereas the polynomial g (l) belongs to the linear span of monomials of weight ≤ b + l. Proof. Assume that the monomial xI comes into the polynomial f (l) with a I nonzero coeﬃcient. Then it generates the component tl+w(x ) xI of the series fξ and, moreover, this component comes from this monomial of f (l) only. Therefore l + w(xI ) ≥ b, which is what we need. Assume that the monomial xI comes into g (l) with a nonzero coeﬃcient. It is generated by the monomial tl+b xI of the series fξ , which, in its turn, can be generated by the monomial xI from the polynomial f α only, where α + w(xI ) = l + b. Q.E.D. for the lemma. Let P∗ = {xj = 0|w(xj ) ≥ 1} = Pej |w(xj ) = 0 , P ∗ = {xj = 0|w(xj ) ≤ m − 1} = Pej |w(xj ) = m be the subspaces of the minimal and the maximal weight, respectively. Lemma 3.4. If b ≥ m + 1, then P∗ ⊂ Sing F2 . If m(d − 1) ≥ b + 1, then P ∗ ⊂ Sing F1 . Proof. Assume that b ≥ m + 1. The ﬁber F2 ⊂ P over the marked point is given by the equation f (0) = 0. By assumption f (0) belongs to the linear span of monomials of weight ≥ m + 1. If a monomial xI comes into f (0) with a nonzero coeﬃcient, then xI contains a quadratic monomial in the variables A \ A∗ (otherwise w(xI ) ≤ m).Thus all the ﬁrst partial derivatives of the polynomial f (0) vanish on P∗ . Thus P ⊂ Sing F2 . Similarly, if b ≤ m(d − 1) − 1, then each monomial xI in g (0) contains a quadratic monomial in A\A∗ , otherwise we get w(xI ) ≥ m(d−1), which gives a contradiction with our assumption and Lemma 3.3. Q.E.D. for Lemma 3.4.

Birational Rigidity

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Now take into account that for d ≥ 3 the inequalities b≤m

and b ≥ m(d − 1)

cannot both be true. Consequently, at least one of the two inequalities of Lemma 3.4 holds. Suppose that b ≥ m + 1. Since V2 is smooth, P∗ is a point. Let A∗ = {x0 }, so that P∗ = (1, 0, . . . , 0). Again we use the fact that V2 is smooth and conclude that f (1) (1, 0, . . . , 0) = 0. Consequently, the monomial xd0 comes into f (1) with a nonzero coeﬃcient. By Lemma 3.3 b ≤ 1. Therefore m = 0, which is a contradiction. In the case b ≤ m(d − 1) − 1 the arguments are symmetric: V1 is smooth, P ∗ is the point (0, . . . , 0, 1), A∗ = {xM } and g (1) (0, . . . , 0, 1) = 0, so that md ≤ b + 1, whence we get m = 0 again, a contradiction. Therefore, nontrivial weights cannot occur and ξ is a ﬁber-wise biregular isomorphism. Consequently χ = ξ|V1 is a ﬁber-wise isomorphism, too. Proof of Theorem 3.1 is complete. Finally, let us prove Proposition 0.2. Let ϕ1 , ϕ2 : PM P1 be two generic projections. Assume that the structures π1 = ϕ1 | V : V P1 and π2 = ϕ2 | V : V P1 are ﬁber-wise birationally equivalent, where V ⊂ PM is a generic smooth hypersurface of degree M − 1 ≥ 4, that is, there exists a birational self-map χ ∈ Bir V such that π2 ◦ χ = π1 . Let P1 , P2 ⊂ PM be the centers of the projections ϕ1 , ϕ2 , respectively. By genericity we may assume that V ∩ Pi is smooth. Let us blow up V ∩ Pi : σi : Vi → V, Ei = σi−1 (V ∩ Pi ) ⊂ Vi being the exceptional divisor. The projections πi extend to the morphisms πi+ : Vi → P1 , the map χ extends to a birational map χ+ : V1 V2 . We get the commutative diagram χ+

π1+

V1 V2 ↓ ↓ π2+ P1 = P1

Now a general ﬁber of πi+ is birationally superrigid. Applying Theorem 3.1, we see that χ+ extends to an isomorphism between V1 and V2 , which maps every ﬁber (π1+ )−1 (t) isomorphically onto the ﬁber (π2+ )−1 (t), t ∈ P1 . Now an easy dimension count shows that for a generic plane P ⊂ PM of codimension 2 there are at most ﬁnitely many planes S ⊂ PM such that P ∩ V ∼ = S∩V. Since E1 ∩ (π1+ )−1 (t) ∼ = P ∩ V , we obtain that

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χ+ (E1 ∩ (π1+ )−1 (t)) = E2 ∩ (π2+ )−1 (t) (otherwise, there would have been a one-dimensional family of planes S ⊂ PM with the property S ∩ V ∼ = P1 ∩ V ). Therefore, χ+ (E1 ) = E2 and the original map χ ∈ Bir V is biregular outside P1 ∩ V and P2 ∩ V , respectively. Therefore, χ ∈ Aut V = {idV }. Q.E.D. for Proposition 0.2.

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