Progress in Mathematics Volume 165
Series Editors Hyman Bass Joseph Oesterle Alan Weinstein
Juan C. Migliore
Introduction to Liaison Theory and Deficiency Modules
Springer Science+Business Media, LLC
Juan C. Migliore Department of Mathematics University of Notre Dame Notre Dame, IN 46556
Library of Congress Cataloging-in-Publication Data Migliore, Juan C. (Juan Carlos), 1956Introduction to liaison theory and deficiency modules / Juan C. Migliore. p. cm. -- (progress in mathematics ; v. 165) Includes bibliographical references and index. ISBN 978-1-4612-7286-1 ISBN 978-1-4612-1794-7 (eBook) DOI 10.1007/978-1-4612-1794-7 1. Liaison theory (Mathematics) 2. Graded modules. 1. Title. II. Series: Progress in mathematics (Boston, Mass.); voI. 165. QA564.M49 1998 98-15680 516.3'5--dc21 CIP AMS Subject Classification: 14M06, 13C40, 14F05, 14M05, 14M07, 14MI0, 14M12, 13D02
Printed on acid-free paper © 1998 Springer Science+Business Media New York Origina\ly published by Birkha.user Boston in 1998 Softcover reprint ofthe hardcover 2nd edition 1998 Copyright is not claimed for works of U .S. Government employees. AlI rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Authorization to photocopy items for internal or personal use of specific clients is granted by Springer Science+Business Media, LLC. provided that the appropriate fee is paid directly to Copyright Clearance Center (CCC), 222 Rosewood Drive, Danvers, MA 01923, USA (Telephone: (978) 750-8400), stating the ISBN, the title of the book, and the first and last page numbers of each article copied. The copyright owner's consent does not include copying for general distribution, promotion, new works, or resale. In these cases, specific wrltten permission must frrst be obtained from the publisher. ISBN 978-1-4612-7286-1 Typeset by the Author in Je..1EX
98 76543 21
To Michelle and Emily and Claire
Preface In the fall of 1992 I was invited by Professor Changho Keem to visit Seoul National University and give a series of talks. I was asked to write a monograph based on my talks, and the result was published by the Global Analysis Research Center of that University in 1994. The monograph treated deficiency modules and liaison theory for complete intersections. Over the next several years I continually thought of improvements and additions that I would like to make to the manuscript, and at the same time my research led me in directions that gave me a fresh perspective on much of the material, especially in the direction of liaison theory. This resulted in a dramatic change in the focus of this manuscript, from complete intersections to Gorenstein ideals, and a substantial amount of additions and revisions. It is my hope that this book now serves not only as an introduction to a beautiful subject, but also gives the reader a glimpse at very recent developments and an idea of the direction in which liaison theory is going, at least from my perspective. One theme which I have tried to stress is the tremendous amount of geometry which lies at the heart of the subject, and the beautiful interplay between algebra and geometry. Whenever possible I have given remarks and examples to illustrate this interplay, and I have tried to phrase the results in as geometric a way as possible. There are really two principal subjects, as indicated in the title: deficiency modules and liaison theory, for subschemes of projective space. Both deficiency modules and liaison theory are used very extensively in algebraic geometry in one form or another. However, there does not seem to be a good general introduction to either subject in the literature, from the point of view in which we are interested, so this was one of the original motivations for this book. It is not entirely self-contained, but I have tried to at least state the necessary background results, and I have included a very extensive list of references (books, research papers and expository papers) which I hope will help the reader look up any material which he or she would like to pursue further. Of course, it is impossible to completely cover such a broad subject.
viii
PREFACE
I have chosen topics which are most interesting to me, and I apologize for any omissions or oversights. When the proofs are reasonable in length I have tried to include them, or at least give the reader an idea of how the proof goes. In some cases I have had to simply state results with no proof. Only a few of the results and observations given here are new, and I have tried to give careful credit where it is due. In this regard as well, I apologize for any oversights. The set of deficiency modules of a subscheme V of pn is a collection of graded modules over the polynomial ring. Under reasonable assumptions, these modules have finite length. The number of modules associated to V is just the dimension of V. One can get a lot of information about V from these modules, and the first part of the book is an attempt to describe some of the techniques used and some of the information that can be obtained from these modules. Liaison is an equivalence relation among subschemes of given dimension in a projective space. Roughly, two schemes are said to be directly CI-linked if their union is a complete intersection, and directly G-linked if their union is arithmetically Gorenstein. In either case, this notion generates an equivalence relation. It is not at all obvious, a priori, that there is any connection between liaison and deficiency modules; but in fact , there is a strong one. Especially in the context of liaison theory (but also sometimes more generally), deficiency modules are also sometimes known as Hartshorne-Rao modules. However, they actually were important in the literature even before Rao's work. Chapter 1 gives the background needed in the following chapters. It defines the deficiency modules, gives important facts about them, and gives a number of examples. It also treats hyperplane and hypersurface sections, Artinian reductions and h-vectors. Chapter 2 gives some applications of deficiency modules, and in particular of submodules of the deficiency module. The term "deficiency module" comes because these modules measure the failure of our scheme V to be arithmetically Cohen-Macaulay. We show that submodules of the (first) deficiency module also measure various types of deficiency. Among the other applications are a generalization of Dubreil's theorem, and a discussion of when the property of being arithmetically Cohen-Macaulay is "lifted" from the general hyperplane (or hypersurface) section of V up to V. In studying graded modules over the polynomial ring, it is important to know something about the module structure. One way of looking at it is to ask what effect "multiplying" by a general linear form has on the module. The extreme case is where multiplication by any linear form
PREFACE
ix
is zero. One would naturally expect that such extreme behavior in the deficiency modules of V would be reflected in many ways in the scheme V. Chapter 3 explores the case where V is a curve with this property, i.e. a so-called Buchsbaum curve. In fact, Buchsbaum curves recur often in this book since they provide interesting examples of many things that we will discuss. Chapter 3 also discusses Liaison Addition, which is an important construction originally due to Phil Schwartau. It is used to construct Buchsbaum curves, and a special case (Basic Double Linkage) is used in liaison theory. Chapter 4 gives the background on arithmetically Gorenstein ideals, including some ways of constructing arithmetically Gorenstein ideals with desired properties, which we will need in liaison theory. It also gives some surprising connections between codimension three arithmetically Gorenstein subschemes of projective space and codimension two arithmetically Cohen-Macaulay subschemes of projective space. Chapter 5 begins the study of liaison theory. Many things can be said about liaison theory for subschemes of projective space in general (i.e. in arbitrary co dimension) , and I have tried to give a good overview of these in this chapter. For instance, we see why it is often more useful to look at even liaison (i.e. restricting to an even number of links). I have also tried to stress similarities and differences between the complete intersection theory and the arithmetically Gorenstein theory. Many of the more powerful results in liaison theory are known at the moment only in codimension two. This is the topic of Chapter 6. We begin with a discussion of the special case where the schemes are arithmetically Cohen-Macaulay. Then we look at Rao's result parameterizing the even liaison classes. Next we turn to the problem of describing the structure of an even liaison class. Finally, in the last part of Chapter 6 we give a number of applications of liaison theory, to give an idea of the breadth of possible ways in which it can and has been used. There are many mathematicians whom I must acknowledge. I am very grateful to Joe Harris for introducing me to liaison theory and for patiently guiding me through my first steps in the field; to Phil Schwartau for teaching me his more algebraic way of looking at it (and for writing such a nice thesis) ; to Giorgio Bolondi and Tony Geramita for their friendship and for the many years in which we have collaborated - many of the results described here come from the "liaison" part of these ongoing collaborations. A special thanks goes to my friends Jan Kleppe, Rosa Marfa Miro-Roig, Uwe Nagel and Chris Peterson - our recent and ongoing joint work has helped mold my view of liaison theory and consequently has fundamentally affected the material in this book and the changes from
x
PREFACE
the first version. I would like to thank Kyung-Hye Cho, Tony Geramita, Heath Martin, Scott Nollet, Yves Pitteloud, Phil Schwartau and especially Chris Peterson for their careful reading of the original manuscript in 1994 and for their helpful suggestions. I would also like to thank Jan Kleppe, Uwe Nagel, Scott Nollet and Chris Peterson for their helpful comments on this revised manuscript. I am grateful to the Mathematics Department and the Global Analysis Research Center of Seoul National University for their hospitality and support. Most of all, I would like to thank my friend Changho Keem for inviting me to Korea to give the talks, and for his great kindness and hospitality while I was there. Without the opportunity that he provided, this book would very likely never have been written. I would like also to thank Ann Kostant and the staff at Birkhiiuser for their professional and courteous manner throughout the preparation of this book. My deepest thanks go to all my family for their love and encouragement through the various stages of my career. Especially to my parents, my wife, Michelle, and my daughters, Emily and Claire, lowe a debt which I can never hope to repay.
Contents vii
Preface 1 Background 1.1 1.2 1.3 1.4 1.5
Finitely Generated Graded S-Modules The Deficiency Modules (Mi)(V) ... Hyperplane and Hypersurface Sections Artinian Reductions and h- Vectors Examples . . . . . . . . . . . . . . . .
2 Submodules of the Deficiency Module 2.1 Measuring Deficiency . . . . . . . .. . 2.2 Generalizing Dubreil's Theorem .. . . 2.3 Lifting the Cohen-Macaulay Property .
3 Buchsbaum Curves and Liaison Addition
1
1
9 16 26 35
47
47 51 55
61
3.1 Buchsbaum Curves . . . . . . . . . . . 3.2 Liaison Addition . . . . . . . . . . . . 3.3 Constructing Buchsbaum Curves in p3
62 68
4 Gorenstein Subschemes of Projective Space
77
4.1 Basic Results on Gorenstein Ideals . . 4.2 Constructions of Gorenstein Schemes 4.2.1 Intersection of Linked Schemes 4.2.2 Sections of Buchsbaum-Rim Sheaves of Odd Rank. . . . . . . . . . . . . . . . 4.2.3 Linear Systems on aCM Schemes 4.3 Co dimension Three Gorenstein Ideals . .
72
77
82 83 84 90
94
CONTENTS
xu
5 Liaison Theory in Arbitrary Codimension 5.1 5.2 5.3 5.4 5.5
Definitions and First Examples .. Relations Between Linked Schemes .. . The Hartshorne-Schenzel Theorem .. . The Structure of an Even Liaison Class . Geometric Invariants of a Liaison Class
6 Liaison Theory in Codimension Two 6.1 6.2 6.3 6.4
The aCM Situation and Generalizations Rao's Results . . . . . . . . . The Lazarsfeld-Rao Property Applications.......... 6.4.1 Smooth Curves in]p>3 . 6.4.2 Smooth Surfaces in ]p>4 and Threefolds in ]p>5 6.4.3 Possible Degrees and Genera in a Co dimension Two Even Liaison Class . . . . . . . . . . . . . . . 6.4.4 Stick Figures . . . . . . . . . . . . . . . . . . . . . 6.4.5 Low Rank Vector Bundles and Schemes Defined by a Small Number of Equations . . . . . . . . . . .
101 103 113 132 137 143
151 152 158 165 177 178 182 184. . . 189 . 191
Bibliography
195
Index
211
Introduction to Liaison Theory and Deficiency Modules
Chapter 1 Background Throughout this book, k = k shall always denote an algebraically closed field. We will occasionally also require that it have characteristic zero, but we will always make it clear when we are making this assumption. All varieties and subschemes will be assumed to be projective. We shall denote by S the homogeneous polynomial ring k[X 0, ... ,X nl and we let lP'n = lP'k = Proj S. Since S is a graded ring, it is the direct sum of its homogeneous components: S = EBd~o Sd, where Sd is the vector space of homogeneous polynomials of degree d. We denote by m the maximal ideal,
m = (Xo, ... ,Xn ) C S.
We assume some knowledge of sheaves and sheaf cohomology. For a sheaf F of Ojp'n -modules we will sometimes use the notation H!(F) for the graded S-module tEZ
(See also page 6.) As usual, we use lower case for dimension
By a free sheaf on F, we mean a direct sum of line bundles, EB~=l Ojp'n (ai) for ai E Z. Such a sheaf is sometimes called dissocie. The main purpose of this chapter is to give some of the necessary background for the material in the subsequent chapters, and to define the deficiency modules.
1.1
Finitely Generated Graded S-Modules
Let M = EBjEZ M j be a finitely generated graded S-module. Notice that Mj = 0 for ] :S ]0 for some ]0. Also, dimkMj < 00 for all ]. How does
J. C. Migliore, Introduction to Liaison Theory and Deficiency Modules © Birkhäuser Boston 1998
2
CHAPTER 1. BACKGROUND
one describe the module structure of M? (It is not enough to know the dimensions of all the components, although there are many things that can be said just from this information - see [10].) In particular, how does "multiplication by a homogeneous polynomial" work? Given F E Sd we get a homomorphism
subscheme of jpn defined by the vanishing of the (r + 1) x (r + 1) minors of
1.1. FINITELY GENERATED GRADED 5-MODULES
3
For example, if 1; is a square matrix then p = q. We expect that p = q = r + 1 and Wr has co dimension (1)(1) , and that in fact Wr is the hypersurface given by the vanishing of det 1;, hence of degree [(q~l) = q. Lemma 1.1.1 is a special case of Porteous' formula. See [5] page 86 and [142] Lemma 1.4 for more details. We will also eventually be interested in the dual module of M. There are two kinds of dual modules that we could define: M Vk = Homk(M, k) and M VS = Homs(M, 5). (This notation is borrowed from [189].) When it is clear from the context which dual is being used, we will sometimes abuse notation and write M V for the dual module. (It will often be the case that our module M has finite length; in this case we leave it as an exercise to check that M VS = 0.) M Vk has the following structure as a graded 5-module: (MVk)i = M"-i (= Homk(M_ i , k), the dual vector space) for all i, and 1;; = t1;_i_l. For a locally free OlP'n-module F , we denote by F V the dual sheaf 1!.omortn (F, OlP'n). Another operation on graded modules which will be important is shifting: we define the module M(d) to be given by M(d)i = M d+i (a shift to the left by d), with the obvious re-indexing of the 1;i. Another way to view the module structure is the following. For each i we have a homomorphism J-li : Mi 0 51 ---t Mi +l . (This takes m 0 L I--t 1;i(L)(m).) This point of view allows us to define the notion of minimal generators of M. We will always assume that such generators are homogeneous. Indeed, in each component Mi+l choose a basis for the image of J-li and extend it with vectors ml,·· ., mt to a basis for Mi+l. Note that in the first non-zero component of M , { ml,··· , mt } is a basis of the whole component. (Before this, J-li is zero.) The set of all such (homogeneous) elements mi of M is a minimal generating set of M. The number of elements in degree d of a minimal generating set is well-defined , and we denote it by vd(M). We denote by v(M) the sum of the vd(M), which is the number of elements in a minimal generating set of M. (Equivalently, a minimal generating set is a set of homogeneous elements of M whose residues in the k-vector space M/mM form a k-basis.) We extend this idea to the notion of minimal free resolutions. (This brief discussion will follow [189].) Let {ml,···, mal be a minimal generating set for our finitely generated graded module M, and assume that the generators are in degree al,···, aa, respectively. Then we obtain a surjective degree zero homomorphism
n8
/ G)]
a
EB 5( -ai)~M ---t 0 i=l
CHAPTER 1. BACKGROUND
4
in the natural way. Note that the map fa is not uniquely determined, but the free module Fa = EB S( -ai) is. Let Kl be the kernel of fa. It can be shown that Kl is invariant up to isomorphism. (That is, if we choose a different f~ with corresponding kernel K~ then Kl ~ K~.) Kl is again finitely generated since Fa is Noetherian, so one can do the same procedure with Kl instead of M, and get a free module Fl surjecting to K 1 : Fl
~ ~
o
/'
Kl
/' ~
Fo~M--tO
o
The kernel Ki obtained in the ith step is called the ith syzygy module of M. Kl gives the relations among the minimal generators of M, K2 gives the relations among the relations, etc. One can continue in this manner, and the Hilbert Syzygy Theorem (cf. for instance [185]) guarantees that, after at most n + 1 steps, the kernel Kn+l will be free. (Recall that n + 1 is the dimension of the ring S. See also the discussion of the AuslanderBuchsbaum theorem on page 10.) Therefore we have a long exact sequence
which is the minimal free resolution of M. Note that we stop as soon as the kernel is free, so some of the Fi above may be zero. The simplest example is the minimal free resolution of a complete intersection; see Example 1.5.1. If Fi = EBf3i,jS(-j), the f3i,j are called the graded Betti numbers of the resolution. Some authors prefer the notation Fi = EB S( - j)f3i,j. An exact sequence ofthe above form (where all but the rightmost module M are free) but not necessarily obtained in the given way is called a free resolution of M. In general, one may desire to know if a given free resolution is minimal. A useful criterion is the following: it is minimal if and only if after choosing bases for the Fi and representing the homomorphisms fi by the corresponding matrices, the matrices fi have no entries which are non-zero scalars (i.e. the entries all lie in m). A useful way of producing free resolutions is the so-called mapping cone procedure (cf. [131] Chapter II, section 4). The basic idea, in our context, is this: given a short exact sequence of finitely generated graded S-modules
1.1. FINITELY GENERATED GRADED 5-MODULES
5
and free resolutions
and
o-+ Gn + 1 9 +l n
Gn ~
...
~ Go ~ M2 -+ 0,
then a free resolution for M3 is given by
We leave it as an exercise to determine the maps of this free resolution, in terms of the ones for Ml and M 2 . Note that this is not necessarily minimal, even if the resolutions of Ml and M2 are. For instance, the length might be too long. But even if the length is not an obstacle, it could violate the criterion for minimality mentioned above. It depends on the map 0: in the short exact sequence of modules. An important construction for us will be the sheafification of a graded module (see Hartshorne [88]). This associates to any graded 5-module M a quasi-coherent sheaf M. If M is finitely generated then M is in fact coherent. Here are some useful examples: Example 1.1.2 (a) If M is a graded module of finite length then
M=
O.
(b) If Y is a closed subscheme of lP'" and I is a homogeneous ideal which defines Y scheme-theoretically (e.g. the saturated ideal of Y - see below for the definition) then j = I y , the ideal sheaf of Yin lP"'. Also, SI I y ~ Oy, the structure sheaf of Y. 0 Sheafification also preserves exactness. exact sequence of graded modules
For instance, given a short
we get a short exact sequence of sheaves
For example, the exact sequence 0 -+ I y -+ 5 -+ 51Iy -+ 0 yields the standard exact sequence of sheaves
(1.1)
CHAPTER 1. BACKGROUND
6
Furthermore, if we begin with a minimal free resolution 0
-t
Fn+l
~
-.!.4
Fn ~
\.
/'
/'
\.
Kn
0
0
\. 0
/'
Kl
Fo
/'
\.
~ M -to
0
we can sheafify and get a corresponding diagram for the sheaves. If M is the coordinate ring S / I of a locally Cohen-Macaulay, equidimensional subscheme of F of dimension r, then Fo = S, Kl = I and Ki is locally free for i ~ n - r. (See Section 1.2.) In the other direction, one can obtain from a sheaf of OlP'n-modules, F, a graded S-module
H2(F)
Note, though, that while H2(F) = F, the reverse is not true: H~(J~1) =f. M in general. (H~(M) is generally bigger than M.) We say that M is saturated if we have equality, and we say, in general, that M = H~(M) is the saturation of M. Furthermore, given a short exact sequence of sheaves, one of course does not necessarily obtain a short exact sequence of cohomology modules. One gets a long exact sequence involving higher cohomology. Specifically, one can obtain graded modules by applying the higher cohomology functor to:F: we define
H!(F)
= EBHi(lpn,F(d)). dE£',
These objects fit into the long exact cohomology sequence arising from the short exact sequence of sheaves
yielding
o
-t
H2(Fd H!(F1 )
-t -t
H2(F2 ) H!(F2 )
-t -t
H2(F3) -t H;(F3) -t
A special case of the above discussion concerns the relation between homogeneous ideals in S and closed subschemes of F. Let I be a homogeneous ideal in S. Then I determines a closed subscheme Y of F.
1.1. FINITELY GENERATED GRADED S-MODULES
7
However, there are infinitely many ideals that define Y , and we define "the" ideal of Y to be the largest such. Indeed, there is a bijective correspondence between closed subschemes of jpn and saturated ideals, defined as follows . Let I be a homogeneous ideal. Then the saturation 1 of I, as defined above for any graded module , is 1 = H~(i). Equivalently, it holds that
1=
{ FE S
I Vi
(0 :S i :S n) there is an r such that X[ . FE I }.
If I defines a closed subscheme Y, then 1 = H~(Iy) 2 I . The Hilbert function of Y is the function Z --t N defined by H(Y, t) = dimk(Sj lk
For large values of t, this function is a polynomial of degree equal to the dimension of the scheme Y. This polynomial is called the Hilbert polynomial of Y. We refer to [88] for details.
Example 1.1.3 (a) If I is the ideal in S
then
= k[Xo , Xl , X 2 , X 3 ] given by
1 = (Xo , Xl) and I defines a line.
(b) In jp3, let V be the four points [0,0,0, 1], [0,0, 1,0], [0, 1,0,0], [1,0,0,0]. The saturated ideal of V has six generators, all in degree 2. In particular, dimk (Iv h = 6. Choose four general elements Fl , F2 , F3 , F4 of (Iv h, spanning an ideal I. Then Fl , F2 , F3 cut out the eight points of a complete intersection, four of which are the points of V, and F4 picks out these points and avoids the other four. Hence one can see from a geometrical point of view that I defines the scheme V but is not saturated. 0 An important tool is the notion of Castelnuovo-Mumford regularity. We first recall
°
Definition 1.1.4 Let F be a coherent sheaf on jpn. Then F is said to be m-regular if Hi(jpn , F(m - i)) = for all i > 0. 0 Note that by a theorem of Serre ([88] Theorem III.5.2) , F is m-regular for some m.
Theorem 1.1.5 (Castelnuovo-Mumford [161]) Let F be an m-regular coherent sheaf on
jpn.
(1) Hi(jpn, F(k)) =
Then
°
whenever i >
°
and k
+i ?
m.
CHAPTER 1. BACKGROUND
8
(2) HO(JP"l,.1'(k)) is spanned by
if k > m.
HO(pn, .1'(k - 1)) 0 HO(pn, 0(1))
In particular, the S -module H~(.1') defined above is generated in degree ::; m. (3) .1'(k) is generated as an Opn-module by its global sections, for all k ~ m. Note that part (1) says that ifF is m-regular, then it is (m+l)-regular. Hence it makes sense to define the regularity of .1', sometimes called the Castelnuovo-Mumford regularity, by
reg(.1')
= min{ml.1' is m-regular}.
Remark 1.1.6 It is worth noting that there is a strong connection between the regularity and minimal free resolutions. Let M = H~(.1') and assume that M is finitely generated as an S-module. Consider the minimal free resolution
o -+ ... ------t Ei'j S( -ai,j) \. /'
------t
Ei'j S( -aO,j) -+ M -+ 0
/'
Ki+1
o
/'
\.
o
o
o
0
with the corresponding short exact sequences, as described starting on page 3. Then one can check, by sheafifying and carefully following the cohomology of these short exact sequences from the resolution, that
reg (F) = II1ax{ ai,j - i}. t ,)
The proof is left as an exercise. It is not trivial, but apart from a careful study of the indices, the main fact that one needs is that Hn (Opn (k)) = 0 for k ~ -no 0
If M is a graded S-module of finite length, we define the diameter of M, diam M, to be the number of components from the first non-zero one to the last (inclusive) . So, for example, a module that is one-dimensional in degrees 1 and 5 but zero elsewhere would have diameter 5. An interesting invariant of a graded S-module of finite length is the least integer k such that all homogeneous polynomials of degree k annihilate M. This has been called the Buchsbaum index of the module M [80J.
1.2.
THE DEFICIENCY MODULES (Mi)(V)
9
See also Remark 1.5.9, Remark 2.2.8 and Remark 3.1.5. It was observed in [150], and is not hard to show, that if k = 2 then M decomposes as a direct sum of modules of diameter k = 2. This is not true for k ~ 3.
Remark 1.1.7 Most of the objects described in this section, and indeed throughout this book, can be computed with computer programs available free of charge. Two of the best known are Macaulay and CoCoA (which stands for Commutative Computer Algebra). We refer to [62] p. 375 for a description of these programs, as well as instructions for obtaining Macaulay. CoCoA can be obtained by visiting the web page located at http://cocoa.dima.unige.it/
Since I am more familiar with Macaulay (and only for this reason), the examples I have chosen for this book use the version of that program which is given in the reference [15]. 0
1.2
The Deficiency Modules (Mi)(V)
In this section we introduce the deficiency modules of a subscheme of projective space, and then in the subsequent sections we give examples and first results. The deficiency modules have been very important in the literature, both in liaison theory and in various papers that have appeared recently. They are a central topic in this book. We will discuss many of these applications later. In the context of liaison, especially for curves in JP3, they are often referred to as the Hartshorne-Rao module(s) (or simply the Rao module(s)) of the scheme. (See Chapters 5 and 6 for more details about the role these modules play in liaison theory.) However, this is somewhat inappropriate outside of curves in 1P'3, since these modules were also heavily studied by others around the same time (or even earlier); for instance, Schenzel [187] and Evans and Griffith [69] played an important role in the theory. In the important paper [128]' Lazarsfeld and Rao use the term "deficiency module" even for curves in JP3. We maintain their terminology here. We will see shortly that the reason for the name "deficiency modules" is that the collection of these modules measures the failure of V to be arithmetically Cohen-Macaulay (see Definition 1.2.2).
Definition 1.2.1 Given a closed subscheme, V, of IF of dimension r ~ 1, we define the deficiency modules of V, (Mi)(V) (1 ::; i ::; r), as the i-th cohomology module of the ideal sheaf of V, namely,
(Mi) (V)
= H! (Iv)
for 1 ::; i ::; r.
10
CHAPTER 1. BACKGROUND
A special case is when V is a curve. In this case we simply write M (V) for the deficiency module. 0 Notice that these modules depend on the embedding of V in pn. It is not hard to check that they can also be obtained as the local cohomology modules of the homogeneous coordinate ring S/ Iv of V: H::l(IV) H~(S/Iv).
The module (Mi){V) can also be expressed in the following way:
We leave the details to the reader. A proof for the case n = 3 can be found on pages 48-50 of [189], and the general case is proved similarly. There is another important module associated to our dimension r subscheme, V, of pn, which is closely related to the deficiency modules. This is the canonical module of V, denoted by K v. This is defined by
Definition 1.2.2 Let Iv be the saturated ideal of a closed subscheme V of lP". Then V is arithmetically Cohen-Macaulay (aCM) if and only if dim SI Iv = depth S/ Iv (where "dim" is the Krull dimension). In this case S / I v is said to be a Cohen-Macaulay ring. 0 From the point of view of Ext, another equivalent formulation of the aCM property is that the projective dimension (i.e. the length of a minimal free resolution of S/ I) is equal to the co dimension of V. This fact also holds for zeroschemes, i.e. zero-dimensional subschemes of pn. In general, the projective dimension is greater than or equal to the codimension, and is related to the codimension, via the depth of S/ I, by the AuslanderBuchsbaum theorem (special case): pd S/ I +depth S/ I = depth S; see for instance [202] Theorem 4.4.15. In the case where S/ I is a Cohen-Macaulay ring (i.e. I defines an aCM scheme), depth S -depth S / I = n+ I-dim S / I is exactly the codimension; in general pd S / I = depth S - depth S / I is larger than the codimension since dim S / I ~ depth SI I. Notice that if V is a zeroscheme, then it is automatically aCM . If V has dimension r ~ 1, we will now describe how the collection of r deficiency modules measures the failure of V to be arithmetically Cohen-Macaulay.
1.2.
THE DEFICIENCY MODULES (Mi)(V)
11
Lemma 1.2.3 If dim V = r 2 1, then V is aCM if and only if (Mi)(V)
o for 1 ::; i ::; r.
=
Proof: This can be shown from the previous paragraph and the connection between the (Mi)(V) and Ext above; it can also be seen directly from Definition 1.2.2, using the point of view of hyperplane sections described in Section 1.3.). 0
Hence as long as V is not a zeroscheme, we may view the modules (Mi)(V) as measuring the failure of V to be arithmetically CohenMacaulay, as indicated above. There is another related kind of deficiency measured by the first deficiency module (i.e. taking i = 1). That is, (Ml) (V) in particular measures the failure of V to be projectively normal, i.e. the failure of the restriction map to be surjective for all d. If dim V = 0, it is still true that H; (Iv) measures the failure of the restriction map to be surjective, even though V is automatically arithmetically Cohen-Macaulay. For this reason we do not define the deficiency module for zeroschemes ; we only define the modules (Mi)(V) for 1 ::; i ::; dim V. We will see some other ways in which this module (and certain submodules) measures the failure of V (and its ideal) to satisfy certain properties. Some of these results will in fact be true also for the case dim V = O. If V is aCM then the rank of the last free module in a minimal free resolution of Iv (or, equivalently of SI Iv) is called the Cohen-Macaulay type of V. In particular, if the Cohen-Macaulay type is 1, then V is said to be arithmetically Gorenstein. If V is aCM and has codimension two, then the minimal free resolution for Iv is a short exact sequence:
~
\.
o
/'
Iv
/' \.
S-+SIIv-+O
o
Here B = [Fl ,···, Frl where the Fi are the minimal generators of Iv, and A is called the Hilbert-Burch matrix of Iv. For more on the Hilbert-Burch matrix and the aCM codimension two situation, see Section 6.1.
12
CHAPTER 1. BACKGROUND
Remark 1.2.4 An important fact is that if V is aCM, then the dual of the resolution for S / Iv is again a resolution, this time for a twist of the canonical module: the resolution
o --* Fn-r --* Fn- r- l --* ... --* Fl --* S --* S / Iv --* 0 yields the resolution
o --* S --* F lv --* ... --* F;:_r+ 1 --* F;:_r --*
Ext ~-r (S / Iv, S) --* 0
II
Kv(n + 1) To see this, use the fact that Ext~(S/ Iv, S) = 0 for i < codim V (cf. [62]; see also the proof of Proposition 5.2.10). In particular, the CohenMacaulay type is the number of minimal generators of the canonical module, and if V is arithmetically Gorenstein then the canonical module is cyclic. (In fact, more is true. See Chapter 4.) 0 Now suppose that V is locally Cohen-Macaulay and equidimensional of dimension r . Then a minimal free resolution of S/ Iv has the form D
L'n-l
\.
fn-l
---t
Kn-
/'"
°
/'"
~ Fl ~ S
\.
l
\.
/'"
°
K2
J:4
/'"
\.
\.
/'"
I
S/Iv--*O
/'"
\.
°
0 0 It then follows from the local version of the Auslander-Buchsbaum theorem that Ki is locally free for i ~ n - r. A related and important function of the deficiency modules is that they determine whether the scheme is locally Cohen-Macaulay and equidimensional. Theorem 1.2.5 Assume that dim V = r ~ 1. Then V is locally eM and equidimensional if and only if the modules (Mi)(V) have finite length for 1 :::; i :::; r . This theorem can be found in [189] Theorem 9 or [95] (37.4). As a very special case, notice that if a scheme V is arithmetically Cohen-Macaulay then it is locally Cohen-Macaulay, but the converse of course does not hold. Notice also that (Mi)(V)j = 0 automatically for all j »0. (This is a theorem of Serre; see for instance [88] Theorem 111.5.2 (b)). Hence V
1.2:
THE DEFICIENCY MODULES (Mi)(V)
13
will fail to be locally Cohen-Macaulay and equidimensional if and only if one of its deficiency modules is non-zero for infinitely many components
in negative degree. These modules are also related to the question of connectedness. For example, we have
Theorem 1.2.6 Let V be a closed subscheme of pn.
(a) If V is reduced and connected then (M1 )(V)o =
o.
(b) If V is reduced then dim(M1)(V)0 = (number of connected components of V) -1. (c) If dim (M1)(V)0 = 0 (e.g. if V is arithmetically Cohen-Macaulay) then V is connected. Proof: (a) is Lemma 4.4 of [63]. Now consider the exact sequence
0-+ HO(Iv) -+ HO(Opn) -+ HO(Ov) -+ (M1)(V)0 -+
o.
If V is reduced and connected then from (a) we get that hO (Ov) = 1. So applying this to each connected component of V and using this exact sequence again gives (b), and we similarly get (c) since in any case hO(Ov) ~ 1. D
We will also prove later that every arithmetically Buchsbaum curve in jp3 is connected, except for two skew lines. (See Definition 1.5.7 and Corollary 3.1.4.) This implies that every Buchsbaum scheme of higher dimension is connected, since the Buchsbaum property is preserved under hyperplane sections (see Definition 1.5.8). In the case of curves in jp3, a useful connection between the deficiency (= Hartshorne-Rao) module M(C) and the ideal, Ie, of the curve is the following theorem of Rao ([179] Theorem 2.5):
Theorem 1. 2.7 Let C be a curve in]P'3 and let M (C) be its deficiency module. Let M (C) have a minimal free resolution
o -+ L4~L3 -+ L2 -+ L1 -+ L o -+ M(C)
-+ O.
Then Ie has a minimal free resolution of the form 0-+ L4 (0"4 ,0) L3 EB for some integers ei, Ii, r, m.
r
m
1
1
EB S( -Ii) -+ EB S( -ei) -+ Ie -+ 0
14
CHAPTER 1. BACKGROUND
Under certain circumstances (for instance if C is minimal in its even liaison class with respect to degree), then it can furthermore be shown that the direct summand EB~ S( -li) does not occur. This is a special case of the work of Martin-Deschamps and Perrin [137], about which we will say more, especially in Chapter 6. Finally, it is natural to ask, what modules of finite length can be the deficiency module of a curve? More generally, what collection of r modules of finite length can be the deficiency modules of a closed subscheme of dimension r? The answer is "all" if you allow for a sufficiently large rightward shift of the modules (making the same shift for each module). In co dimension two this is due to Evans and Griffith [69], who showed that in fact the subscheme can be taken to be reduced and irreducible. Rao [179] also proved this for curves in ]p'3 , and showed that regardless of the choice of module, the curve can be taken to be smooth (again, allowing a sufficiently large rightward shift) . Rao's approach motivated similar constructions by Martin-Deschamps and Perrin [137] (again curves in ]p'3) and by Bolondi [27] (for surfaces in JIM). Finally, in the case of arbitrary codimension, this was proved by Migliore, Nagel and Peterson [153], and again the subscheme can be taken to be reduced and irreducible. The question remains whether, given a collection of graded S-modules of finite length, it is necessarily the collection of deficiency modules of a scheme of appropriate dimension (without shifting to the right). The answer is "no." The point (which is an important ingredient in the structure theorem for codimension two even liaison classes, as we will see) is that there is a "leftmost" shift of the modules for which they form the deficiency modules of some scheme. We will see in Chapter 3, with Basic Double Linkage, that once the collection of modules actually is the set of deficiency modules of a scheme then any rightward shift is also the collection of deficiency modules of some scheme (see Remark 3.2.4 (b)). This is not true for leftward shifts. We summarize this with the following result.
Proposition 1.2.8 ([34]) Given a collection {Ml' .. . ,Mr } of graded Smodules of finite length (1 ::; r ::; n - 2), there is a scheme X of dimension r with the following properties: (a) There is an integer d such that (Mi)(X) l::;i::;r .
(b) If Y is a scheme of dimension r with (Mi)(y) = Mi(e) for all 1 ::; i ::; r then e ::; d.
1.2.
THE DEFICIENCY MODULES (Mi)(V)
15
(c) For any integer e with e :::; d, there exists a scheme, Y, of dimension r with (Mi)(y) = Mi(e) for all 1 :::; i :::; r. Note that the integers d and e above are the same for all i; that is, the modules in the collection are all shifted together.
Recall that M(d) is a shift d places to the left, for d positive. We will not prove this proposition here, but see Remark 1.3.11 (c) where we give the main idea in the case of curves; however, the main idea for the general case is to reduce to the case of a curve, by taking hyperplane sections.
Remark 1.2.9 The original statement of Proposition 1.2.8 in [34] assumed that we were given an even liaison class rather than a collection of graded S-modules of finite length. As stated here, part (a) actually relies on the more recent result of [153] mentioned above. The point of the result, though, is that once a scheme exists with the given deficiency modules (up to shift), there is a limit to how far you can shift the modules to the left and still hope that a scheme will exist with deficiency modules in the new shift. Note that the scheme, X , guaranteed by Proposition 1.2.8 need not be reduced or irreducible. 0 In the context of curves, Proposition 1.2.8 says that once the module is known (up to shift) then there is a lower bound on the degree in which it can start. In Remark 1.3.11 (c) we give the details. It is given purely in terms of the dimensions of the components of the module. Sharp examples can be given (e.g. a double line in Jr4; cf. [143]), but it is not sharp for most modules. Martin-Deschamps and Perrin have given a more detailed analysis [137] of the leftmost shift attained by any given module, for curves in jp3, by studying the minimal free resolution of the module. Another interesting problem is to bound the module (in various senses) in terms of the degree and (arithmetic) genus of the curve. For curves in jp3, some useful results can be found in the paper of Martin-Deschamps and Perrin [139]. Here they give explicit bounds on the dimensions of the components of the deficiency module in terms of the degree d and (arithmetic) genus g of the curve C c jp3. They do not assume that the curve is reduced or irreducible. In particular, they give a lower bound for the degree ra (C) in which the deficiency module starts, and an upper bound for the degree r 0 (C) in which it ends, and they give a bound for the dimension of any component. Specifically, they show r a (C) ~ 9 + 1 -
(d - 2)(d - 3) 2
ro (C) :::;
d(d-3)
2
- g,
CHAPTER 1. BACKGROUND
16 and if C is not a plane curve then 'r/n E Z, h1(Ic(n)) ~
(d - 2)(d - 3) 2 - g.
This was generalized to curves in jpn in [51]. Again, sharp examples exist ([139] gives an example; double lines provide another; see Example 1.5.10 and [143]). In fact, those curves for which these bounds are achieved are well understood. They are the so-called extremal curves described in [66] and [140] . Both [66] and [140] apply these results to obtain interesting consequences for the Hilbert scheme H(d, g). For instance, it is shown in [140] that if d 2 6 and g ~ (d-3~d-4) + 1 then H(d, g) is not reduced. In fact, there exists an irreducible component which is not generically reduced, but such that the underlying reduced scheme is smooth of dimension ~d(d - 3) + 9 - 2g. Ellia [66] showed furthermore that if C is neither aCM nor extremal then stronger bounds hold. Nollet [170] then improved these bounds and showed that they are sharp, and he classified the curves (called subextremal curves) which achieve these stronger bounds. In the case of an integral curve C in jpn , ra(C) 2 1 (cf. [63] and Theorem 1.2.6) and ro(C) ~ d - n (cf. [82]). (In fact, in [82] it is shown that r o( C) = d - n if and only if C is smooth and rational and either d = n + 1 or else d > n + 1 and C has a (d + 2 - n )-secant line.)
1.3
Hyperplane and Hypersurface Sections
Given a subscheme V C jpn of dimension 2 1 with defining (saturated) ideal Iv, and given a sufficiently general homogeneous polynomial F of degree d, we would like to define the subscheme Z = V n F cut out by F. (In Section 1.4 we will see what happens when dim V = 0.) We would also like to specify what "sufficiently general" means. We can view V n F as a subscheme of jpn, or as a subscheme of F (thought of as a hypersurface) . By abuse of notation, we will use the letter F for both the hypersurface and the polynomial. If d = 1 we can also view Z = V n F as a subscheme of jpn-l. Throughout this section we will use the notation R = Sj(F) . Here "define V n F" means to give the saturated ideal, either in S or in R. The natural "guess" is to form the ideal Iv + (F) c S, or Iv(t\F) C R. These aren't quite right: in general we need to take the saturation of these ideals. (In Chapter 2 we will discuss "how far" these ideals are from already being saturated, and show that the failure is measured by a submodule of the first deficiency module.)
1.3. HYPERPLANE AND HYPERSURFACE SECTIONS
So we define the hypersurface section, Z, of V by F in scheme with saturated ideal
17 ]pn
= Iv + (F),
fz
to be the
(1.2)
and in the hypersurface F we have the saturated ideal
I
_ (Iv
ZIF -
+ (F)) (F)
(1.3)
in the ring R. If d = 1, this is called the hyperplane section of V by F. How are these related to each other and to Iv? We first observe that
Iv
+ (F) (F)
~
Iv Iv n (F) Iv F·Iv
Iv F · [Iv: FJ
(for any F)
(1.4)
(for F sufficiently general).
Indeed, the isomorphism is standard (cf. [7] p. 19) . For the first equality, recall that [Iv : FJ = {G E S IF· GElv} ; this is a special case of an ideal quotient (see page 47). Then the first equality is a tautology. The second equality is by generality of F, and is not true for all F. "General" here will mean "not vanishing on any component of V." Suppose that
is the primary decomposition of Iv in S. Since Iv is saturated, m is not an associated prime. Choose F so that it is not in any associated prime Pi (i.e. it does not vanish on any component of V). Now, if FG E Iv, it is in each primary ideal Qi' But no power of F is in Qi, so G E Qi for each i, hence G E Iv. As a result , we get two useful exact sequences of sheaves:
o -+ Q IPn( -d) ~ Iz -+ IZIF -+ 0 (from sheafifying 0 -+ (F) -+ Iv
+ (F) -+ Iv(;)Fl -+ 0)
o -+ Iv( -d) ~ Iv -+ IZIF -+ 0 (from sheafifying 0 -+ Iv( -d) ~ Iv -+
i1v
-+ 0).
(1.5) and
(1.6)
CHAPTER 1. BACKGROUND
18
Remark 1.3.1 (a) Notice that (1.5) implies H!(Iz ) ~ H!(IzlF ) for 1 ::; i ::; n - 2.
(1.7)
This fact is often useful. (b) From (1.5) we also have the short exact sequence
o -t S( -d)
~ Iz -t IZIF -t O.
Hence one can verify that there is a one-to-one correspondence between the minimal generators of IZIF and those of Iz other than (possibly) F itself. (c) We now claim that F is a minimal generator of I z. Indeed , first let VI be an irreducible component of V, of degree d l , corresponding to a primary ideal Ql with associated prime Pl' Let Zl C Z be the hypersurface section of VI cut out by F. By assumption, dim Zl = dim Vl -1 and deg Zl = dd l . Now suppose that F = L X iFi where Fi E I z . If Fi ~ PI then Fi cuts VI in a subscheme Z' of dimension dim Zl and degree (deg Fi)(dd < dd l . But on the other hand, Fi E Iz C IZ1 so Zl C Z' (exercise). But then deg Zl ::; deg Z' , contradicting the degree calculations above. Thus Fi E Pl. The same is true of all the other components of V. That is, Fi vanishes on the support of each component of V . This holds for all i, so F also vanishes on the support of each component of V. This contradicts the generality of F. (d) Notice that if H;(Iv) = 0 (e.g. if V is aCM), then in particular we have
Iv / [Iv ( -d)]
~
lzlF
~
lz/(F) .
The second isomorphism comes from (1.5) and is true regardless of whether or not (Iv) = O. The first isomorphism comes from (1.6), and it implies that the generators of lzlF are in bijective (degree preserving) correspondence with those of Iv. 0
H;
Remark 1.3.2 The point of the exact sequence (1.6) , and indeed most of this section, is to study what happens when we restrict elements of Iv to the hypersurface F . The sequence (1.6) holds whenever F does not vanish on any component of V . However, even in this case it is not necessarily true that if G l , ... , G k E Iv form a regular sequence in S then their images in lzlF form a regular sequence in R = S/(F). This is only true if F is sufficiently general. For instance, if V is a twisted cubic in jp3, there is a regular sequence of two quadrics G l and G 2 containing V, and
1.3. HYPERPLANE AND HYPERSURFACE SECTIONS
19
the complete intersection of these two quadrics defines the union of V and a line A. Suppose deg F = d = 1. For a general F E 51, the images of G 1 and G 2 in IzIF do form a regular sequence in R. However, if F is a plane which contains A then these images have a common component (namely A itself) and so do not form a regular sequence in R. See also Theorem 1.3.6. 0 Now that we have the notion of a hyperplane or hypersurface section, it is natural to ask what properties are passed from a closed subscheme Vof ]pn to its (general) hyperplane or hypersurface section, and what properties of the general hyperplane or hypersurface section force a conclusion about V. In particular, when can we deduce that V is arithmetically CohenMacaulay? In some sense the most interesting aspect of this question is the case when V is a curve, and we will discuss this more in Chapter 2. However, now we give a result for higher dimension. This is also a special case of a result in [105].
Theorem 1.3.3 Let V be a locally Cohen-Macaulay, equidimensional closed subscheme of]pn and let F be a general homogeneous polynomial of degree d cutting out on V a scheme Z eVe ]pn. Assume that dim V 2 2. Then V is arithmetically Coh en-Macaulay if and only if Z is. Proof: Consider the exact sequence in cohomology obtained from (1.6):
where 1 :S i :S (dim V) - 1. Notice that Z has dimension equal to (dim V) - 1. If V is aCM then (Mi)(V) = 0 for all 1 :S i :S dim V so H!(IzlF) = 0 for aliI :S i :S dim V -1. But then by (1.7) H!(Iz) = 0 for 1 :S i :S dim Z, so Z is aCM. Conversely, if Z is aCM then Hi(Iv(t - d)) .:::..4 Hi(Iv(t)) is surjective for all t and Hi+1(Iv(t - d)) .:::..4 Hi+1(Iv(t)) is injective for all t. This is impossible unless V is aCM, since (Mi)(V) has finite length by Theorem 1.2.5. 0
It is interesting to note that Chang has shown in [48] that at least in the case where V has co dimension two and d = 1, this theorem does not hold if "aCM" is replaced by "Buchsbaum" (defined in Section 1.5). Indeed , she gives an example of a non-Buchsbaum 3-fold whose general hyperplane section is Buchsbaum. (This is the only direction in which such an example
20
CHAPTER 1. BACKGROUND
is possible - as we will see in Definition 1.5.8, the general hyperplane or hypersurface section of a Buchsbaum scheme is again Buchsbaum.) Note also that without the assumption that V is locally CohenMacaulay and equidimensional, Theorem 1.3.3 would not be true. For instance, V could be the union of an aCM surface and a point. We also have the following basic result . Proposition 1.3.4 Let V be a locally Cohen-Macaulay, equidimensional, closed subscheme of IF' of dimension 2': 1, and let F be a general homogeneous polynomial of degree d cutting out on V a scheme Z eVe IF'. The following are equivalent.
(a) H!(Iv)
+ (F) is Iv + (F) .
(b) Iv (c)
= 0;
(F)
a saturated ideal in S;
.
.
zs a saturated zdeal zn R .
Moreover, if these conditions hold then the Hilbert function of the hypersurface section is
H(Rj IZIF, t) = H(Sj Iv, t) - H(Sj Iv, t - d).
Proof: The equivalence of (b) and (c) comes from taking cohomology on the exact sequence (1.5) and comparing with the exact sequence
o ---t (F)
---t
Iv
+ (F)
---t
Iv
+ (F) (F)
---t
o.
The equivalence of (a) and (c) comes from the exact sequence (1.6) and the fact that both (a) and (c) imply that
o---t Iv( -d) ---t Iv ---t H2(IzlF ) ---t 0
(1.8)
is exact. To prove that (c) implies (a) we use the consequent injectivity of the map H;(Iv)(t - d) ---t H;(Iv(t)) for all t, and argue as in Theorem 1.3.3. The Hilbert function computation comes from the exact sequence (1.8). 0 Corollary 1.3.5 Under the hypotheses of Proposition 1.3.4, the Hilbert function of the hypersurface section is independent of the choice of hypersurface, as long as the degree of the hypersurface remains fixed and the hypersurface meets V properly.
21
1.3. HYPERPLANE AND HYPERSURFACE SECTIONS
We would now like to compare the graded Betti numbers of an aCM scheme V with those of its hypersurface section, Z. The latter can be taken to mean the graded Betti numbers of the S-resolution of lz or those of the R-resolution of IzlP (which are not the same). We mention two results.
Theorem 1.3.6 Under the hypotheses of Proposition 1.3.4, the minimal free S-resolution of Iv and the minimal free R-resolution of lzlP have the same graded Betti numbers. Proof: The basic idea is to start with the minimal free resolution of Iv and taking the quotient by F. One way to see it is to build up the resolution of lzlP as was done in Section 1.1, as follows. Since H; (Iv) = 0, we have the short exact sequence
o -7 Iv( -d) -7 Iv -7 lzlP -7 0 (this is just (1.8)). Suppose we have a minimal free resolution
o -7 Fk -7 Fk - I
---+
-7
\. /'
\. /'
K k-
o
-7
SjIv
-70
Iv
2
/' \.
S
o
0 .. ·0
/' \.
o
Note that the Ki are reflexive modules (cf. [90]), so in particular they are saturated. Since Iv is saturated it follows from the exact sequence
o -7 KI -7 FI -7 Iv -7 0 that H;(K I ) = 0 (where the sheaves are the sheafifications of the corresponding modules arising from a short exact sequence from the resolution above). Since the other Ki are saturated, we similarly get the vanishing of the first cohomology of all the K i . We can now construct the minimal free resolution of lzlP as follows. First we build up a commutative diagram 0
t
0
xF
t
-7
KI(-d)
--+ KI
0 -7
FI ( -d)
0 -7
Iv( -d)
0
t t t
0
-7
(KdF
-7
0
--+ FI
-7
(FdF
-7
0
~ Iv
-7
lzlP
-7
0
xF
t t
t
0
22
CHAPTER 1. BACKGROUND
where the last column is the restriction to the hypersurface F, and in particular (Fl)F is a free R-module of the same rank as that of Fl . Since Kl is saturated with H;(Kd = 0, it follows that (KdF is saturated. The Snake Lemma gives the short exact sequence
Playing the same game with Kl we build up the commutative diagram 0
t
0 -+ K2(-d)
t
0 -+
F2(-d)
t
0 -+ Kl (-d)
t
0
t
~ K2 -+ (K2)F -+ 0 ~
t
F2 -+ (F2)F -+ 0
t
~ Kl -+ (KdF -+ 0
0
t
0
Applying the Snake Lemma and patching gives
0-+ (K2)F -+ (F2 )F -+ (FdF -+ IzIF -+
o.
Continuing in this manner, the resolution is built up as in Section 1.1 (see page 3). The only remaining question is that of minimality. But using the criterion for minimality on page 4, it is clear that in restricting to F, no scalars will arise in the matrices if they were not there originally. 0 Now we will view the hypersurface section Z of V (still aCM) as a subscheme of IF. The main fact which we would like to mention is that at least the Cohen-Macaulay type is preserved.
Proposition 1.3.7 Under the hypotheses of Proposition 1.3.4, the CohenMacaulay type of V is the same as that of Z (viewed as a subscheme of IF ). Proof: From the exact sequence
0-+ Iv( -d) ~ Iv -+ IzIF -+ 0 (cf. (1.6)) , a mapping cone (p. 4) gives a free S-resolution of IzIF from two copies of that of Iv , one twisted by -d. Then in a similar way the exact sequence 0-+ S( -d) -+ Iz -+ IzIF -+ 0
23
1.3. HYPERPLANE AND HYPERSURFACE SECTIONS
(cf. (1.5)) together with the Horseshoe Lemma (cf. [202]' 2.2.8, p. 37) gives the result. 0 We summarize some of these results as follows.
Corollary 1.3.8 Let V be an aCM subscheme of]p'n of dimension 2: 1.
Let F be a general homogeneous polynomial of degree d cutting out on V a scheme Z eVe r. Then (a) Z is also aCM; (b) in S we have the equality of saturated ideals /z
= Iv + (F) ; Iv + (F)
. . . (c) m R we have the equalzty of saturated zdeals /zIP =
(F)
;
(d) the Hilbert function of R/ /zIP is given by H(R/ /zIP, t)
= H(S/ Iv, t) -
H(S/ Iv, t - d);
(e) the graded Betti numbers of /zIP as an R-module coincide with the graded Betti numbers of Iv as an S-module; (f) the Cohen-Macaulay type of Iv is the same as that of I z . Proof: The only small additional observation is that if dim V = 1, then Z is automatically aCM since it is a zeroscheme. 0 Remark 1.3.9 The assumption that F is general, in particular that it is a non-zero divisor on S/ Iv, is important in Theorem 1.3.6, since the exact sequences used in the proof do not apply otherwise. Furthermore, consider the following examples as illustrations of some interesting behavior if this generality condition is not met. First, let C1 C H = ]p'3 C jp4 be a degenerate curve, where H is the hyperplane defined by X 4 . One cannot get the minimal free resolution of Ie! , as a R = k[Xo, .. . , X 3 J-module, just by taking a quotient by X 4 in a minimal free resolution of Ie! as a S = k[Xo , ... , X 4 J-module. For example, if C 1 is a line then its minimal free resolution over S is
0--+ S(-3) --+ 3S(-2) --+ 3S(-1) --+ Ie! --+ 0 while its minimal free resolution over R is
0--+ R(-2) --+ 2R(-1) --+ Ie! --+ O.
24
CHAPTER 1. BACKGROUND
Notice that the hyperplane is a minimal generator of the ideal I Cl as an S-module. (See also Example 1.5.1.) The former resolution can in fact be obtained from the latter via a tensor product. Now let C 2 be a rational normal curve in JIM, and let Q be a quadric hypersurface in JIM containing C2 • For convenience we denote by Q also the quadratic form defining the quadric hypersurface. Let R' = S/Q. Again, we cannot get the minimal free resolution of IC2 over R' by taking the quotient by Q in the minimal free resolution of I c2 . This time, though, the resolution over R' is not even finite! (See [61].) Here is a way to find the minimal free resolution with the computer program Macaulay [15]. First produce the ideal of the rational normal curve (e.g. using the script <monomial_curve). Then choose a random element Q in this ideal, and use the commands qring, fetch and nres. In using nres, though, be sure to specify how far in the resolution you want it to compute: nres I IRES 10, for example. In this example, the minimal free resolution has the form
... ---+ 8R'(-5) ---+ 8R'(-4) ---+ 8R'(-3) ---+ 5R'(-2) ---+
IC2
---+
o.
See also [62] pp. 377-378. D Remark 1.3.9 leads us to an observation about the deficiency modules of a degenerate subscheme of Jpn. The point is that if V c H = Jpn-l C Jpn then we may view V as a subscheme of Jpn or as a subscheme of Jpn-l. The natural question is whether the corresponding deficiency modules are the same, and what role the linear form defining H plays. Part (c) gives a way, in the case of curves, of building up V to a non-degenerate curve with the same deficiency module (possibly in more than one step).
Proposition 1.3.10 Let V c Jpn be degenerate, i.e. V c H = some hyperplane H. Let L be the linear form defining H. Then
Jpn-l
for
(a) The deficiency modules of V, viewing V as a subscheme of Jpn, are isomorphic to those of V viewed as a subscheme of Jpn-l . (b) L annihilates (Mi)(V) for alll
:S i :S dim V.
(c) Suppose that V is a curve. Let oX be a line such that oX n V is a reduced point, P, and such that oX is not contained in H. Let V' = V U oX. Then M(V) ~ M(V').
1.3. HYPERPLANE AND HYPERSURFACE SECTIONS
25
Proof: In a manner similar to (1.5), we get an exact sequence
o -t OlP'n(-I)
~ Iv -t IVIH -t O.
(1.9)
Then (a) follows immediately. For (b) we want to show that the multiplication induced by L on (Mi)(V)(-I) -t (Mi)(V) is zero. This follows from the exact sequence
o-t Iv(-I) -t OlP'n(-I) -t Ov(-I) -t 0, the exact sequence (1.9), the observation that the map Iv(-I) -t Iv factors through OlP'n (-1), and taking cohomology. This idea was used in [142] and [145]. Part (c) was proved in [51]. The main steps are (i) observing that Ip is generated by n linear forms, namely the n - 1 which define A plus L; (ii) observing that then Iv
+ 1;. =
Ip ;
(iii) using the exact sequence
o -t Iv'
-t Iv EEl 1;.. -t Iv
+ 1;. -t 0;
(iv) sheafifying and taking cohomology. Special care has to be given to the case of negative degrees, and we refer to [51] for the details. 0
Remark 1.3.11 (a) Many strong results are known about the general hyperplane section of an integral curve, especially from the point of view of the Hilbert function or the resolution of the points of the hyperplane section. The main results say that the general hyperplane section is a set of points such that two subsets of these points, of the same cardinality, are indistinguishable numerically - they have the same Hilbert function (i.e. they have the Uniform Position Property), and in fact the same graded Betti numbers (i.e. they have the Uniform Resolution Property). See for instance [59], [77], [83], [86], [87], [133], [186] among others. See also Remark 1.4.6 and Section 6.4.1. (b) We will now prove the main part of Proposition 1.2.8 for the case of curves (for simplicity). This was proved in [142] and independently in [189] by a different method. Specifically, we claim first that for any d ::; 0, dim M(C)d-l ::; dim M(Ck (But see (c) for a stronger statement.)
26
CHAPTER 1. BACKGROUND
Let C c Jpn be a locally Cohen-Macaulay, equidimensional curve. Let H be a hyperplane not containing any component of C. Following convention, we will denote by IcnH the ideal sheaf of the points of the hyperplane section in the projective space H = Jpn-I. (Above, this was denoted by IZIF ') Then for any d :::; 0, HO(IcnH(d)) = 0 and so we have the exact sequence (from (1.6))
It follows that hI (Ic(d - 1)) :::; hI (Ic(d)) . This proves the claim. In particular, M(C) has finite length so it has a last non-zero component. What the above argument shows is that this last component cannot be in negative degree. Hence given a graded S-module M of finite length, there is a leftmost possible shift for which M can be the deficiency module of some curve, M = M(C), and at the very worst this shift cannot place M entirely in negative degrees. (Recall that Rao [179] showed that a sufficiently large rightward shift of M is the deficiency module of some curve.) Of course, there may be no curve actually achieving the crude bound just given. The simplest example is the case where M is a 2-dimensional vector space concentrated in one degree. The above argument says that if M = M(C), then M cannot occur in degree:::; -1. But in fact, Corollary 3.1.4 says that this module must occur in degree ~ 2. This turns out to be sharp (see Section 3.3). See [137] for a detailed discussion of how to find the leftmost shift that actually occurs for a given graded module M. Furthermore, it was shown in [12] (in the generality of codimension two subschemes of projective space), [137] (for curves in JP3) and [36] (for codimension two subschemes of a smooth Gorenstein variety), that the schemes which achieve the leftmost possible shift are very special from the point of view of liaison. They are the so-called minimal schemes of the even liaison class. We will discuss this much more in Chapters 5 and 6. (c) In fact, it was shown in [142J that if M(C) has components in negative degree, then we must have dim M(C)i :::; dim M(C)i+I -1 for all i :::; -1. A nice example that achieves this bound for arbitrarily many negative degrees is a double line (see Example 1.5.10). 0
1.4
Artinian Reductions and h- Vectors
Throughout this section we consider only aCM subschemes of Jpn, and we shall make frequent use of Corollary 1.3.8. We also restrict ourselves
1.4.
ARTINIAN REDUCTIONS AND h-VECTORS
27
to hyperplane sections, i.e. taking deg F = 1. To stress this, we use the letter L rather than F for our general linear form. Let H be the hyperplane defined by L. Let V be an aCM subscheme of pn of dimension 2 1 , and let Z c H c pn be its general hyperplane section. Since we will usually only consider Z as a subscheme of H = pn-l, we usually will abuse notation and just think of Iz as an ideal in R = S/(L) . If f : Z -+ Z is a function , we denote by 6 the first difference function : 6f(t) = f(t) - f(t-1). We recursively define 6 i f(t) = 66 i - 1 f(t). Then Corollary 1.3.8 implies, in particular, that H(R/ lz , t) = 6H(S/ Iv, t). Furthermore, the aCM property and the graded Betti numbers are preserved in passing from V to Z . Notice that conversely, because V is aCM, knowledge of the Hilbert function and graded Betti numbers of Z let us deduce the corresponding information about V. Notice also that the co dimension of V in pn and that of Z in H = pn-l are the same. The idea of an Artinian reduction is to continue this process of taking hyperplane sections as long as possible. To set notation , let us now start with an aCM scheme V c pn of dimension c and let S = k[Xo, ... , Xn]. Using the ideas of Section 1.3 we can successively cut V with c hyperplanes defined by general linear forms L 1 , •.. , Le (or intersect with a general linear space of dimension c) , since we assumed at the beginning of Section 1.3 that dim V 2 1.) Let us define VI to be the hyperplane section of V cut out by L 1 , and Rl = S/(L 1 ) . Inductively we define 11;+1 to be the hyperplane section of V; cut out by Li+l' and Ri+l = R;/(Li+l) ' We have that Ve is a zeroscheme. Notice that Re/ I Vc has Krull dimension 1 and depth 1. In particular, a general linear form, L e+1 , will be a non-zero divisor on Re/lvc and we can consider the quotient as we did in Section 1.3. The corresponding hyperplane will miss the points of V, so the geometry of "hyperplane sections" is somewhat obscured and we will have to approach the problem algebraically. To simplify notation we now let L = Le+l and R = Re/(L). Let J = Iv + (L) 9:! Iv = Iv (L) Ivn(L) L·lv as in (1.4) on page 17. R/ J is again a Cohen-Macaulay ring, this time of dimension O. Its Hilbert function is given by H(R/ J, t) = Ll e+ 1H(S/ Iv, t). However, since the vanishing locus of J is empty, for high enough degree i, the degree i component Ji coincides with the "maximal" ideal of R, by Hilbert's Nullstellensatz (e.g. [53] p. 370). Hence the Hilbert function of R/ J is eventually zero, so it can be viewed as a finite sequence of positive
CHAPTER 1. BACKGROUND
28
integers: 1
Cl
C2
C3
...
Cr
0 ...
The sequence (l,Cl, . •. ,Cr ) is also known as the h-vectorofV, and R/J is called the Artinian reduction of V (or of the coordinate ring of V). In particular, it is an Artin ring. The same proof as in Theorem 1.3.6 shows that J has the same graded Betti numbers, as an R-module, as Iv has as an S-module, and the Hilbert function of V can be computed from the h-vector by "integrating." We can make this more formal as follows.
Definition 1.4.1 The series 00
Hv(t) = I>iti, i=O
with ai
= H(S/ Iv, i), is called the Hilbert series of V.
0
One can check that if d is the Krull dimension of S/ Iv then it is possible to write Hv(t) = P(t)/(l- t)d, where the coefficients of P(t) are the entries of the h-vector: P(t) = 1+c1t+C2t2+ . . ·+crtr . It is not hard to see, from this point of view, that the h-vector is obtained as described above, by taking the difference function d times beginning with the Hilbert function H(V, t). (In fact , the theory of Hilbert series is much more general than is described here. We refer the reader to [62] for more details.) We would like to stress, in this section, the great amount of information about V that is preserved in passing to the Artinian reduction (and also some idea of what is not preserved). We have seen that the graded Betti numbers are preserved, but they are not the only invariants of V which can be determined from knowledge of R/ J. In fact, it is not hard to verify that deg V = LCi and if V is non-degenerate then codim V
= Cl.
If V is an aCM curve, we can also determine the arithmetic genus of V as follows. This should be compared with (6.11) on page 178 for non-aCM curves, and the discussion following it, where an application is described. It should be noted that the first equality in the formula for the genus below does not depend on the aCM property, and instead holds for any curve. The aCM property gives the second and third equalities, and in particular the very geometric fact that to determine the genus in the aCM case, we only have to study the points of the hyperplane section.
1.4.
ARTINIAN REDUCTIONS AND h-VECTORS
29
Proposition 1.4.2 Let V c F be an aCM curve of degree d with Hilbert function H(V, t). Let r be the general hyperplane section of V, with Hilbert function H(r, t) = b.H(V, t) and h-vector (1, C1, C2,···, cr ), where Ci = b. 2 H(V, i). Let C:» 0, so that b. 2 H(V, C) = 0 (and hence b.H(V, C) = d). Then the arithmetic genus of V is given by l
Pa(V) = 2]d - b.H(V, i)J i=l
l
I]d - H(r, i)J i=l
l
L(i - l)ci
i=O
Proof: Let g = Pa (V). Then
dC - g + 1
Therefore
= H°(V,Ov(e))
(by Riemann-Roch & e:» 0)
=
(since H(C, -1) = 0) (since V is aCM)
H(V,C) Ef=o b.H(V, i) Ef=o H(r, i) g
= dC + 1 - Ef=o b.H(V, i) = dC+ 1- Ef=oH(r,i)
from which the first and second equalities follow since b.H(V, 0) = 1. The third equality was observed by Harris [86J, and follows from the first one. The details are left to the reader. 0 If dim V ~ 2, the genus above is called the sectional genus of V (cf. [21J p. 11). It is interesting to study which h-vectors can arise under reasonable restrictions on V, such as assuming that V is integral. This kind of question will be discussed further in Section 6.4.1, and we will use this information to ask which genera can occur for integral curves, and describe some work which has been done in this area, using liaison theory. Also, the question of which sequences of positive integers can arise as the h-vector of some aCM subscheme of F has been answered (cf. [192]' [74]). There are (at least) two other invariants of our aCM scheme V which can be read immediately from the h-vector. First, dearly the least degree of a hypersurface containing V (i.e. the initial degree of the saturated ideal of V) can be read off since up until that point, Rj J is equal to R. This invariant is important, for example, in liaison theory. It is often denoted by a(V) or a (Iv ). See Section 2.2 for more on this invariant.
30
CHAPTER 1. BACKGROUND
A second invariant that can be determined is the Castelnuovo-Mumford regularity of Iv (also referred to as the Castelnuovo-Mumford regularity of V) (cf. Definition 1.1.4 and Theorem 1.1.5). It is not hard to show, since V is aCM, that if the h-vector of V is (1, CI, C2, ••• ,cr ) then reg (Iv ) = r + 1. So, in particular, the homogeneous ideal of V is generated in degree ~ r+ 1. One invariant that is very important in liaison theory, but which cannot be determined from the h-vector, is the least degree i in which the degree i component of the ideal of V has a regular sequence of length equal to the codimension of V. However, this invariant is preserved under general hyperplane sections (see also Remark 1.3.2), and so it can be read off from the Artinian ring Rj J itself (and not just the dimensions of its components). Also, note that it is at least true that the regularity of Iv is an upper bound for this number. Example 1.4.3 Consider aCM curves in jp3 with h-vector (1 , 2, 1). Any such curve has degree 4 and arithmetic genus 1, and it is non-degenerate but lies on a 2-dimensional family of quadrics. Its regularity is 3, so its homogeneous ideal is generated in degree ~ 3. However, there are two kinds of aCM curves with this h-vector. The general one is the complete intersection of two quadrics. Its homogeneous ideal is thus generated in degree 2, rather than 3. However, there are also aCM curves with this h-vector consisting of the union of a line and a plane cubic curve, meeting at a point. For such a curve, the family of quadrics containing the curve all have a common component (namely the plane containing the cubic), and there is a third generator in degree 3. This is the maximum possible number of generators; see Dubreil's theorem, Corollary 2.2.3. So the least degree in which there is a regular sequence of length 2 is 3. See also Example 1.4.7. Notice that the assumption that V be aCM is very important. For example, ageneral choice C of four skew lines also has, as general hyperplane section, a set of four points with h-vector (1,2,1). However, C does not even lie on one quadric surface! Notice also that there exist hyperplanes meeting C properly, but in four collinear points. Hence those hyperplane sections have h-vector (1 , 1,1, 1). (Compare with Corollary 1.3.5.) The study of non-aCM curves (or higher dimensional objects) via their general hyperplane section is central in algebraic geometry, and deficiency modules are very important in this study. 0 Remark 1.4.4 There is one situation when this invariant, the least degree i in which the degree i component of the ideal of V has a regular
1.4.
ARTINIAN REDUCTIONS AND h-VECTORS
31
sequence of length equal to the codimension of V, can be read off from the h-vector. This is when V is aCM of co dimension two and the two generators of least degree form a regular sequence. This happens, for example, when there is an irreducible element of least degree in Iv. This in turn is guaranteed if V is integral. See Remark 1.4.6. 0 The following notion was mentioned in Remark 1.3.11, and we now formally make the definition. See also Definition 4.1.8 for a related (weaker) notion. Definition 1.4.5 A reduced zeroscheme r in IF has the Uniform Position Property (U.P.P.) if any two subsets of r with the same cardinality have the same Hilbert function. 0 Remark 1.4.6 In the study of co dimension two subschemes of projective space, by passing to hyperplane sections it is often necessary to study zeroschemes in 1P'2. In this remark we collect many useful facts about zeroschemes in 1P'2, both in general and under the assumption that they are the general hyperplane section of an integral curve (possibly non-aCYl) in JP"l. Let r be a zeroscheme in ]P'2 with defining ideal I r = (Fl' F2 , ... , Fm), where deg Fi = di and d l :=:; d2 :=:; •• . :=:; dm . Assume that the degree of r is d. Let (1, Cl, C2, ... , cr ) be the h-vector of r, where Ci = tlH(r, i). If r is the hyperplane section of an aCM curve C in ]P'3, then the minimal generators of Ie have the same degrees as those of fr , and H(r , i) = tlH (C, i). If r is the hyperplane section of a non-aCM curve C then there is much less that can be said about the generators of Ie, but nevertheless many things can be said about C from knowledge of r. We have the following facts about r. (1) 1 :=:;
Cl
:=:; 2. We have
Cl
= 1 if and only if
r lies on a line.
r
(2)
L Ci = d = deg r.
(We saw this fact above.)
i=l
(3)
Ci
= i
+ 1 for 0 :=:;
i :=:; dl
-
1.
(4) Ci = d l for d l - 1 :=:; i :=:; d 2 - 1. In particular, the value of d l can be read from the Hilbert function.
(5) Cd2 < Cd2-l. In particular, combining with (4), we see that the value of d2 can be read from the Hilbert function.
32
CHAPTER 1. BACKGROUND
(6) Cj :::; Cj-l for all d1 :::; i :::; r. That is, the h-vector is non-increasing from degree dl on. This is a consequence of Macaulay's growth condition [130] for the Hilbert function of the Artinian reduction of the coordinate ring of r. See [54] and [22] for interesting geometric consequences of the situation where there is an equality CHI = Cj. In particular, it was first shown in [54] that if there is such an equality then the forms in Ir of degree i + 1 have a common factor. (7) Let j3 be the least degree in which Ir has a regular sequence of length 2. If j3 :::; i :::; r, then Cj < Cj-l. This is a consequence of (6) since the existence of a regular sequence precludes a common factor. (8) In any case we must have j3 :::; r + 1 because of the regularity. If j3 = r + 1, this just reflects the fact that Cr is the last non-zero entry in the h-vector. (9) If j3 = d2 then Cj < Ci-l for all i 2: d2 , by (7). In this case, we say that the Hilbert function of r is of decreasing type. This happens for instance if Ir has a form of degree d1 which is irreducible. (10) Iff has the Uniform Position Property (U.P.P.) then Ir contains an irreducible form of degree d1 ([73], [132]), and hence it contains a regular sequence of type (d 1 , d2 ) and its Hilbert function is of decreasing type. (11) ([86]) If C is any integral curve in IF (not necessarily aCM, but assume that the field has characteristic zero) and f is the general hyperplane section, then f has U.P.P (12) If Cis aCM and integral then the Hilbert function of its general hyperplane section (hence also that of C itself) is of decreasing type. (13) If C is integral and aCM then d2 is the least degree i in which the degree i component of Ie has a regular sequence of length 2; hence this degree can be read off from the h-vector. Most of these facts are easy to check. See for instance [77] for details and further references. 0
Example 1.4.7 We would like to illustrate some of the above facts and the importance of U.P.P. A simple example of a set of points without U.P.P. is a set Z with the following configuration:
• • • •
1.4. ARTINIAN REDUCTIONS AND h-VECTORS
33
(that is, three points on a line and one point off the line). The h- vector of Z is (1 , 2, 1) , which is also that of a complete intersection of type (2, 2) . Z clearly does not have U.P.P. since one subset of three points lies on a line while the ot her subsets of three points do not have this property. Here, d1 = d2 = 2, but the points do not lie on a complete intersection of type (2 , 2). The smallest complete intersection containing Z is one of type (2,3). See also Example 1.4.3. Notice again that the values of d1 and d2 can always be read off directly from the Hilbert function, i.e. from the h-vector. For instance, if Z is a zeroscheme in JP'2 with h-vector (1 , 2, 3, 3, 3, 3, 1) then d1 = 3 and d2 = 6. The U P . .P. property guarantees that there is a regular sequence of type (3,6). Without the assumption of U.P.P. the best we can do is use the fact that the regularity of I z is ::; 7 (cf. page 30), so at worst Z lies in a complete intersection of type (3,7). Finally, the h-vector (1 , 2,3,4,4,4, 4,2 , 2, 1) does not have decreasing type (because of the consecutive 2's) , so it cannot correspond to a set of points with U.P.P. 0 Remark 1.4.8 We end t his section wit h an observation about the connection between the h-vector and the canonical module of an aCM scheme. First, suppose that Rj J is zero-dimensional, where R = k[X 1 , . . . ,Xnl (note that we have n variables, not n + 1) . Hence one can view it as a graded R-module of finite length. Consider the minimal free resolution of RjJ,
° Fn -7
-7 ... -7
Fl
-7
R
-7
Rj J
-7
0,
and consider the dual complex
On the one hand, we have seen in Remark 1.2.4 that this is the minimal free resolution of the canonical module of Rj J, twisted by n. On the other hand, one can show that for a graded module M of finite length, the dual of the resolution is a minimal free resolution of the twisted k-dual module Homk(M, k)(n). (This fact is used in the proof of Theorem 1.2.7.) Hence the canonical module is the k-dual of Rj J. More precisely, if we denote by K the canonical module of Rj J , we have K ~ ExtR(RjJ, R)( -n) ~ Homk(Rj J, k)
(1.10)
as graded R-modules. In particular, the h-vector gives the dimensions, in reverse, of the components of the canonical module.
34
CHAPTER 1. BACKGROUND
Now let V be an aCM subscheme of F and let H be a general hyperplane. Combining Remark 1.2.4 and Theorem 1.3.6 (and its proof), we see that the "restriction to H" of the canonical module of V is the canonical module of V n H, twisted by -1. (See also [62].) Compare this with the classical adjunction formula (cf. [81], p. 147) for the canonical bundle Kv of a smooth analytic hypersurface V in a manifold M: Kv = (KM 0 [V]) Iv .
In this case [V] = OM(I). See Example 1.4.9. We make the following conclusion. If V is aCM then the h-vector of V gives the dimensions, in reverse, of the components of the canonical module of the Artinian reduction of V.
In particular, one can compute the dimensions of the components of Kv from the h- vector of V (or, more precisely, from the reverse of the h- vector) by "integrating" and shifting by 1. Again, see Example 1.4.9. 0 Example 1.4.9 We illustrate the observations in Remark 1.4.8 with a simple example. Let C be a twisted cubic curve in 1P'3, let Z be a hyperplane section of C, and let J be the ideal of the Artinian reduction. Notice that the h-vector of C is (1,2) . Setting Sl = k[Xo, Xl, X 2 , X 3 ], S2 = k[Xo, Xl, X 2] and R = k[Xo, Xd we have the resolutions 0-+ 2Sl ( -3) -+ 3Sl ( -2) -+ Ie -+ 0 0-+ 2S2 ( -3) -+ 3S2 ( -2) -+ lz -+ 0 0-+ 2R( -3) -+ 3R( -2) -+ J -+ o.
These give the following resolutions for the corresponding canonical modules: 0-+ Sl( -4) -+ 3Sl ( -2) -+ 2Sl ( -1) -+ Ke -+ 0 0-+ S2(-3) -+ 3S2 (-I) -+ 2S2 -+ Kz -+ 0 0-+ R( -2) -+ 3R -+ 2R(I) -+ K -+ O. One can then compute the dimensions of the components of the canonical modules: -2 -1 o 1 2 3 o 0 o 2 5 8 dimk(Ke)i o 0 2 3 3 3 dimk(Kz)i o 2 1 o o o dimk(K)i Without knowing the resolutions we could compute the dimensions of the components of the canonical modules as follows, using Remark 1.4.8. Knowing that the h-vector is (1,2) immediately gives us the last line, and any line above it is obtained by "integrating" and shifting to the right. 0
1.5. EXAMPLES
1.5
35
Examples
In this section we give a number of examples to illustrate the ideas in the preceding sections. Example 1.5.1 Complete intersections If V is a subscheme of IF of codimension r, then the saturated ideal of V clearly has at least r generators. If the number of generators is equal to the codimension then V is said to be a complete intersection. From the algebraic point of view, V is a complete intersection if and only if the saturated ideal Iv = (FI , " ' , Fr) where (FI , " ' , Fr) is a regular sequence. (In fact , it is not hard to show that a regular sequence always defines a saturated ideal.) The minimal free resolution of a complete intersection, known as the Koszul resolution, is particularly simple. It basically says that the only relations are the "obvious" ones, FiFj - FjFi, = 0, and similarly for the second and higher syzygies (see page 4). Formally, let di = deg Fi . Then the minimal free resolution is given by 2
0--+ S( -
L di ) ~ ... ~ 1\( EB S( -di)) ~ EB S( -di ) ..!4 Iv --+ 0
where
If we use the standard bases for t\2(EBI
2 is represented by an r x -matriX, each of wh;se entries is either zero or an Fi (up to ±1) . Similarly, the ith free module occurring in this resolution (1 ::; i ::; r) has rank (:) and the matrices all have entries which are either zero or an Fi (up to ±1). In particular, the projective dimension of Iv is equal to the codimension, r , so V is aCM. Hence the deficiency modules of V are zero (unless dim V = 0, in which case there are no deficiency modules, by definition). In fact, notice that in particular V is arithmetically Gorenstein (see page 11 and Chapter 4). One can check directly from the above observations that this resolution is self-dual, and in particular that the last matrix, CPr, is an r x 1 matrix whose entries are precisely the minimal generators of the ideal Iv. Compare with Proposition 4.1.1 and Corollary 4.1.3.
(;j
36
CHAPTER 1. BACKGROUND
The degree of a complete intersection is simply the product of the degrees of the defining polynomials (by Bezout's theorem). In the case of a complete intersection curve C, the arithmetic genus is given by
To see this, one can for instance take the general case first. Take each hypersurface section to be smooth, and so one can use the adjunction formula ([81] p. 147) to compute 2g - 2, and hence g. But the arithmetic genus depends only on the Hilbert polynomial, which is determined by the graded Betti numbers of the resolution. These depend only on the degrees di , and not on smoothness. Hence this value of 9 is correct for any such complete intersection curve. A special case of a complete intersection is a hypersurface (assuming that it is locally Cohen-Macaulay and equidimensional) since in this case the ideal is principal (hence isomorphic to S( -d 1 ); this was used in Section 1.3 to get (1.5)) and the codimension is 1. In co dimension two, say Iv = (F1' F2 ) is a regular sequence (i.e. F1 and F2 have no common component). Then the Koszul resolution for Iv is given by
The case of higher codimension proceeds analogously. D We now pass to the case of curves and compute some deficiency modules. Example 1.5.2 Let C be a set of two skew lines Al and A2 in IF. Consider the exact sequence
For i < 0 we get that M(C)i = 0 and for i = 0 we get dim M(C)o = 1. On the other hand, for i 2: 1 we claim that C imposes 2i + 2 independent conditions on hypersurfaces of degree i. Indeed, choose i + 1 distinct points on each of the two lines Al and A2. Clearly a hypersurface of degree i contains C if and only if it contains all of these 2i + 2 points. But excluding anyone of these points, it is a simple exercise to find a union of i hyperplanes which contains all of the remaining points.
1.5. EXAMPLES
37
Hence hO(Ic(i)) = (i~n) - (2i + 2) while of course hO(Ojpn(i)) = (i~n) and hO(Oc(i)) = 2i + 2. Therefore M(C)i = 0 for all i =J o. In particular, M( C) is non-zero but it is annihilated by the maximal ideal m of S. Because of this, C is an example of a so-called Buchsbaum curve, which we will discuss shortly (and in Chapter 3). 0 Example 1.5.3 Let C be a smooth rational quartic curve in JP3 . Consider again the exact sequence
By the Riemann-Roch theorem, hO(Oc(i)) = 4i+l for i ~ O. On the other hand, hO(Ic(i)) = 0 for i ~ 1 so this exact sequence gives M(C)i = 0 for i ~ 0 and dim M (Ch = 1. Now let H be a general hyperplane and consider the exact sequence
0-* HO(Ic(i - 1)) -* HO(Ic(i)) -* HO(IcnH(i)) -* M(C)i-l -* -* M(C)i -* Hl(IcnH(i)) -* H2(Ic(i - 1)) -* ... where IcnH is the ideal sheaf of the hyperplane section in the hyperplane H. (This is the sequence (1.6) of Section 1.3.) Observe that C n H is a set of four points in the plane with no three on a line. It is in fact a complete intersection, so the cohomology of IcnH is not hard to compute (from the Koszul resolution, which is a short exact sequence since the codimension is two). In particular, hI (IcnH(i)) = 0 for i ~ 2 (which gives us hO(IcnH(i)) = (i~2) -4 for i ~ 2). Furthermore, C has degree four and arithmetic genus 0 so it is not a complete intersection - from the formula in Example 1.5.1 we know that the complete intersection of two quadrics in JP3 would have arithmetic genus 1. Therefore, since C is smooth, it can lie on only one quadric and hO(Ic(2)) = 1. Then setting i = 2 in the above exact sequence gives M (Ch = 0, and proceeding to i = 3, 4, . .. gives that M(C) is zero in every degree other than 1, and it is one-dimensional in degree 1. Therefore a rational quartic has the same module as does a set of two skew lines, but shifted. This will be very relevant when we discuss liaison theory. Notice, incidentally, that there are deeper (but quicker) reasons why M(C)i = 0 for i ~ 2. One could apply the theory of liaison, which we will discuss later. Alternatively, one could use the main theorem of [82] (which says, in the special case of a reduced, irreducible curve in JP3, that M(C)i = 0 for i ~ deg C - 2), or the fact [98] that a general rational curve C in F of degree d has maximal rank, i.e. that the restriction map
CHAPTER 1. BACKGROUND
38
HO(OJl> n(i)) -+ HO(Oe(i)) has maximal rank for all i. (Using the exact sequence (1.1) and taking cohomology, one sees that this is equivalent to the condition that hO(Ic(i)) . hI (Ic( i)) = 0 for all i.) 0 Example 1.5.4 Let C be the disjoint union in ]p3 of a line .x and a plane curve Y of degree d (so Ie = h n Iy) . A special case is the disjoint union of two lines, which we discussed from a geometric point of view above. Now we show how to derive the deficiency module in a more algebraic way. We have an exact sequence
o -+ Ie -+ Iy EB h, -+ I y + h. -+ 0 and we can sheafify it and take cohomology to obtain
o
-+
Ie
-+
I y EB h.
--+ ~
/'
/'
~
S
-+
M(C)
-+
0
Iy +h
+
o
o
(Note that H~(Iy h) = S since Y and .x are disjoint - use the Nullstellensatz. Also, in particular .x and Yare complete intersections so as we have seen, H; (Iy EBI>.) = 0 .) Notice that the ideally + I>. has three (independent) generators in degree 1. Hence M(C) ~ k[xJl(F) where F E k[x] has degree d. In particular, ifO::Si.::Sd-1; otherWIse. Consider for instance the case d = 2. Here the deficiency module is one-dimensional in each of degrees 0 and 1, and zero otherwise. From the point of view of Section 1.1, what is the module structure? We have a homomorphism
o
0-+ HO(Ie) -+ HO(Ie(l)) -+ HO(IenH (l)) -+ M (C)o
4>o(L)
---'--'-t
M (Ch -+ ...
39
1.5. EXAMPLES
As long as H meets the curve in finitely many points, we have that rk ¢o(L) = 0 if and only if hO(IcnH (l)) = 1 if and only if the three points of intersection of C with H are collinear, i.e. if and only if H passes through the point P of intersection of A with the plane of Y. So the degeneracy locus Va is the plane in the dual projective space which is dual to the point P. Notice that initially we only considered planes meeting C properly. However, since the degeneracy locus is closed we get in this case that any linear form L vanishing on a component of C also gives a rank 0 homomorphism on M(C), because it also vanishes at P. See also Examples 5.5.1 and 5.5.2. One can also check that this plane determines the module structure of M (C) (although in general the degeneracy loci are not enough). A similar analysis can be done with the case d 2: 3 as well, with the same conclusions about Vo. Notice that the fact that the curve lies in jp3 is important here. For example, if d = 2 a similar curve lying in jp4 has a one-dimensional deficiency module. See Example 5.4.B. D Example 1.5.5 Suppose, in the last example, that we had let A and Y meet. If they lie in the same plane then C is a plane curve, hence a hypersurface and so arithmetically Cohen-Macaulay. The only other possibility is that they do not lie in the same plane and meet in exactly one point P. We claim that this curve is also aCM. In this case H?(Iy 1;..) = Iy + 1;.. = I p . One can see this since Iy C Ip and h, C Ip so I y + 1;.. C Ip. But on the other hand, Ip is generated by three independent linear forms and I y + h. contaim three independent linear forms, so they agree in degree 1 (we already haw one inclusion) and so Ip C I y + 1;... Therefore we get the exact diagram
+
Ip
o
-+
M(C)
-+
0
o
so M(C) = 0 and Cis aCM as claimed. This is a special case of Proposition 1.3.10 (c). 0 Example 1.5.6 The simplest case in Example 1.5.4 is the disjoint union of two lines in jp3, which we already discussed, and the second simplest is the disjoint union of a line and a conic. The latter has deficiency module which has dimension 1 in each of two components, and the module
40
CHAPTER 1. BACKGROUND
structure is nontrivial. (The multiplication from the first nonzero component to the second, induced by a general linear form, is an isomorphism.) For comparison, we give an example of a curve with the same module dimensions as this one, but different structure.
Figure 1.1: A curve with trivial multiplication Let C be the union of lines in the above picture. It is understood that lines which do not intersect in the picture in fact do not intersect, so it is an example of a so-called stick figure (see the discussion preceding Remark 1.5.9). One can verify from geometry that three "horizontal" lines and the five leftmost "vertical" lines lie on a smooth quadric surface, and that the last two lines do not lie on this surface. Then one can give elementary but tedious arguments to show that this curve has deficiency module which is one-dimensional in degrees 2 and 3, and 0 elsewhere. (This curve was constructed using Liaison Addition, which we will describe in Chapter 3. These dimensions come for free as a result of that theorem.) Also, it is not hard to check that the curve lies on no quadric or cubic surfaces, by geometrical considerations. However, the general hyperplane section of this curve does lie on a cubic plane curve. Then from the exact sequence
--+ M(Ch o(L~ M(Ch --+ ... we get that the general linear form, and hence arbitrary linear form, induces the zero homomorphism on the deficiency module. 0 The curve in Example 1.5.6 is the simplest non-trivial example of an arithmetically Buchsbaum curve. Specifically:
Definition 1.5.7 A curve C c F is arithmetically Buchsbaum (or simply Buchsbaum) if its deficiency module is annihilated by the maximal ideal (Xo,'" ,Xn) of S. 0
1.5. EXAMPLES
41
By "simplest non-trivial example" we mean that the curve of Example 1.5.6 is a curve of least degree which is Buchsbaum and which has a deficiency module with components that are non-zero in at least two degrees. (Obviously if the module is non-zero in at most one degree then the curve must be Buchsbaum for "trivial" reasons.) The minimality of this curve is not obvious: it is a consequence of the Lazarsfeld-Rao property which will be described later. See also [31], [32] and [75]; some relevant facts from these papers will be discussed in Chapter 3. Of course this curve is not the unique curve with this minimality property, since there is some flexibility in the lines chosen for the components of the curve. Note that if C is Buchsbaum then its deficiency module, as a graded S -module, is a direct sum of twists of copies of the field k; that is, the module structure is just that of a k-vector space. So, for example, the resolution of such a module is the direct sum of twists of copies of the resolution of k, and so is easy to write down. We will use this fact in the proof of Corollary 2.2.6. There is also a notion of arithmetically Buchsbaum subschemes of higher dimension, which is not quite what one would expect. The analogous definition, extending the one just given, is an inductive one:
Definition 1.5.8 A scheme V C lP'" of dimension r 2: 2 is arithmetically Buchsbaum if all of its deficiency modules (in the range 1 ::; i ::; r = dim V) are annihilated by the maximal ideal, and if its general hyperplane section is arithmetically Buchsbaum. A scheme for which the deficiency modules are all annihilated by the maximal ideal, without the assumption about the general hyperplane section, is said to be quasi-Buchsbaum. 0 Definition 1.5.8 is not the original definition, but it is the one that fits in most neatly with the topics in this book. See [195] for an excellent overview of the theory of Buchsbaum rings. See also [46] and [47] for a description of the arithmetically Buchsbaum subschemes of IF of co dimension two, and the connection to vector bundles. The condition that a scheme be quasi-Buchsbaum is not sufficient for the scheme to be Buchsbaum, since examples exist of schemes whose deficiency modules are annihilated by the maximal ideal, but for which the deficiency modules of the general hyperplane section are not annihilated by the maximal ideal (cf. for instance [159]). One can check, though, that an r-dimensional scheme V C lP'" is Buchsbaum if and only if its deficiency modules are all annihilated by the maximal ideal, and the same is true for the deficiency modules of the scheme
CHAPTER 1. BACKGROUND
42
V n A, obtained by intersecting V with a general linear space A of any dimension k satisfying n - r + 1 :S k :S n - 1. In this book, for the most part the only Buchsbaum schemes that we will consider are Buchsbaum curves. Buchsbaum curves can be viewed in many ways as a generalization of the aCM curves. Certainly from the module point of view it is clear that an aCM curve is Buchsbaum. There are also some other ways that come up, especially for curves in jp3, which we will discuss in Chapter 3 once we prove a striking result of Amasaki (Corollary 2.2.6) relating the k-vector space dimension of the deficiency module and the least degree of a surface containing the curve. We will also discuss a surprising fact about Buchsbaum curves in the last chapter: every Buchsbaum curve in jp3 specializes to a stick figure (i.e. a union of lines with at most double points). This will be an application of the Lazarsfeld-Rao property and Liaison Addition, giving a partial answer to a classical question attributed to Zeuthen: does every smooth curve in jp3 specialize to a stick figure? This question was recently answered negatively by Hartshorne [91] . See also Section 6.4.4.
Remark 1.5.9 As mentioned at the end of Section 1.1 , it is interesting to study the Buchsbaum index of a graded module M of finite length, i.e. the least degree k such that all forms of degree k annihilate M. If M is the deficiency module of a curve C, we also say that C is k-Buchsbaum. As a special case, a Buchsbaum curve is I-Buchsbaum (and an arithmetically Cohen-Macaulay curve is O-Buchsbaum). Notice that every curve is k-Buchsbaum for some k. k-Buchsbaum curves (and generalizations to higher dimensions) have been studied extensively. See for instance [8], [9], [38], [65], [80], [99], [100], [101], [150], [160], [164]. (This is not intended to be a complete list.) 0 Example 1.5.10 Another very nice curve in jp3 is a double line, i.e. a curve of degree 2 supported on a line. These are described in [87] pp. 32-34 and [143], and here we give the main points about its ideal and deficiency module. The paper [143] also gives a geometric description of its liaison class. A double line Yin ]p'3 has arithmetic genus -d for some d ~ O. Without loss of generality assume that Y is supported on the line A with defining ideal!;. = (X2' X3)' Then the homogeneous saturated ideal of Y is of the form
Iy
(I~,X2A(Xo,Xd
- X3 B (XO,Xd)
(Xi, X 2X 3, Xj, X2A(Xo , Xl) - X3 B (XO, Xd)
1.5. EXAMPLES
43
where A and B are homogeneous of degree d = -Pa(Y) with no common zero. If d = 0 then A and B are constants, and Y is a plane curve and hence is aCM. So from now on assume that d 2:: 1. The deficiency module M (Y) is of the form
M(Y)
rv
=
S (X2 ' X3, A, B) (d - 1)
rv
k[Xo, Xl]
= (A, B) (d - 1).
It follows that
0,
d'
1m
M(C). = { d + i t d- i
o
if i :::; -d; if -d:::; i :::; 0; if 0 < - i < - d', if i 2:: d.
For instance, if d = 1 then Y is linearly equivalent to two skew lines on a smooth quadric surface, and its deficiency module is one-dimensional; see Example 1.5.2 and also Corollary 5.3.4. More generally, the dimension and diameter of the deficiency module can be made arbitrarily large. 0 Example 1.5.10 was generalized in [155], where curves in Ipm were studied which were defined by (not necessarily saturated) ideals of the form (J2, Fl , ... , Fk ), with I the saturated ideal of a smooth curve and F E I. A special case of the results of [155] is the following: Theorem 1.5.11 Let C be the complete intersection in]p'3 of forms G l and G 2 , with deg G l = d l and deg G 2 = d2 • Let F = AG I + BG 2 be a form of degree d + 1 which is smooth along C, and let Y be the scheme defined by the ideal (lb, F). Then
(a)
(lb , F)
is saturated and Y has no embedded points;
(b) deg Y = 2d l d2, Pa(Y) = (3 - d)(d l d2) + 2d l d2(d l
+ d2 -
4)
+ 1;
(c) If d < d2 then Y is aCM; (d) A ssume that d 2:: d2 . Then
Note that the condition that A and B have no common zeros, in the defining ideal of a double line, corresponds in the above theorem to the assumption that F is smooth along C.
CHAPTER 1. BACKGROUND
44
Example 1.5.12 One can use the computer program "Macaulay" [15J to compute the deficiency module of a subscheme of projective space. For example, the following script computes (not necessarily in the most efficient way) the deficiency module of a curve in projective space, using scripts already written by D. Eisenbud and M. Stillman. The name of the script is def _mod.
incr-set prlevel 1 if #0=2 START HERE: incr-set prlevel -1 ••• Usage: <def_mod I def
••• •• •
... ••• •••
Computes the deficiency module of I . This assumes that I is an ideal that defines a variety of dimension one. The script returns the deficiency module in def. The deficiency module is the first cohomology module of the ideal sheaf .
•• • incr-set prlevel 1 jump END ERONE: shout echo The ideal needs to define a one dimensional variety kill ~zz ~I ~i ~tt jump HERE ERTWO: shout echo The first entry needs to be an ideal kill
~zz
jump HERE •• • Parameters and output values: I is an ideal . def is a presentation for a module . •• •
•• • START: nrows #1 ~zz if (~zz
l) ERTWO std #1 ~I
1.5. EXAMPLES
45
nvars @I @i codim @I @zz int @tt @i-@zz-1 if (@tt>1) ERONE if (@tt<1) ERONE int @j @i-1 <ext(-,R) @j @I @C std @C @CC <ext( - ,R) @i @CC @D std @D @DD copy @DD #2 std #2 #2 <prune #2 #2 std #2 #2 kill @zz @I @tt @i @j @C @CC @D @DD END: incr-set prlevel -1 A quick way to read the dimensions of the components of the module is to use hilb, as illustrated by the following simple Macaulay session (compare with Example 1.5.4):
%
%
46
CHAPTER 1. BACKGROUND
% hilb def 1 t
-3 t 3 t
0 1
2
-1 t -1 t 3 t
5 6
-3 t
7
1 t
8
t t t t t
0 1
1 1 1 1 1
codimension degree genus
3
2 3 4 4
5 6
Here, line is the ideal of a line, plane_quintic is the ideal of a (nonreduced) plane quintic disjoint from line, curve is their union and def is the deficiency module. To read off the dimensions and degrees of the nonzero components simply look at the part of the printout above codimension: this module is one-dimensional in each of degrees 0,1,2,3 and 4. Many extremely useful scripts have been produced by Eisenbud, Stillman and others, and are available with the program "Macaulay." 0
Chapter 2 Submodules of the Deficiency Module In this chapter we will define certain submodules of the first deficiency module of a closed subscheme Z ofm, and see how they give a more refined measure of the failure of the ideal of Z to have certain nice properties. We then describe some applications of these submodules from the literature.
2.1
Measuring Deficiency
Recall that we defined the first deficiency module of a closed subscheme
V of m to be H; (Iv). If dim V = 1 then this is unambiguously called the deficiency module of V. Since this chapter deals only with this particular module (and submodules thereof), we denote it simply by M(V). In Section 1.2 we saw one reason for the name "deficiency module": it measures the failure of V to be projectively normal. We now define a submodule of this module, which will be one of the main objects of study in this chapter, and we will see other reasons for the name. For now we do not require that V be locally Cohen-Macaulay or equidimensional, and we allow dim V = 0 (although in this case M(V) will not have finite length). Recall that if M is a graded S-module, A C M a submodule and J C S an ideal then by definition [A
:M
J] = { mE M
IF· mEA for all F
E
J }.
This is a submodule of M. If M = S then A is an ideal and we often write [A : J] for [A :s J] . This is called an ideal quotient. Now the submodule referred to above is given by
J. C. Migliore, Introduction to Liaison Theory and Deficiency Modules © Birkhäuser Boston 1998
48
CHAPTER 2. SUBMODULES OF THE DEFICIENCY MODULE
Definition 2.1.1 Let FE Sd be a homogeneous polynomial of degree d. Then we define Kp = [0 :M(V) (F)] . 0 In the special case where V is a curve in IP'" and deg F = 1, Kp is a key component in a formula for the arithmetic genus of V; see page 178. Proposition 2.1.2 Let F E Sd be a general homogeneous polynomial of degree d. Then (a) Kp has finite length. (b) If J
Iv
rv
= F. Iv =
Iv
+ (F)
then Kp( -d)
(F)
rv
!zIP
=J
rv
=
Iv
!z
+ (F)
Proof: Note that we have not assumed that V is locally Cohen-Macaulay or equidimensional. Let Z be the subscheme cut out on V by F and consider the exact sequence in cohomology obtained from (1.6) of Chapter 1
0-+ Iv( -d)~Iv -+!z1P
-----+ ~
/'
/'
~
~
M(V)( -d)
M(V)
Kp(-d)
o
o
Since IzlP is an ideal, (Kp)i is zero for i :S 0, and it is zero for i » 0 since this is true of M(V)i (regardless of whether or not V is locally CohenMacaulay or equidimensional). Hence Kp has finite length. This proves (a). For (b), the isomorphism defining J comes from (1.4) on page 17. The first isomorphism in the conclusion of (b) then also comes from the above exact sequence. For the second isomorphism consider the exact sequence
o -+ Iv n (F) -+ Iv EB (F) -+ Iv + (F)
-+ 0
which, after sheafification and taking cohomology, gives
[~]
0-+ Iv( -d) ---+ Iv EB (F)
-----+
\.
Iv
o
/'
!z
-----+
M( -d) ~ M
/' + (F) ~
o
o
where M = M(V) (note that (F) ~ S( -d) so its higher cohomology vanishes) from which (b) follows immediately. 0
49
2.1. MEASURING DEFICIENCY
Remark 2.1.3 The last two isomorphisms of Proposition 2.1.2 say that the submodule KF simultaneously measures the failure to be saturated for Ivt;.\F) in R = S/(F) and Iv + (F) in S. If V is aCM (for example) then these are automatically saturated. In the case of a curve V in ]pn , they are saturated if and only if V is aCM. In general, they are saturated if and only if H; (Iv) = O. 0
We now consider a submodule of K F , which was introduced in [148]. Definition 2.1.4 Let Fl , F2 E S be general homogeneous polynomials. Let A = (Fl' F2). Then KA = [0 :M(V) A] = KFl n K F2 . 0 It turns out that KA also measures a "deficiency." For two ideals I and J it is always true that I J c In J. Then it is natural to ask when we have equality, and furthermore whether there is a natural measure of the failure of equality to hold. In the special case where A is a codimension two complete intersection, we have the following result. See Remark 2.1.6 for comments about ways in which this has been generalized.
Proposition 2.1.5 ([148]) Let deg Fi = di (i = 1,2) and assume that A is sufficiently general (that is, A is the ideal of a complete intersection meeting V in codimension two (or disjoint from V)). Let R = S / (Fr) and J = Iv /(F1 • Iv) as above. Then K
A
~ [J
:R
J
F2 ] (d ) ~ Iv 1
n A (d
Iv . A
1
+
d )
2 ·
The first part of the proposition is a technical result which will be used in Section 2.2. The second one gives an answer to the question posed above: the module KA measures the failure of Iv· A to be saturated, i.e. to equal the intersection. (Recall that the intersection of two saturated ideals is again saturated.) The proof of this proposition is somewhat technical, but to a large extent it is a diagram chase using the commutative diagram
where M = M(V), F2 is the image of F2 in S/(F1 ) and r is the restriction of Iv to the ideallzlFl of the hypersurface section Z in S/(Fl ). See [148] for details.
50
CHAPTER 2. SUBMODULES OF THE DEFICIENCY MODULE
Remark 2.1.6 It might be tempting to conjecture that a more general version of Proposition 2.1.5 holds: if A is the ideal of an aCM subscheme which is disjoint from V and KA = [0 :M(V) A] then KA So! Ilv~; (with some shift). Unfortunately this is not true. For example, let A be the ideal of a twisted cubic curve Y in ]p3 (which is aCM) and let V be a smooth rational quartic curve in ]p3 disjoint from Y. Then from Example 1.5.3 we know that M(V)i = 0 for i "# 1 and dim M(Vh = 1. Therefore KA So! k, occurring in degree 1. On the other hand, we can check that ~V~; is at least 2-dimensional in degree 4. Indeed, knowing that M(V)4 = 0 allows us to use the first exact sequence in Example 1.5.3 to compute that hO(Iv(4)) = 35 -17 = 18. Then Y imposes at most (4)(3) + 1 = 13 conditions on this linear system, so dim (Iv n A)4 2 5. But dim (Iv . A)4 = (1)(3) = 3 2 2 (compute the number of quadrics in Iv and in A) so dim (~V~ and hence this cannot be a submodule of M(V) with any shift. Notice that this does not yet give any information about degrees other than 4, or whether our estimate in degree 4 is sharp. However, it turns out that Ilv~; is isomorphic to a submodule of a direct sum of copies of shifts of M(V). In [148] a generalization of Proposition 2.1.5 is given in the case where dim V ~ 1. Let A be the ideal of a codimension two aCM subscheme which is disjoint from V, and consider the minimal free resolution of A
t
r-l
0--+ EBS(-bj j=l
r
)
~ EBS(-ai) --+ A --+ 0 i=l
where ¢ is the Hilbert-Burch matrix of A (see page 11). Consider the induced homomorphism r-1
r
¢': EBM(V)(-bj ) --+ EBM(V)(-ai)' j=l
i=l
Then it is shown that ~V~; So! ker ¢'. In the example above, one can very quickly check that then ~V~; is isomorphic to EBL1 M(V)(-3), which agrees with the estimate above and shows that the ideals agree in all degrees other than 4. Proposition 2.1.5 has been generalized in a different way in [135]. Let I = Iv and J = Iw be the saturated ideals of any two subschemes of JIM (no longer assuming one is aCM) which are disjoint. Then II~f fits in a certain long exact sequence, from which many things can be deduced. In the case where J = A is aCM of co dimension two, we recover the result
51
2.2. GENERALIZING DUBREIL'S THEOREM
from [148] mentioned above. However, this is a very special case in the context of [135]. In particular, [135] gives a new proof of the following version of a theorem of Serre ([190] p. 143). Let I and J define disjoint subschemes V and W of IF as above. Then I J = In J if and only if dim V + dim W = n - 1 and both V and Ware arithmetically Cohen-Macaulay. Notice that Proposition 2.1.5 only assumed that the complete intersection defined by A meets V in the expected dimension. On the other hand, the results mentioned in this remark assume that V is disjoint from the scheme defined by A. It is natural to ask, for two homogeneous ideals I and J of 5, when is it true that I J = In J? One can see from Proposition 2.1.5 that this can be true sometimes even if the schemes are not disjoint and one is not aCM. For example, if W is a complete intersection but H;(Iv) =j:. 0, and surface in p4 and V is a surface with M(V) = if V and W meet in a finite number of points, then it is still true that IJ = In J. 0
°
2.2
Generalizing Dubreil's Theorem
The goal of this section is to show how the submodule KA introduced in Section 2.1 can be used to generalize a classical result of Dubreil. A special case of this result is due to Amasaki (see below) . In order to state Dubreil's theorem we first make
Definition 2.2.1 Let I be a homogeneous ideal of 5. Then (a) 0:(1) = min { i E Z I Ii =j:.
°}.
(b) 1/(1) = number of minimal generators of I.
0:(1) is sometimes called the initial degree of I. 0 Dubreil 's theorem, in its simplest form, is the following (cf. [55]).
Theorem 2.2.2 (Dubreil) Let R = k[x, y] and let I neous ideal. Then 1/(1) ~ 0:(1) + 1. 0
c R
be a homoge-
The standard application of Dubreil's theorem is the
c IF be a codimension two aCM subscheme with defining saturated ideal Iv. Then I/(Iv) ~ o:(1v) + 1.
Corollary 2.2.3 Let V
52
CHAPTER 2. 5UBMODULE5 OF THE DEFICIENCY MODULE
Proof: We saw in Section 1.4 that the graded Betti numbers, and in
particular the number (and degrees) of minimal generators and the initial degree, are preserved in passing to the Artinian reduction of the coordinate ring of an aCM sub schemeof]pn . Then the result follows immediately from Dubreil's theorem. 0 Another proof of Corollary 2.2.3 can be found in Remark 6.1.2. The following generalization of this result can be found implicitly in [1] and explicitly, with a different proof, in [76]. We remark that for a Buchsbaum curve C, the dimension N = dimkM(C) is sometimes called the Buchsbaum type of C.
Theorem 2.2.4 (Amasaki) Let C be a Buchsbaum curve in]p3 (cf. Definition 1.5.7)' Let N = dimkM(C). Then v(Ie) :s: o:(Ie) + 1 + N. 0 Note that formally this implies the corollary above in the case of curves in ]p3, since aCM curves are trivially Buchsbaum. In this section we will prove a generalization from [148] of Theorem 2.2.4. From now on, assume that V is a closed subscheme of ]p3, but of any co dimension ~ 2 and not necessarily locally Cohen-Macaulay or equidimensional. Assume that A = (L1' L 2 ) where deg Li = 1 (i = 1,2) and the Li are chosen generically. Let J = ~ L, ·Iv ~ Iv+(LJ} (L,) as above. Let J = 1+(L2) (L2) C k[x , y]. We have (2 .1)
Theorem 2.2.5 Let V C ]p3 be a closed subscheme of dimension :s: 1. Let L 1, L2 E 51 be generically chosen linear forms and let A = (L1' L2)' Then v(Iv) :s: o:(Iv) + 1 + V(KA)'
v(Iv)
v(J) = v(L;J) J) :L21) < V ( LdJ:L21 + V (h[J L2 J = v(J) + v (lJ;21)
<
v(J) + V(KA) 0: (Iv ) + 1 + V(KA)
(by (2.1)) (by Prop. 2.1.5) (by Dubreil)
0
This generalizes Amasaki's theorem above. This result has in turn been generalized in [135] to arbitrary co dimension in]pn, using Koszul homology,
2.2. GENERALIZING DUBREIL'S THEOREM
53
but the statement requires a good deal of explanation and notation and we will not give it here. We now show, following [76], how to deduce from Theorem 2.2.4 Amasaki's important theorem concerning the least degree of a surface containing a Buchsbaum curve in jp3 . (Amasaki's proof is along different lines.)
Corollary 2.2.6 (Amasaki) Let C c jp3 be a Buchsbaum curve and let N = dimkM(C) be the Buchsbaum type. Then a(Ie) ~ 2N.
Proof: We first recall a result of [39]. That is, v(Ie) ~ 3N + 1. Assuming this, the result follows immediately from Theorem 2.2.4 above: 3N + 1 ::; v (Ie ) ::; a (Ie ) + 1 + N. To prove that v(Ie) ~ 3N + 1, the idea of [39] is to use Rao's result, Theorem 1.2.7. Indeed, note that as an 5-module, k has a minimal free resolution
o -t 5(-4) -t 45(-3) -t 65(-2) -t 45(-1) -t 5 -t k -t 0 (it is isomorphic to 5/(Xo, Xl, X 2 , X 3 ), a complete intersection, so this is just the Koszul resolution). But since C is Buchsbaum, M (C) is isomorphic to a direct sum, with twists, of copies of k. (See page 41.) Hence the minimal free resolution of M (C) is just a direct sum, with twists, of this resolution. That is, a minimal free resolution of M(C) has the form
o -t F4 -t F3 -t F2 -t FI -t Fo -t M (C)
-t
0
where rk Fi = N . (~) . Now, Theorem 1.2.7 says that a minimal free resolution of Ie has the form r m
o -t F4 -t F3 EB EB 5( -Ii) -t EB 5( -ei) -t Ie -t 0 I
I
where m = v (Ie ) = rk F3 + r - rk F4 + 1 ~ rk F3 - rk F4 + 1 = 3N + 1. 0 C. Peterson [176] has used Theorem 2.2.5 to study the deficiency module of powers of certain ideals of curves in jp3 . In particular he has shown that the deficiency modules of these curves grow very quickly:
Corollary 2.2.7 (Peterson) Let C be a reduced, locally CohenMacaulay curve in jp3 which is not a complete intersection, and such that 12: is saturated with no embedded components for n »0. Let Cn be the curve defined by 12: . Then v(KA(Cn )) > sn 2 for some s > 0 and all
n» o.
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CHAPTER 2. SUBMODULES OF THE DEFICIENCY MODULE
Remark 2.2.8 (a) What can we say about r in the resolution above (without recourse to [137])? Observe that if C lies on a surface of minimal degree 2N then 3N + 1 ~ v(Ie) ~ a(Ic) + 1 + N = 3N + 1 so v(Ie) = 3N + 1. Therefore in this case r = O. This was observed in [76]. (b) Since Theorem 2.2 .5 is much more general than Theorem 2.2.4, and Theorem 1.2.7 holds for any (locally CM equidimensional) curve in 1P'3, the approach described above will always work, in principal, to get a lower bound for the least degree of a surface containing a curve in 1P'3 with given deficiency module (up to shift). However, to get a result as "clean" as Corollary 2.2.6 it must be possible for the invariants in the lower and the upper bounds of v(Ie) to be combined in a nice way. This was easy for the Buchsbaum case but not so easy in general. One case in which a reasonably nice answer was obtained (in [148]) is in the 2-Buchsbaum case. Recall from Remark 1.5.9 that a curve C is said to be k-Buchsbaum if M(C) is annihilated by all forms of degree k, but not by all forms of degree k - 1 (see also [65], [150]). In particular, consider k = 2. A simple example of such a curve is any non-Buchsbaum curve whose deficiency module is non-zero in exactly two components, in consecutive degrees (i.e. diameter two). Suppose C is such a curve, and suppose that the non-zero components of the module have dimensions a and b respectively. By liaison, which we will discuss later, without loss of generality we may assume that a ~ b. Then it can be shown that a(Ie) ~ 2b - a. Furthermore, for any choice of a and b, sharp examples can be constructed. (There is one exception, which we have already seen. If a = b = 1 then of course 2b - a = 1 cannot be sharp since a plane curve is always aCM. The disjoint union of a line and a conic is an example of a curve with these dimensions.) A complete answer can be given in the 2-Buchsbaum case in general, similar to but slightly more complicated than the Buchsbaum case. It is based on this special case of diameter two, and the fact (which is an interesting exercise) that the module of a 2-Buchsbaum curve decomposes as a direct sum of modules of diameter ~ 2. See [148] for details. (c) Note that Theorem 2.2.5 holds for any curve in ]p>3, not necessarily locally Cohen-Macaulay or equidimensional. Furthermore, it even holds for a finite set of points in 1P'3, which has codimension three. (d) A natural question is whether the converse of Dubreil's theorem holds: if C is a curve in ]p>3 with v(Ie) ~ a(Ie) + 1, then does it follow that C is aCM? The answer is "no." A simple example is the following. Let C1 and C2 each be the complete intersection of two quadric surfaces in
2.3. LIFTING THE COHEN-MACAULAY PROPERTY p3, and assume that C 1 and C2 are disjoint. Let C
55
= C1 U C2 •
Note that C 1 and C2 are each aCM, but C is not (since it is not connected; cf. Theorem 1.2.6). But it follows immediately from Proposition 2.1.5 (letting Iv = IC I and A = I c2 , and noting that KA = 0) that Ic = IC I • I c2 ; hence v(Ic) = 4 < 4+ 1 = a(Ie) + 1. Many other counterexamples exist, even with smooth irreducible curves. For instance, one can also check that if C is a smooth curve of degree 7 and genus 4, not lying on a quadric surface, then Ic has 4 minimal generators, and this is a(Ic) + 1. But C is not aCM. (e) Another natural question is whether this result as stated for subschemes of p3 would also hold for curves in jp4. The answer is "no." For example, take V to be a set of two skew lines in jp4. Here v (Iv ) = 5 but a(Iv) = 1 and V(KA) =dimkM(V) = 1. 0
2.3
Lifting the Cohen-Macaulay Property
For technical reasons, in this section we will assume that the characteristic of the base field is zero. (This is mainly used in the proof of a lemma which leads to Proposition 2.3.2; cf. [193], [105], [123], [149]). We saw in Theorem 1.3.3 that for dimension ~ 2, V is aCM if and only if any hypersurface section Z of V , not containing a component of V , is aCM. Of course one direction of this statement is not true for curves, since every finite set of points is aCM while not every curve is. So one would naturally like to know if there are any conditions on the hypersurface section Z which force the curve to be aCM. Furthermore, we saw in Corollary 1.3.8 that if V is aCM and F is any hypersurface not containing a component of V , then the subscheme Z cut out by F has the same Cohen-Macaulay type as V, whether viewed as a subscheme of F (hence considering the resolution over R = Sj(F) of IZIF) or as a subscheme of]pn (hence considering the S-resolution of Iz) . So any condition of the sort described in the last paragraph will force, as a result, that the curve have the same Cohen-Macaulay type as does the hypersurface section. In particular, we will shortly be interested in the problem of when the condition that Z is arithmetically Gorenstein forces the curve to be as well. Notice, incidentally, that if we know that Z is a complete intersection (in any dimension) and that V is aCM, then V must in fact be a complete intersection as well, not simply arithmetically Gorenstein (which is all that the Cohen-Macaulay property would guarantee). The "cleanest" result is the following (but 2.3.2 is more general). We
56
CHAPTER 2. SUBMODULES OF THE DEFICIENCY MODULE
will give a proof shortly.
Theorem 2.3.1 ([149]) Let V c F be a non-degenerate, locally CohenMacaulay, equidimensional curve. Let F E Sd be general, cutting out a zeroscheme Z. Assume that Z is a complete intersection. Then V is a complete intersection, unless one of the following holds. (a) n
= 3, d = 1, V
(b) n
= 3, d = 1, deg V = 4
lies on a quadric surface and has even degree. and V does not lie on a quadric surface.
(c) n> 3, d = 1, V is a double line. "Quadric" here means simply "of degree 2," with no implication about irreducibility. Hence for instance, (a) includes the possibility that V is a double line in JP3; such a curve lies on the union of two planes ([143]) and its general hyperplane section is clearly a complete intersection. As stated, the list is not intended to give necessary and sufficient conditions, however. For instance, the union of two plane curves (in different planes) in JP3 whose degrees are different, but of the same parity, satisfies the condition in (a) but does not have a general hyperplane section which is a complete intersection. It would not be too difficult to precisely identify the curves in (a), (b) and (c) whose general hyperplane section is not a complete intersection, but we will content ourselves with noting that "most" such curves do have such general hyperplane sections. What is really interesting is the fact that any curve which is an exception to the theorem must fall into one of these three categories, and we will prove this fact below. In particular, as soon as d ~ 2 then there are no exceptions. A nice application of this result can be found in Proposition 5.2.23. [149] is just one of a series of papers on this general subject. In [75] a very special case of this theorem was proved, and the more general question was asked for curves in JP3 and d = 1. This question was answered by Strano in [193]. Then Strano's student, Re, proved the analogous result for curves in F and d = 1 in [182]. This was re-proved in [105], and some generalizations were proved for Gorenstein curves, but always assuming d = 1. To our knowledge [149] is the first work in this direction for d ;::: 2. More recently, some results were obtained in [151] for higher Cohen-Macaulay type, and we describe one of these below. [149] is based on a "translation" of the preparatory results of [105] to allow for higher d, and it follows the same approach. Interestingly, it uses in a heavy way the module KF described in Section 2.1.
2.3. LIFTING THE COHEN-MACAULAY PROPERTY
57
Theorem 2.3.1 is a consequence of the following result from [149J (analogous to one in [105]), which gives a nice condition on the hypersurface section that forces the curve to be aCM. The notation is as follows. Let V c IF be a non-degenerate, locally Cohen-Macaulay equidimensional curve, and let F E Sd be a general homogeneous polynomial of degree d cutting out on V a zeroscheme Z. Consider the minimal free resolution of Iz (in IF), o -t Fn -t ... -t Fl -t Iz -t 0, and write Fn = $~~l S( -mi)' Let Kp be the kernel of the multiplication map on M(V) induced by F, as in Definition 2.1.1 and let K = Kp( -d) (K is the kernel of the map induced by F : M(V)( -d) -t M(V)). Let b = min{ j I K j =I O}. Equivalently, b is the least degree in which IZIF contains an element (necessarily a minimal generator) which does not lift to Iv (apply the cohomology functor to the exact sequence (1.6) of Chapter 1; see for instance the proof of Proposition 2.1.2) . We stress this last interpretation since it will be used several times in the proof of 2.3.1.
Proposition 2.3.2 Assume that V is not aCM, so that K =I O. Then b + n ~ min {mi} (where the mi are defined in the paragraph above). 0 The proof of this result is based on the approach of [105]; see [149] for details. The key technical step is the extension to higher degree d of their "Socle Lemma." We now give this extension (the Socle Lemma is just the case d = 1 of this result). For any graded S-module N we define the initial degree
a(N) =
inf {t E Z
I Nt =I O} .
See also Section 2.2. The initial degree is sometimes denoted by i(N) .
Lemma 2.3.3 Let M be a non-zero, finitely generated, graded Smodule. For d ~ 1 let F E Sd be a general homogeneous polynomial of degree d. Let
o -t K be exact. If K
-t
M( -d)-4M -t
=I 0 then let b = a(K)
= min{ j
I K j =I 0 } as above.
Then
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CHAPTER 2. SUBMODULES OF THE DEFICIENCY MODULE
We refer the reader to [149] for the proofs of Lemma 2.3.3 and its corollary, Proposition 2.3.2. See Example 5.5.3 for an illustration of another way in which Lemma 2.3.3 can be applied.
Proof of 2.3.1: Say Z is the complete intersection of hypersurfaces (in IF) of degrees al,"', an with al :S ... :S an. Since Z is a complete intersection, the minimal free resolution of Iz is the Koszul resolution. In particular, the last free part of this resolution has rank one, and the twist is - Lai (see Example 1.5.1). Hence the integer min{mi} in Proposition 2.3.2 is simply - L ai· Suppose V is not a complete intersection. Then V is not arithmetically Cohen-Macaulay. (If it were, the Cohen-Macaulay type would be preserved in passing to Z, so Z would not be a complete intersection.) Hence by Proposition 2.3.2 we have b + n ~ L ai . Notice that one of the ai is d (by Remark 1.3.1(c)) and one is b (by the discussion preceding 2.3.2, and Remark 1.3.1(b)) . We will denote by LO ai the sum al + ... + an - b, so we have LO ai :S n. Since lz has n minimal generators, we have two possibilities: either LO ai = n - 1 or LO ai = n. In the first case all the ai other than bare 1. But b is the least degree in which lzlP has a generator which does not lift to V, so here b = 1 as well since V is non-degenerate, and so deg Z = 1. It follows that deg V = 1. Impossible (V is non-degenerate) . Now suppose that 2::° ai = n. Then n - 2 of the ai are 1, one is 2 and one is b. Furthermore, one of these (other than b) is d. If n ~ 4 then at least two of the ai are 1. At most one of these corresponds to F (if d = 1) so at least one of the generators of IzlP has degree 1. Hence by definition b = 1 as well, and so deg Z = 2. Then the non-degeneracy of V means in particular that deg V> 1, so this forces d = 1, deg V = 2, V non-reduced (i.e. a double line). The last possibility is n = 3. Then Z is the complete intersection of surfaces of degree 1,2 and b. There are two cases. If b = 1 then the same reasoning as the last paragraph gives d = 1 and V is a double line. As mentioned already, this falls into category (a). The second case is b ~ 2. We claim that the definition of b then forces d = 1. Indeed, d is either 1 or 2; if it were 2, then lzlP has a generator of degree 1, and since b ~ 2,his means that this generator must lift to Iv, contradicting the non-degeneracy of V. So Z is a zero-scheme in ]p>2 which is the complete intersection of a conic (not necessarily reduced) and a plane curve of degree b. If b = 2 then deg Z = deg V = 4 and V may or may not lie on a quadric. If b > 2 then by the definition of b, V lies on
2.3. LIFTING THE COHEN-MACAULAY PROPERTY
59
a quadric and clearly has even degree. 0 A similar, but not as complete, analysis can be done for the case where Z is merely assumed to be arithmetically Gorenstein (i.e. having CohenMacaulay type 1). The following theorem summarizes much of what is known in this case (beyond Theorem 2.3.1):
c 11M be a curve and let Z be its general degree d hypersurface section. Assume that Z is arithmetically Gorenstein.
Theorem 2.3.4 Let X
(a) ([105]) If d = 1 and X is reduced and connected, not lying on a quadric hypersurface, then X is arithmetically Gorenstein. (b) ([203]) If d = 1 and X is integral and non-degenerate, and if X is not arithmetically Gorenstein, then Z is contained in a rational normal curve in the hyperplane and deg X == 2 (mod n - 1) . Conversely, given any integer m 2: n + 1 such that m == 2 (mod n - 1), there exists a smooth irreducible curve X C 11M of degree m which is not arithmetically Gorenstein but whose general hyperplane section is arithmetically Gorenstein. (c) ([149J) Assume that X is reduced, not lying on a quadric hypersurface. If n 2: 5 or d 2: 2 then X is arithmetically Gorenstein. (d) ([203]) Assume that X is reduced, irreducible and non-degenerate, and d 2: 2. Then X is arithmetically Gorenstein.
The proof is done via Proposition 2.3.2 and an analysis of the minimal free resolution of an arithmetically Gorenstein ideal. We conclude this section by illustrating how these kinds of results can be extended to higher Cohen-Macaulay type, giving a result from [151] concerning maximal rank curves. The point that [151] tried to develop is that the above results about arithmetically Gorenstein curves illustrate the more general phenomenon that if d is strictly greater than the CohenMacaulay type, then V is often forced to be aCM (and hence have the same Cohen-Macaulay type as its general hyperplane section) .
Definition 2.3.5 Let Z C 11M be aCM with codimension c and minimal free resolution r
o -+ EB S( -ai) -+ Fc - 1 -+ . .. -+ Fl
-+ lz -+ 0
i=l
where r is the Cohen-Macaulay type. We say that Z is level if all the ai are equal. 0
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CHAPTER 2. SUBMODULES OF THE DEFICIENCY MODULE
Theorem 2.3.6 ([151]) Let V c IP'n be a reduced, connected curve with maximal rank and assume that its general degree d hypersurface section, Z, is level. Let r be the Cohen-Macaulay type of Z. Assume that d ~ r. (a) If n = 3 then V is aCM unless d = 1 and V is a quartic curve with arithmetic genus 0 (in which case Z is a complete intersection). (b) If n ~ 4 then V is aCM.
Chapter 3 Buchsbaum Curves and Liaison Addition One of the goals of this book is to show how much information one can get about a scheme from knowledge of its deficiency modules. This is especially true for curves in jp3. (For instance see Rao's theorem, Theorem 1.2.7.) As an illustration of this connection, in this chapter we consider the case of Buchsbaum curves. Recall that a Buchsbaum curve is one whose deficiency module structure is trivial, i.e. it is annihilated by all linear forms (cf. page 40). In this chapter we will first apply some of the results and techniques of the previous chapters to describe some aspects of the theory of Buchsbaum curves. In Section 3.1, most of the results in fact hold for Buchsbaum curves in any projective space. We will then describe in Section 3.2 a useful construction known as Liaison Addition. This was introduced by P. Schwartau in his Ph.D. thesis (Brandeis University, 1982) and generalized in [78]. The basic goal of this construction is to construct (reducible) schemes from given ones, so that the various deficiency modules of the new scheme are all direct sums of the corresponding modules for the component schemes (with twists) . This has been applied in many ways, but for the moment we will focus on one application in Section 3.3: constructing "nice" Buchsbaum curves. An important observation (which we will use later) will be that the curves we produce with this construction are "the best possible" from the point of view of the general results obtained in Section 3.1. A special case of Liaison Addition, Basic Double Linkage, is very important in liaison theory. We will introduce this notion in this chapter (Remark 3.2.4), and discuss it more carefully in Chapters 5 and 6.
J. C. Migliore, Introduction to Liaison Theory and Deficiency Modules © Birkhäuser Boston 1998
CHAPTER 3. BUCHSBAUM CURVES & LIAISON ADDITION
62
3.1
Buchsbaum Curves
One of the themes that will emerge in this section is that there are several ways in which the Buchsbaum property can be viewed as a generalization of the aCM property. In fact, the definition we are using was not the original one. The notion of a Buchsbaum ring was first introduced by J. Stiickrad and W. Vogel after a negative answer by Vogel to a question of D. Buchsbaum as to whether, for an ideal generated by a system of parameters, there is a constant value (depending only on the ring and not on the system of parameters) for the difference between the length and multiplicity of the ideal. (The answer is "yes" for a CM ring; in fact, the difference is zero.) For a very complete and useful description of Buchsbaum rings, beginning with this point of view and developing all the cohomology theory, we refer the reader to [195]. For a classification of codimension two Buchsbaum subschemes of IP'n see [46], [47]. The first result, which is classical for aCM schemes (cf. Corollary 1.3.5), is the following:
Proposition 3.1.1 ([75]) Let C c pn be a Buchsbaum curve and let H be any hyperplane which contains no component of C. Then the Hilbert function of the hyperplane section CnH is independent of the choice of H.
Proof: Consider the exact sequence
0--+ (Ie)i-l --+ (Ie)i --+ (IenH)i --+ M(C)i-l ~ M(C);. The dimension of the third term (equivalently, the Hilbert function of C n H) depends only on the dimension of the first, second and fourth terms. But none of these depends on H. (Of course this result holds for Buchsbaum schemes of higher dimension, and in fact all that is needed is that the first deficiency module be annihilated by the maximal ideal, so it holds even more generally.) D Probing slightly deeper, we have
Proposition 3.1.2 ([75]) Let C c IP'n be a Buchsbaum curve. Let H be a general hyperplane, let C n H be the hyperplane section and let IenH be its ideal in the hyperplane H. Let a = a(Ie). Then
(a) a-I ~ a(IenH) ~ a. (b) a-I = a(IenH) if and only if M(C)o:-2 hO(IenH(a - 1))
=
i- O.
In this case,
dim M(C)o:-2'
3.1. BUCHSBAUM CURVES
63
(c) M(C)i = 0 for all i :=:; a - 3. (d) C is locally Cohen-Macaulay and equidimensional. Proof: Let L be a linear form defining the hyperplane H and let L' be another linear form that is not a scalar multiple of L. By abuse of notation we will also denote by L' the restriction of L' to the hyperplane H. As usual, for any integer d we denote by
1
xL'
1
xL'
The second inequality of (a) is always true, regardless of whether C is Buchsbaum or not. Then (a) , (b) and (c) follow from the following observation. Let FE HO(IenH(d)) . Then L'F E im rd+!. This follows from commutativity of the diagram, exactness and the fact that
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CHAPTER 3. BUCHSBAUM CURVES & LIAISON ADDITION
Proposition 3.1.2 shows that a Buchsbaum curve deviates just slightly from this, in that the initial degree of the ideal of the hyperplane section differs by at most 1 from that of the ideal of the curve. In fact, the proof of Proposition 3.1.2 shows the following amusing fact. While it is not the case that any element F E Iz lifts to an element in Ie, what is true is that for any linear form L in the coordinate ring of H and any F E I z, L . F lifts to an element of Ie. Geometrically, this says the following. Let F now represent a hypersurface in H = JP"l-1 which contains Z. Then F does not necessarily lift to a hypersurface containing C , but the union (counting multiplicity) of F with any hyperplane does lift. For example, if C is a set of two skew lines in p3, then Z is a set of two points in H = jp'2. The line F joining these two points does not lift to a surface (a plane) containing C , but the union of this line with any other line (including itself, suitably viewed) does lift to a quadric surface F containing C. Note that for any r ~ 0, one can find a curve C (of large degree and of course not Buchsbaum) for which the difference between the initial degree of Ie and that of the general hyperplane section of C is r. So this really is another sense in which the Buchsbaum property generalizes the aCM property. In general, it is a very interesting problem to study the difference between these initial degrees (for the general hyperplane section) and to relate that to properties of the curve (for instance its degree). Work along these lines has been done by Laudal [127], Strano [194] , Mezzetti [141], etc. The work in Section 2.3 can also be viewed as a contribution to this problem: in that case the difference was forced to be zero (since the curve was forced to be aCM). 0 Proposition 3.1.2 also has some striking consequences, which give information about the liaison classes of Buchsbaum curves in p3, but which can be stated simply in terms of the deficiency module, so we present them here.
Corollary 3.1.4 ([75]) Let C c JP"l be Buchsbaum and as usual let N = dimkM(C) . Then (aJ If C c p3 then M(C)i = 0 for i ~ 2N - 3. That is, the left-most component of M(C) occurs in degree ~ 2N - 2. (b) If C c p3, and if C is not the disjoint union of two lines, then C is connected. (c) If diam M (C) ~ 3 then C does not have maximal rank.
3.1. BUCHSBAUM CURVES
65
Proof: We remark that (b) and (c) were also proved in [67] in the case of integral curves in p3. (b) first appeared in [144] but with a complicated proof. The proof in [75] was very quick but used some results involving liaison. The proof given here is in the same spirit but uses the ideas of Section 2.3. Now, (a) follows easily from part (c) of Proposition 3.1.2 and from Amasaki's theorem (Corollary 2.2.6). As for (b) , suppose C is a disconnected Buchsbaum curve. Then in particular dimkM(C)o ~ 1 (see Theorem 1.2.6). Now let H be a general hyperplane and consider the usual exact sequence
(recall that C is Buchsbaum). Since C is locally Cohen-Macaulay and equidimensional but not aCM, hO(Ic(I)) = 0 (otherwise C would be a hypersurface in 1P'2 , hence aCM). We conclude that hO(IcnH (I)) = dim M(C)o ~ 1. By Proposition 3.1.2(c), a = 2 so by Amasaki's theorem N = 1 and M(C) occurs in degree O. In particular hO(IcnH (I)) = 1. Now, C n H is a set of points in 1P'2 lying on a line; hence it is a hypersurface in that line, and so it is a complete intersection in Hand thus also in p3. Then use the argument in Section 2.3: b = 1 and C n H is a complete intersection of surfaces of degrees al = 1, a2 = 1 and a3 ~ 2. Proposition 2.3.2 then says that 1 + 3 ~ 1 + 1 + a3 , so a3 = 2. Therefore deg C = deg (C n H) = 2 and in order for C to be disconnected, it must consist of two skew lines. For (c) , recall that for C to have maximal rank means that the restriction map has maximal rank, i.e. is either injective or surjective for all d. Equivalently, for each d we have either hO(Ic(d)) = 0 or h1(Ic(d)) = 0 (or both) . Then the result follows from the definition of a and from (c) of Proposition 3.1.2. 0
Remark 3.1.5 An analog, for k-Buchsbaum curves, of many of the results in this section can be found in [150]. 0 Remark 3.1.6 (a) Part (c) of this corollary is stated for any projective space, while parts (a) and (b) are stated for p3 only. It is natural to ask whether they are true in higher projective spaces. Note that both of them use Amasaki's theorem in the proof, so one suspects right away that there
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CHAPTER 3. BUCHSBAUM CURVES & LIAISON ADDITION
will be problems. And indeed, both are false in higher projective space. For (a), a counterexample can be found in [78] Proposition 2.7. Here a Buchsbaum curve Cr is constructed in p4 using Liaison Addition (see Section 3.2), where dimkM(Cr) = r and the module begins (and in fact is concentrated) in degree r - 1. As for (b), consider for instance three general skew lines in ]p5. The module is 2-dimensional and concentrated in degree 0, hence C is Buchsbaum. However, using a similar argument, it should be possible to classify the disconnected Buchsbaum curves in pn. (b) One of the main ideas of the structure of an even liaison class of codimension two subschemes of pn, which we will discuss in Section 6.3, is that the schemes whose modules have the leftmost possible shift are very special. We have already discussed possible shifts in Chapter 1. Notice that part (a) of Corollary 3.1.4 gives the leftmost possible shift for a Buchsbaum curve in jp3 (with specified Buchsbaum type N, i.e. with the dimension N of the deficiency module specified). One of the consequences of the structure theorem, called the LazarsJeld-Rao Property (Theorem 6.3.1), is that for a fixed deficiency module (up to shift), any Buchsbaum curves which are extremal with respect to this bound have the same degree. For example, if the module has dimension k and is supported in one degree, then the leftmost possible shift would be when the non-zero component occurs in degree 2k - 2. In this case, the curve necessarily has degree 2k2 (cf. [31]). (If k = 1 then two skew lines give an example of an extremal curve.) D In Chapter 2 we discussed Dubreil's theorem bounding the number of minimal generators of the ideal of an aCM curve in jp3, and we generalized this result to non-aCM curves. A related problem is to bound the degrees of the minimal generators of the ideal of the curve. One tool for doing this is Castelnuovo-Mumford regularity (Theorem 1.1.5). Another is to relate this to the generators of the ideal of the general hyperplane section. It is already clear that the degrees of the generators of the ideal of the curve cannot be closely related to those of the general hyperplane section without some extra information (for instance something about the deficiency module). For example, if C is a double line (i.e. a non-reduced scheme of degree two supported on aline) then its general hyperplane section has degree 2 and in fact is a complete intersection, so its ideal is generated in degrees :s; 2. However, the ideal of C can have a minimal generator of arbitrarily large degree (and correspondingly the deficiency module can get arbitrarily large) . See Example 1.5.10 and [143] for details. Now we will prove a result for Buchsbaum curves which generalizes a
3.1. BUCHSBAUM CURVES
67
standard result for aCM curves. We need the following lemma.
Lemma 3.1. 7 Let Z be a zeroscheme in IF with saturated ideal lz. Let t = min { i E Z I h1(Iz(i)) = o}. Then lz is generated in degree ~ t+ 1. Proof: This follows immediately from Theorem 1.1.5 (Castelnuovo-Mumford), but we can also give a simple direct proof. Let L be a general linear form ; in particular, L does not vanish at any of the points on which Z is supported. Thus the saturation of lz+(L) is all of S by the Nullstellensatz, so there is no hyperplane section, and using the notation of Section 1.3 we have IZIL ~ OlP'n-1. Hence the exact sequence (1.6) of that section becomes 0-+ I z (-l) ~ Iz -+ O lP'n-l -+ O. Twisting by d 2 t
+ 1 and letting R =
k[Xo, · .. , X n -
1 ],
we get
Now, let F E (lz)d where d 2 t + 2. We want to show that F is not a minimal generator. Note that rd(F) E Rd can be written as 2:: XjG j where Gj E R d- 1 . Since rd-l is surjective (since now d 2 t + 2) we get rd(F) = 2::Xjrd-l(Fj) = rd(2::X jFj) for some Fj E (lz)d-l. Then by exactness, F - 2:: X jFj = LG for some G E (lz )d- l so F is not a minimal generator. 0
Corollary 3.1.8 ([75]) Let C c IF be a Buchsbaum curve, and for a general hyperplane H let t = min { i I h I (IenH (i)) = 0 } . Then the saturated ideal Ie of C is generated in degree ~ t + 1. Proof: Consider the exact sequence
(where the last 0 is from the definition oft). This guarantees that M(C)t = 0, and since h1(IenH(i)) = 0 for all i 2 t, we also have that M(C)j = 0 for all i 2 t. With this information and using the above exact sequence (twisted by 1), the same argument as in Lemma 3.1.7 gives the result. 0 Note that Corollary 3.1.8 could also be proved using CastelnuovoMumford regularity: the argument given in the proof gives hI (Ic(t)) = 0, and a similar argument can be given to show h2 (Ie(t - 1)) = O.
68
3.2
CHAPTER 3. BUCHSBAUM CURVES & LIAISON ADDITION
Liaison Addition
Liaison Addition was introduced in the Ph.D. thesis of Phillip Schwartau [189] in 1982. His work was motivated by the following naive question: Let C l and C2 be curves in Jr3 with deficiency modules Ml and M2 respectively. Does there exist a curve C with M( C) ~ Ml 81M2? The following example shows that, as stated, this is too ambitious. Example 3.2.1 Let C1 and C2 each be a set of two skew lines in Jr3, so as we saw in Example 1.5.2, Ml and M2 are both one-dimensional kvector spaces, and as graded modules they are non-zero only in degree zero. Hence Ml $ M2 ~ k 2 , also occurring in degree zero. The simplest reason that no curve can exist with precisely this deficiency module is that such a curve would be Buchsbaum with Buchsbaum type 2, so by Corollary 3.1.4 (a) the module has to occur in degree 2': 2. Here is a more direct argument. Suppose there were to exist such a curve C . Let H be a general plane and consider the exact sequence
(where IcnH is the ideal sheaf of the hyperplane section in the hyperplane H). We know from Theorem 1.2.5 that C is locally Cohen-Macaulay and equidimensional. On the other hand, if C lay on a plane it would be a hypersurface in that plane and hence (as noted in Example 1.5.1) a complete intersection, which contradicts the fact that C is not aCM. Therefore hO(Ic(l)) = o. The exactness of the above sequence then gives that hO(IcnH (l)) = 2. But C n H is a set of points in the plane, and hO(IcnH (l)) = 2 implies that C n H consists of exactly one point. Hence deg C = 1, and because C is locally Cohen-Macaulay and equidimensional, this means that it is a line. Again this contradicts the fact that C is not aCM. 0 Example 3.2.2 Schwartau remarks that however the question is phrased, simply taking the union of the given curves cannot work. Indeed, if C1 and C2 are disjoint lines then both are aCM (hence have trivial deficiency module), while their union is not. 0 Schwartau discovered the correct way to rephrase the question for curves in p3, and in fact his theorem is stated in the more general context of "adding" two codimension two subschemes of projective space. This theorem was generalized in [78] to allow higher codimension and a greater
69
3.2. LIAISON ADDITION
number of "added" schemes (rather than only two). We will not give this theorem in its full generality here, but rather give a simplified version which will suit our purposes. We refer the reader to [78] for the stronger version of the theorem . We will now state the theorem. Note that we use the cohomology notation H! (Iv) rather than the notation (Mi) (V) . This is because technically (Mi)(V) is only defined for 1 :S i :S dim V , while in this theorem for some V we may need i to be larger.
Theorem 3.2.3 (cf. [78]) Let VI"'" v,. be closed subschemes of rn, with 2 :S r :S n. (We allow the possibility that Vi = 0, in which case Iv; = S.) Assume that codim Vi ~ r for all i. By making general choices (possibly of large degree), we may choose homogeneous polynomials
Fi
E
n IYj
(3.1)
I~j ~ r
#i
for 1 :S i :S r, such that (FI' .. . , Fr) forms a regular sequence and hence gives the saturated ideal of a complete intersection scheme V of codimension r. (This is possible because of our assumption that codim Vi ~ r for all i). Let deg Fi = di . Let I = FIIvl + ... + FrIvr and let Z be the closed subscheme of rn defined by I . We denote by H (Z, t) the Hilbert function of Z (see page 7). Then (a) Assets, Z=VIu···uv,.uV. (b) For each 1 :S j :S n - r
= dim V,
we have
Ht(Iz ) ~ Ht(Iv1)(-dd EEl··· EEl Ht(IvJ(-d r ).
(c) I is saturated (I = lz) . (d) H(Z, t) = H(V, t)
+ H(VI, t -
dd
+ ... + H(v,. , t -
dr ).
Proof: If P E Vi for some i, or if P E V, then clearly from the way I is defined, every F E I vanishes at P. So as sets, Z:;2 VIU·· ·UVrUV. Now let P be any point of rn not on VI U ... U v,. U V. In particular there is some Fi not vanishing at P (since P V), and since (for that i) P Vi , we have some G E IV; not vanishing at P. So GFi E I does not vanish at P, hence P Z. This proves (a). From Example 1.5.1 we know that the resolution for Iv begins with
tt
tt
tt
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CHAPTER 3. BUCHSBAUM CURVES & LIAISON ADDITION
where (PI = (F1 , .. . , Fr) and (P2 is the matrix of relations: it is an r x (;) matrix of homogeneous polynomials, and each column C t consists of exactly two non-zero entries (each being one of the Fi up to ±l) so that the matrix product ¢l . Ct is of the form FiFj - FjFi = O. In particular, for 1 ::; i ::; r , each entry ofthe ith row of ¢2 is an element of Iv; (possibly 0) because of the assumption (3.1). (What we are saying is that Fi never occurs in the ith row, but all the other generators do.) The significance of this observation is that the image of ¢2 is actually contained in EEh:S;i:s;r Iv; (-d i ). Therefore we also have a long exact sequence
l
l:S;i<j:S;r
If we let K be the cokernel of ¢3, exactness gives that K is also the kernel of ¢l (in either (3.2) or (3 .3)). Therefore we have two short exact sequences
and
o -t K
-t
o -t K
-t
EB
S( -di )
...5:.4 Iv
-t
0
(3.4)
EB
Iv; (-d i )
...5:.4 I
-t
o.
(3.5)
l:S;i:S;r
Let K be the sheafification of K and sheafify these two exact sequences. Note that the sheafification of I is I z . The idea is to use (3.4) to get information about K, and then apply this to (3 .5). Since ¢l is surjective in (3.4) and Iv is saturated, we get that H;(K) = O. Similarly we get that H~(K) = K and that Hi(K) = 0 for 2 ::; j ::; n - r + 1 (using the fact that V has dimension n - r and is aCM). Now apply this information to (1.5) (sheafifying and taking cohomology) . Part (b) follows immediately from the vanishing of the higher cohomology of K. As for (c) , this follows from the facts that H;(K) = 0, Iv; is saturated for each i, and H~(K) = K. (The point is that when you sheafify the short exact sequence and then take cohomology, you again get a short exact sequence and the first two terms have not changed; hence I = H~(Iz) = I z .) 0
Remark 3.2.4 (a) As we noted above, this theorem is a special case of a more general result in [78]. (One can allow V to be much more general than simply a complete intersection, and one can "add" more than r schemes.) But this version seems to be the simplest to apply in any case. The thesis of Schwartau proved this theorem in the special case where r = 2, codim Vi = 2 and Vi are not trivial. This thesis has not been published elsewhere.
3.2. LIAISON ADDITION
71
On the other hand, [78] was written with the advantage of hindsight: knowing the work of [189] made it somewhat natural to guess what ideal would most likely produce the desired scheme (once one decided in which direction it should be generalized). As a result, the main theorem in [78] is not only much more general but the proof is much simpler. The approach of [189] is more constructive. (b) As mentioned in the theorem, some of the Vi can be trivial. In the case where VI is non-trivial and all the rest of the Vi are trivial, the resulting scheme Z is called a Basic Double Link of VI, and this construction is called Basic Double Linkage. So in this case we have FI E S, a general homogeneous polynomial of any degree dl ?: 1, and F2, . .. , Fr E IVll such that (FI' F2, ... , Fr) form a regular sequence and hence define acorn plete intersection V. See Section 5.4 for an explanation of this terminology, and for generalizations. Basic Double Linkage was introduced for curves in jp3 by Lazarsfeld and Rao [128], and extended to the current generality in [34] without recourse to the above theorem. The connection to Liaison Addition was made for curves in ]p3 in [128] and for the general case in [78]. Basic Double Linkage in co dimension two plays a crucial role in liaison theory, and especially in the structure theorem for an even liaison class (cf. [128], [12], [137], etc.). This will be discussed later. For now notice three things: (1) As sets, Z
(2) Iz
= VI U V.
= FIlvi + (F2,···, Fr).
(3) (Mi)(Z) ~ (Mi)(Vl)( -d 1 ).
In particular, this gives the proof of part (c) of Proposition 1.2.8. (c) In Section 3.3 we will see one important application of Liaison Addition: the construction of Buchsbaum curves. This theorem can also be used to create quick and simple examples of saturated ideals of schemes with embedded components. For example (from [78]), consider the case of curves in jp3 and r = 2. It is fairly easy to show using part (b) of the theorem that Z is locally Cohen-Macaulay, equidimensional and onedimensional if and only if VI and V2 are (but allowing V2 to possibly be trivial as well). (Use Theorem 1.2.5.) A simple example is the following. Let VI be a point, say with ideal (Xl, X 2 , X 3 ) and let V2 be trivial. Form a basic double link taking FI and F2 to be linear forms in I Vi: say FI = X I, F2 = X 2. The ideal
72
CHAPTER 3. BUCHSBAUM CURVES & LIAISON ADDITION
Iz = X l (Xl ,X2 ,X3 ) + (X 2 ) = (Xl, X l X 3 , X 2 ) is the saturated ideal of a line Z with an embedded point (at Vd. 0 Remark 3.2.5 It is worth commenting on the geometric meaning of Basic Double Linkage in Remark 3.2.4. Let S be the complete intersection defined by the ideal (F2 , .•• , Fr). Note that dim S = dim Vl + 1, so Vl is a divisor on S, in the sense of Hartshorne's generalized divisors [90]. Let V = HpJ be the hypersurface section on S cut out by F l ; this is also a divisor on S. Then Z is just the divisor Vl + HpJ on S. 0
3.3
Constructing Buchsbaum Curves in JP3
In this section we show how to use Liaison Addition to construct Buchsbaum curves, especially in p3. We will follow [35] for the most part because we would like to show that the curves we construct are "nice" from a number of viewpoints used in that paper (and we will use these properties later when we use liaison techniques to show that every Buchsbaum curve in p3 specializes to a stick figure, which is one of the main results of [35]). However, similar ideas for constructing appropriate Buchsbaum curves in p3 via Liaison Addition were used in [189], [31], [32] and [33]. A similar construction for Buchsbaum curves in p4 with some "nice" properties was done in [78], thanks go the generality of the construction described in Section 3.2; see Remark 3.3.2. Let M be a graded S-module of finite length which is annihilated by all linear forms. Our goal is to construct a Buchsbaum curve C in p3 for which M(C) is some shift of M. For such a curve C, let N = dimkM(C) and let Ie be the saturated ideal of C. For the curves we construct, the bounds of Theorem 2.2.4, Corollary 2.2.6 and Corollary 3.1.4 will be sharp. That is, our curves will satisfy (a) v(Ie) = a(Ie)
+ N + 1 (Theorem
2.2.4).
(b) a(Ic) = 2N (Corollary 2.2.6). (c) The leftmost nonzero component of M(C) occurs in degree 2N - 2 (Corollary 3.1.4). In fact, it follows from arguments of liaison theory (especially the Lazarsfeld-Rao Property) that our curves have the least degree among all curves with the given deficiency module (up to shift). They are examples of so-called minimal elements in their even liaison classes, about
3.3. CONSTRUCTING BUCHSBAUM CURVES IN p3
73
which we will say more in Chapters 5 and 6. The curves we construct in this section will also be hyperplanar stick figures (cf. [35]). This means that each curve we construct is (d) a reduced union of lines with no three meeting in a point (i.e. a stick figure) and (e) contained in a (reduced) union of 2N hyperplanes such that the intersection of any three of the hyperplanes is a point and the intersection of any two of them is not a component of C (i.e. hyperplanar). We will use the fact that our curves are hyperplanar in Section 6.4.4 when we show that any Buchsbaum curve in p3 specializes to a stick figure. The basic idea here is that we will use sets of two skew lines in p3 as "building blocks" to construct Buchsbaum curves with larger deficiency modules. We can use Liaison Addition to do this. Recall that a set of two skew lines has deficiency module which is just one-dimensional (in degree zero); cf. Example 1.5.2. But any graded S-module, whose multiplication by linear forms is trivial, is isomorphic (as an S-module) to a direct sum (with twists) of copies of this one-dimensional vector space (see page 41), so Liaison Addition lends itself naturally to this problem. Suppose we are trying to construct a Buchsbaum curve C whose deficiency module has components of dimension nl > 0, n2 2: 0, ... , n r-l 2: 0, nr > (where these are the dimensions of all the components of the module M, from the first non-zero one to the last). The proof works by induction on N = nl + ... + n r . Note that we would like the first component to occur in degree 2N - 2. For N = 1, C is the disjoint union of two lines. It is trivial to verify that all the conditions (a)-(e) are verified. So assume that N > 1. For convenience we will assume that nl > 1. The case nl = 1 is similar and is left to the reader. (The main difference is that the choice of Fl below will be slightly different.) Suppose that C1 is a curve whose module components have dimensions nl - 1, n2, .. . , nr and which satisfies all the conditions above (with now N replaced by N - 1). Let F2 E I C 1 be the union of 2 (N - 1) planes guaranteed by (e). The singular locus Sing F2 is a union of lines having no component in common with C 1 • Let C2 be a disjoint union of two lines, also disjoint from C1 and from Sing F2. Note that Sing C l , (C1 n Sing F2) and {triple points of F2} are all finite sets disjoint from C2 • Let Fl be a union of two planes that contains C2 but avoids these three finite sets, and such that the line Sing Fl avoids
°
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CHAPTER 3. BUCHSBAUM CURVES & LIAISON ADDITION
Yl , Sing F2 and C2 n F2. Choosing one plane first gives an additional finite number of points for the second plane to avoid. Notice that the fact that Sing Fl avoids C2 n F2 implies that F2 avoids C2 n Sing Fl , which is the analogue of the condition that Fl avoids Cl n Sing F2 which we required in the choice of Fl. Let C be the result of applying Liaison Addition to C l and C2 using F2 E ICI and Fl E IC2' so lc = FllCI + F2 1c2 · Clearly a(Ic) = 2N, and by Liaison Addition M(C) ~ M(Cd(-2) EB M(C2 )(-2N + 2). Then by the inductive hypothesis, one can check that (b) and (c) hold for C, and so by Remark 2.2.8(a) we also have (a) above. We now check that C is a stick figure. By the generality of the choices made, C is a reduced union of lines. (We need the fact that Fl contains no component of C l or of Sing F2 and F2 contains no component of C2 or of Sing Fl , which follow from the above set-up.) As for the absence of triple points, it is somewhat tedious to check all the details, but it follows from the fact that both Cl and C2 are hyperplanar with F2 and Fl the corresponding unions of planes, and these are sufficiently general so that they meet well. C is the union of C l , C2 and the complete intersection of Fl and F2, and the two conditions of (e) are exactly what are needed to avoid triple points. (The first condition avoids triple points lying on a plane and the second avoids triple points that are not on a plane.) We refer to [35] for details. To see that C is hyperplanar, we need to exhibit a reduced union E of 2N planes containing C such that the two conditions in (e) hold. One would be tempted to take for ~ the union of Fl and F2, but since C contains in particular the complete intersection of Fl and F2, the second condition of (e) fails. However, by the way things were chosen, F2 does almost all of the job (it contains C l and the complete intersection of Fl and F2)' One can check that by taking for ~ the union of F2 and a general union of two planes containing C2 , this will do the trick. Again there are several details to check, for which we refer the reader to [35]. As mentioned above, the case nl = 1 is very similar. Here, instead of taking Fl to be a union of two planes containing C2 , we take it to be a union of two planes containing C 2 and an appropriate number of general planes. The number is chosen in order to construct the right module with Liaison Addition.
Remark 3.3.1 (a) One can also use Liaison Addition to construct Buchsbaum curves in the other extreme: for any "Buchsbaum" module M whose components have dimensions nl, . . . ,nr as above, one can construct a
3.3. CONSTRUCTING BUCHSBAUM CURVES IN p3
75
Buchsbaum curve whose deficiency module (up to shift) is M, and which is sharp with respect to conditions (a), (b) and (c), but which is supported on a line. The idea is that rather than using sets of two skew lines as the "building blocks" one can use a double line of arithmetic genus -1 (cf. Example 1.5.10). Note that at each step we use the same double line rather than a general one. Such a double line has one-dimensional deficiency module, so the same ideas apply. Notice that one can pick any line A C p3 and construct the desired Buchsbaum curve supported on A. See also Remark 6.4.15. (b) Similar constructions (for subschemes which are reduced unions of co dimension two linear varieties with "good" properties) are possible in higher dimensions as well. These subschemes were called "good linear configurations" in [35], where some work was also done for the case of surfaces in ]p4 . (c) One can check that the degree of the curve C constructed in this section (in particular satisfying condition (c) on page 72) is
2N2 - N
+ nl + 3n2 + ... + (2r -
l)n r .
This can be shown directly using induction, or it can be shown using Corollary 2.18 (b) of [33]. (d) See Example 1.5.6 for the case r
= 2 and
nl
= n2 = 1.
0
Remark 3.3.2 While Buchsbaum curves in p3 are rather well understood, the case of Buchsbaum curves in ]pn, n 2: 4, is more difficult. In [78] it was shown that the same type of construction described here for curves in p3 in fact gives a non-degenerate Buchsbaum curve in ]pn with any prescribed module M (of course with trivial multiplication). We would like to briefly compare the situation in p3 with that in higher projective space, especially ]p4. The first question is what should the "building blocks" look like. Clearly we want such a curve to have a deficiency module which is one-dimensional, and we would like this module to occur as far to the left as possible. An easy argument using the sequence (1.6) of Section 1.3 gives that this leftmost shift is in degree O. In 1P'3 it follows from the Lazarsfeld-Rao Property (cf. Definition 5.4.2 and Theorem 6.3.1) that all curves with this shift of the module have degree 2. In]p4 we know that a set of two skew lines, although degenerate, still has a one-dimensional deficiency module which occurs in degree 0 (by Proposition 1.3.10). On the other hand, the curve in ]p4 consisting of a non-degenerate disjoint union of a line and a plane conic also has a
76
CHAPTER 3. BUCHSBAUM CURVES & LIAISON ADDITION
one-dimensional deficiency module occurring in degree 0 (either use the method of Example 1.5.4 or use Example 5.4.8). In IF there are other curves with one-dimensional module occurring in degree 0, such as the non-degenerate union of two conics and the non-degenerate union of a line and a twisted cubic (this list is not complete). In higher projective space the list continues to grow. Hence it probably would be necessary to have more than one kind of curve for our building blocks, in order to be as complete as possible. (In [78], only sets of two skew lines were used.) We can also consider the question of whether we can get "all" Buchsbaum curves in any sense. For curves in jp3, using liaison theory (in particular the Lazarsfeld-Rao Property), it turns out that, up to deformation, we can get all Buchsbaum curves whose module is some shift of our given module M, by starting with the curves constructed in this section and then using Basic Double Linkage. This is described in Section 6.4.3; see especially Example 6.4.11. It is not clear what can be said in F', n 2: 4. A special kind of Buchsbaum curve in jp3 is one whose module is supported in one degree. These were studied in [31]. The smallest such curve, which we constructed in this section, has the module occurring in degree 2N - 2, and the least degree of a hypersurface containing it is 2N. Since in this case N = nl and n2 = ... = nr = 0 in the formula given in Remark 3.3.1 (c), we see that the degree of the smallest such curve is 2N 2 . It is interesting to compare with the case of the analogous curves constructed in ]p4 in as small a way as possible (using only pairs of skew lines as building blocks). In [78] it is shown that these curves have degree (N 3 + 5N)/3, the module occurs in degree N - 1 and the least degree of a hypersurface containing the curve is N. 0
Chapter 4 Gorenstein Subschemes of Projective Space There is a huge literature on arithmetically Gorenstein ideals, and it is well beyond the scope of this book to give a thorough treatment of the subject. The purpose of this chapter is to collect the definitions, facts and results that we will need in the coming chapters when we talk about liaison theory for subschemes of projective space, in the generality of "linking" by arithmetically Gorenstein ideals rather than only by complete intersections.
4.1
Basic Results on Gorenstein Ideals
The purpose of this section is give some basic facts on arithmetically Gorenstein subschemes of projective space, in preparation for our discussion of liaison theory in the coming chapters. A much more thorough treatment can be found in [62], Chapter 21, and in [40], Chapter 3, and we strongly encourage the reader to look there for further information. See also Remark 4.3.3. We have already seen on page 11 that if V is an arithmetically CohenMacaulay (aCM) subscheme of projective space, then its projective dimension (Le. the length of a minimal free resolution) is equal to its codimension, and that V is arithmetically Gorenstein if it is aCM with CohenMacaulay type 1. We saw that this property is preserved under hyperplane and hypersurface sections (Corollary 1.3.8), and we saw that interesting questions can be asked about lifting the Gorenstein property from the hyperplane section of a curve up to the curve itself (Section 2.3). We also saw in Chapter 1 that the canonical module of an arithmeti-
J. C. Migliore, Introduction to Liaison Theory and Deficiency Modules © Birkhäuser Boston 1998
CHAPTER 4.
78
GORENSTEIN SUBSCHEMES OF 1pm
cally Gorenstein subscheme of pn is cyclic (page 12). In fact, more is true (this is the graded version of [62] Theorem 21.16):
Proposition 4.1.1 Let V be an aCM subscheme of pn. The following are equivalent: (a) V has Cohen-Macaulay type 1 (i.e. V is arithmetically Gorenstein); (b) SI Iv ~ K v(£) for some £ E Z; (c) The minimal free resolution of SIIv is self-dual, up to twisting by n + 1. Proof: The fact that (b) implies (c) comes from Remark 1.2.4, which says that the dual of the resolution of S I Iv is the resolution of K v , up to twisting by n + 1. Also, (c) clearly implies that the Cohen-Macaulay type of V is 1, which is (a). The heart of the proposition is the implication that (a) implies (b) . Consider a minimal free resolution of SI Iv:
o --t S( -t) --t Fc - I --t . .. --t FI --t S --t S I Iv --t 0 (where c is the co dimension of V). Dualizing this resolution, using the fact that the Cohen-Macaulay type is 1, and applying Remark 1.2.4, we get that Kv(£) ~ SIJ for some ideal J and some £ E Z. On the other hand, since V is aCM (hence equidimensional - cf. Theorem 1.2.5) and Kv = Exts(SI Iv, S)( -n - 1),
we have from [64] Theorem 1.1 (3) that Iv = anns(Kv) . Combining these facts gives that J = Iv, which is what we need for (b). 0
Remark 4.1.2 We defined arithmetically Gorenstein subschemes of pn in terms of the Cohen-Macaulay type, but in fact, in the literature they are often defined according to condition (b) of Proposition 4.1.1. Here we see that these definitions are equivalent. 0 We now give several corollaries of Proposition 4.1.1 which will be useful in the coming sections and chapters.
4.1. BASIC RESULTS ON GORENSTEIN IDEALS
79
Corollary 4.1.3 Let V be an arithmetically Gorenstein subscheme of]pn with minimal free resolution 0-+ S(-t) ~ Fc- 1 -+ ... -+ Fl -+ S -+ S/Iv -+ O. Then the entries of the matrix cPc are precisely the minimal generators of the saturated ideal Iv.
Corollary 4.1.4 If V is an arithmetically Gorenstein subscheme of]pn then the h-vector of V is symmetric. Proof: This follows from Proposition 4.1.1 and Remark 1.4.8. 0
Corollary 4.1.5 Let V be an arithmetically Gorenstein subscheme of]pn. Then Ov ~ wvU') for some £ E Z. Many of the integers described in this section are related, as shown in the following result.
Corollary 4.1.6 Let V be an arithmetically Gorenstein subscheme of]pn of codimension c with minimal free resolution 0-+ S(-t) -+ Fc - 1 -+ ... -+ Fl -+ S -+ S/Iv -+ O. Let Ov ~ wv(£). Assume that the (necessarily symmetric) h-vector of V has r + 1 entries: (1, c, C2, .. . , Cr -2, C, 1) . Then t-
C
+1=
reg (Iv ) = r
and
£= n
+1
+ 1- t .
Proof: This follows from Remark 1.1.6, the discussion of h-vectors and regularity on p. 30, and the definition of the canonical module (p. 10). 0
Remark 4.1.7 If V is arithmetically Gorenstein as above, a related integer is the so-called socle degree of S / Iv, which is the integer t - c = r. The name comes from the following. First, let V be any aCM subscheme of ]pn and let A be the Artinian reduction of S/Iv . Note that A is a graded S-module of finite length (in addition to being a ring). Then the socle of A is defined as the annihilator in A of the maximal ideal. Now assume that V is arithmetically Gorenstein. It is not hard to show that then the socle of A is simple (i.e. in this case it is a one-dimensional k-vector space). See [62] Proposition 21.5 for details. Clearly the last non-zero component of A is annihilated by the maximal ideal, so the fact that the socIe is simple implies that it is concentrated in this degree, which is r = t - c. 0
80
CHAPTER 4.
GORENSTEIN SUBSCHEMES OF 1pm
A natural question is to determine more geometric conditions which guarantee that a subscheme of jpn is arithmetically Gorenstein. It follows from Theorem 1.3.3 and Theorem 1.3.6 that for any subscheme V of jpn of dimension ~ 2, V is arithmetically Gorenstein if and only if its general hyperplane section is. Furthermore, if V is an aCM curve then the same conclusion holds. Also, as we saw in Section 2.3, interesting questions can be asked about when the Gorenstein property for the general hyperplane section of any curve (not necessarily aCM) forces the curve itself to be arithmetically Gorenstein. So in some sense, the most natural question is to seek a set of conditions on zeroschemes which force them to be arithmetically Gorenstein. One necessary condition is given by Corollary 4.1.4. Example 1.4.3 shows that this condition by itself is not sufficient. However, we can give the complete answer after we make one more definition.
Definition 4.1.8 Let Z be a zeroscheme in jpn and let r + 1 = reg(Iz ) (cf. pages 8 and 30). Then Z has the Cayley-Bacharach property if, for every subscheme Y c Z with deg Y = deg Z - 1, we have that hO(jpn,Iy(r - 1)) = hO(jpn,Iz(r - 1)). 0 Example 4.1.9 Consider subschemes ofJP2 with h-vector (1 , 2, 1) (cf. Example 1.4.3 and Example 1.4.7). Here the regularity is 3, so the definition requires us to consider the linear system of curves of degree 1 passing through the points. For convenience we will consider only reduced zeroschemes. If Z is the complete intersection of two conics, then Z consists of four points, no three of which are on a line. Then any subscheme Y of degree 3 satisfies hO(JP2,Iy(1)) = 0 = h°(lP2,I z (1)) . Hence a complete intersection has the Cayley-Bacharach property. On the other hand, if Y is a set of four points with three on a line and one off the line, then choosing Y to be the three points on the line gives hO(JP2, Iy(1)) = 1 # h°(lP2,I z (1)). Hence Z does not have the Cayley-Bacharach property, and is not a complete intersection. 0 The following theorem then answers the question of which zeroschemes are arithmetically Gorenstein. The reduced case was proved in [56], and the picture was completed for the non-reduced case in [124] . See also [62] Exercise 21.24.
Theorem 4.1.10 A zeroscheme Z in jpn is arithmetically Gorenstein if and only if it has symmetric h-vector and it has the Cayley-Bacharach property.
4.1. BASIC RESULTS ON GORENSTEIN IDEALS
81
Example 4.1.11 (a) The simplest example of an arithmetically Gorenstein subscheme of IF of any codimension is a complete intersection (cf. Example 1.5.1). (b) It is not hard to check, using Theorem 4.1.10, that a set of n + 2 points in IF in linear general position (i.e. no n + 1 on a hyperplane) is arithmetically Gorenstein. (c) In codimension two , the converse to (a) holds. If V is arithmetically Gorenstein, then from the minimal free resolution
0-+ S(-t) -+ Fl -+ S -+ S/Iv -+ 0, one quickly sees that the rank of Fl must be 2, i.e. that V must be a complete intersection. This was originally observed by Serre [191]. Example (b) above shows that this is not true in higher codimension. (d) In general, it is difficult to construct arithmetically Gorenstein subschemes of IF, apart from complete intersections, with prescribed properties (e.g. with given Hilbert function or containing a given scheme) . In Section 4.2 we will give some constructions which have proved useful in this regard. (e) Let V be an arithmetically Gorenstein subscheme of IF and let F be a form of degree d not vanishing on any component of V. Let Z be the subscheme cut out on V by F, as in Section 1.3, viewed as a subscheme of IF. We saw in Proposition 1.3.7 that Z has the same Cohen-Macaulay type as V, and hence is also arithmetically Gorenstein. This can be viewed as a special case of Theorem 4.2.8, in view of Proposition 4.1.1. See also Theorem 5.2.27. 0 In Example 1.5.1 we considered a special type of arithmetically Gorenstein scheme, namely the complete intersections, and we described a number of facts about them which we have now generalized in this section. We conclude this section with one more generalization, namely the arithmetic genus of an arithmetically Gorenstein curve. Recall that if a curve C is the complete intersection of hypersurfaces of degrees d1 , . . . , dr, then
Recall also that in the minimal free resolution of a complete intersection, the twist in the last free module in the resolution is L. di . We now generalize this fact with the following formula, which will be useful in liaison theory (cf. Corollary 5.2.14).
82
CHAPTER 4.
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Proposition 4.1.12 Let C be an arithmetically Gorenstein curve in with minimal free resolution
]pn
0-+ S(-t) -+ Fn - 2 -+ ... -+ FI -+ S -+ S/Iv -+ O. Then Pa(C) = ~(deg C)(t - n - 1)
+ 1.
Proof: Without loss of generality we may assume that C is non-degenerate. One can then check that we necessarily have t 2:: n + 1. If t = n + 1, then the regularity is 3 and the h-vector must be (1, n, 1). The genus formula (Proposition 1.4.2) then gives that Pa(C) = 1, as required. So now assume that t > n + 1. Let D be a canonical divisor, i.e. a regular section of We = Oe(t - n - 1). Clearly deg D = (t - n - 1) . deg C. The result will be proved if we show that deg D = 2Pa (C) - 2. (This is obviously true for smooth curves, for instance.) We have
Pa(C)
1 - X(Oe) hl(Oe) hO(Oc(t - n - 1)) hO(Oe(D)) deg D - Pa(C) + 1 + hl(Oe(t - n - 1)) deg D - Pa (C) + 2
(Serre duality)
from which the result follows immediately. 0
4.2
Constructions of Gorenstein Schemes
In Chapter 5 we will see that Gorenstein ideals can be used to define an equivalence relation on subschemes of projective space. In Chapter 6 we will see that the picture in co dimension two is rather complete, and a number of applications are outlined. A big part of the reason for the theory in this situation to be so complete is that, as we saw in Example 4.1.11 (c), in co dimension two the Gorenstein ideals are precisely the complete intersections, and these are well understood and easy to produce. In higher codimension, though, we will see that the theory is still being developed. In order for this theory to "catch up," and in order for meaningful applications to be found, we need useful constructions of Gorenstein ideals. This section is intended to describe some constructions which have already proved useful in the context of liaison theory. There are certainly other constructions of Gorenstein ideals which we will not describe in detail but which have played an important role in other questions. An important
4.2. CONSTRUCTIONS OF GORENSTEIN SCHEMES
83
example is Macaulay's construction of inverse systems. See [62] and especially [72] for a description of this construction and for further references and applications.
4.2.1
Intersection of Linked Schemes
Despite the title, this construction can be understood without using liaison theory. It was introduced in [174], and the idea is to start with two aCM schemes whose union is arithmetically Gorenstein (this notion will be extremely important starting in the next chapter!) and which have no common component, and to look at the intersection of those schemes. More precisely, we have the following result, an application of which will be described in Section 4.3.
Theorem 4.2.1 Let VI and V2 be aCM schemes of codimension c with no common component, and assume that Iv! nIv2 is arithmetically Gorenstein. Then Iv! + IV2 is arithmetically Gorenstein of codimension c + 1. Proof: This was proved in [174] for the case where Iv! n IV2 is a complete intersection, but the proof is the same in the general case. We have a diagram 0
0
.!-
.!-
Fe EB Gc
S(-t)
.!-
H c-
.!-
F c- I
I
.!HI
0
~
.!Iv! n IV2 .!0
~
.!-
EB G c-
.!-
.!-
GI
FI
EB
Iv!
EB IV2
.!-
.!-
I
~
Iv!
+ IV2
~
0
0
where the Hi give a minimal free resolution of Iv! n I v2 , the Fi give a minimal free resolution of Iv! and the Gi give a minimal free resolution of I v2 . Taking a mapping cone (p. 4) gives the result. 0
Remark 4.2.2 We will see in the next chapter that the assumption that the union of VI and V2 is arithmetically Gorenstein forces a relation be-
CHAPTER 4.
84
GORENSTEIN SUBSCHEMES OF lpm
tween the free resolution of IVl and that of I v2 , and this relation is connected to the resolution of IVl n I v2 . This is what gives the self-duality of the minimal free resolution of IVl + I v2 . 0
4.2.2
Sections of Buchsbaum-Rim Sheaves of Odd Rank
Our second construction, introduced in [154] and [152]' comes from taking sections of certain reflexive sheaves. Let
F
t+r
= EB O!P'n( -ai)
t
and
i=l
Q = EB°!P'n(-bj ) j=l
and let ¢ : :F -+ Q be homogeneous. The map ¢ can be expressed as a homogeneous tx (t+r) matrix, and we assume further that the degeneracy locus of ¢ defined by the ideal of maximal minors of ¢ has the expected codimension, r + 1, in pn. This forces r ~ n. The kernel of ¢, denoted 8"" is reflexive of rank r, and in [152] it was given the name BuchsbaumRim sheaf (This comes from the fact that the map ¢ can be viewed as the first map of a longer complex, called the Buchsbaum-Rim complex, and the assumption on the codimension of the ideal of maximal minors implies that this complex is in fact exact. See [62] for more details.) In [152] a number of theorems are given to study the degeneracy locus of a set of multiple sections of a Buchsbaum-Rim sheaf. A special case is the degeneracy locus of one section, s, assumed to be regular (i.e. defining a scheme of the expected codimension r = rk 8",). The Gorenstein property arises here as a very pleasant surprise. We will focus on the situation in which we get Gorenstein schemes, but quote the result slightly more generally (although not in full generality). We want to study the ideal I corresponding to s E H~(pn, 8",), and we approach it as follows. Let F = H~(pn, F) and G = H~(pn, Q) be the free modules whose sheafifications are F and Q, respectively. We also view ¢ as a homomorphism from F to G. Let M", be the cokernel of ¢ and B", its kernel. One can verify that B", = H~(pn, 8",). Consider a minimal free resolution (the Buchsbaum-Rim complex) of M",:
...
-+ H
0
~
~ ? B", ? ~
F
~ G -+ M", -+ 0 (4.1)
0
4.2. CONSTRUCTIONS OF GORENSTEIN SCHEMES
85
It turns out that H has rank G!~), but what is important is that A has t+r rows. Then the module B¢ is simply the column space of A , and scan be viewed as an element of this column space, i.e. a linear combination (with coefficients that are homogeneous polynomials) of the columns of A. That is, I is the transpose of the corresponding element of the column space. The number of generators of I is precisely n + r, but it is not clear a priori that these are minimal generators or that I is saturated. (In fact , the latter is often not true, as we shall see shortly.) The result we want to cite is summarized in the following theorem.
Theorem 4.2.3 ([152]) Let s be a regular section of a Buchsbaum-Rim sheaf B¢ of rank r on IF, and let I be the ideal corresponding to s. Let Y be the subscheme of IF defined by I. Let J be the top dimensional component of I, defining a scheme X. (a) If r of I.
= n,
then X
=Y
is a zeroscheme and J is the saturation
(b) If r is even, r :::; n , then I = J is saturated and Y = X is arithmetically Cohen-Macaulay. Its Cohen-Macaulay type is :::; 1 + (r/~~~-l) . (c) If r is odd, r :::; n, then I is not saturated. If r < n then Y is not arithmetically Cohen-Macaulay. However, the top dimensional part, X, is arithmetically Gorenstein.
The paper [152] also gives a free resolution, usually minimal, for the schemes obtained. We will not present it here, except in the case of codimension three, which was proved in the earlier paper [154]' and codimension five (to give the idea of how the pattern goes in higher codimension). Let Cl = cl(B¢). In the case of co dimension three, rk F-rk G = rk B¢ = 3 and I y has a free resolution 0
.!-
R( -Cl)
+
G*
$
F*
$
.!.!Ix .!-
o.
F( -Cl)
G(-cd
86
CHAPTER 4.
GORENSTEIN SUBSCHEMES OF IP'n
For codimension five, rk F-rk G = rk B¢ = 5 and I y has a free resolution
0
-!.
R(-cd
-!.
S2(G)*
EB
-!.
F* 0G* /\2 F*
EB
F( -Cl) G( -Cl) EB
-!.
EB G*
EB
F*
EB
-!.
/\2 F(-Cl) F0G(-cd S2(G)( -Cl)
-!.
Ix -!.
o. The self-duality is evident from this description of the resolutions. From the point of view of liaison theory, it is important to be able to produce arithmetically Gorenstein schemes containing a given scheme, V, of the same codimension, c. We will assume that V is equidimensional. The easiest situation where this can be done is to produce a complete intersection containing V. Indeed, just take a regular sequence of length C in Iv. This can be done using polynomials of sufficiently large degree, but the question is always to determine whether this can be done in sufficiently small degree for the purposes at hand. We will see in the chapters on liaison theory that in codimension > 2, there are usually "not enough" complete intersections and it is important to look for other arithmetically Gorenstein schemes containing V. See for instance Example 5.1.7. We now show how this can be done for arithmetically Gorenstein schemes produced via Buchsbaum-Rim sheaves as above. We follow §6 of [152]. We first treat the case where the codimension, c, of V is odd. Let B¢ be a Buchsbaum-Rim sheaf on ~ of rank c. Sheafifying (4.1) , we get an exact sequence of sheaves
o -+ B¢ -+ :F ~ 9 -+ M¢
-+ 0
where Mq, is the sheafification of Mq,. There are two steps to consider. First, we want a regular section of Bq,(j) (for some j E Z) which is also in H~(~,:F 0Iv). This can be done if j is sufficiently large. Doing this will guarantee that we get a scheme,
4.2. CONSTRUCTIONS OF GORENSTEIN SCHEMES
87
Y, which contains V and has codimension c, and whose top dimensional part, X, is arithmetically Gorenstein. Since V is equidimensional of cod imension c, V must be contained in X. The second step is to isolate X, i.e. to get the saturated ideal of the top dimensional part of Y. For this we again use [64] Theorem 1.1 (3) to say that Ix = annsExts(Sj I y , S). This can be computed using standard scripts in macaulay [15] or using CoCoA [183]. However, an alternative method is described in Remark 4.2.4 below. Now suppose that the codimension, c, of V is even. We make a small deviation from §6 of [152]' although our construction is essentially the same. Notice that above, we were free to choose the Buchsbaum-Rim sheaf B¢ as we liked. We will assume then that the degeneracy locus of has codimension c as well (so that is a t x (t + c - 1) homogeneous matrix and B¢ has rank c -1, which is odd) , and assume furthermore that the degeneracy locus of ¢ has no component in common with V. Choose a regular section, s, of B¢ as above, defining a subscheme Y of F that contains V. Then [152] Proposition 4.5 and Theorem 4.7 force the conclusion that V lies inside the top dimensional part, X, of Y , and X has codimension c -1 and is arithmetically Gorenstein. Now take a hypersurface, F, containing V but meeting X properly. The hypersurface section of X cut out by F is again arithmetically Gorenstein (Corollary 1.3.8) of codimension c, and it contains V.
Remark 4.2.4 Another way to isolate the top dimensional part, X, of Y is by using liaison theory. The procedure is very simple. Choose a regular sequence, J, in I y of length equal to the codimension of Y. The ideal J is the saturated ideal of a complete intersection of co dimension c. Then the double ideal quotient [J : [J : Iy]] (cf. page 47) gives the saturated ideal Ix of the top dimensional part of Y (cf. Remark 5.2.5). In many calculations this is faster (sometimes much faster) than using the annihilator anns Exts (S j I y, S) as above. 0 Example 4.2.5 As an example, we give an edited (for space reasons) and annotated Macaulay session which produces an arithmetically Gorenstein curve X of degree 5 containing a given monomial quartic curve V in jp>4. The step to find the top dimensional part uses Remark 4.2.4.
%
R
;set the ring R = k[a,b,c,d,e]
%<monomial_curve 1 2 3 4 IV ;first produce the monomial ;quartic and call the ideal IV
88
CHAPTER 4. GORENSTEIN SUBSCHEMES OF Ipm
% std IV IV
;get a standard basis
%type IV
;here are the elements of IV ; d2-ee cd-be e2-ae bd-ae be-ad b2-ae
%mat PHI
;produce the homogeneous ;matrix [a bed]
number of rows ? 1 number of columns ? 4 (1,1) ? a (1,2) ? b (1,3) ? e
(1,4) ? d
%nres PHI RESPHI
;minimal free resolution ;of phi
% iden 4 14
;4 x 4 identity matrix
% tensor 14 IV TEN
;produee F (tensor) IV
% submat TEN FTENI !
rows? 1 .. 4 columns ? 1 . 24 .
%type FTENI
d2-ee cd-be c2-ae bd-ae be-ad b2-ac 0 0 0
0 0 0
0 0 0
0 0 0
o
o o
0
0
0
0
o
o o
o o
o o
o o
o o
o
0
0
0
0
0
0 0
0 d2-ee cd-be c2-ae bd-ae be-ad b2-ae 0 0 0 o 0 0 o 0 0 0 0 0
o o
d2- ce cd-be e2-ae bd-ae be-ad b2-ae 0
4.2. CONSTRUCTIONS OF GORENSTEIN SCHEMES
o o o
o o o
o o o
o o o
o o o
89
o o o
d2-ce cd-be c2-ae bd-ae bc-ad b2-ac
%std FTENI FTENI % intersect FTENI RESPHI.2 INT % std INT INT %
;get a regular section of ;the Buchsbaum-Rim sheaf, ;of suitable degree, ;also in F (tensor) MON
% flatten s IY
;make it an ideal, IY
% std IY IY % hilb IY 1 t
0
-4 t
2
t 6 t -5 t 1 t
3
1
1 t 3 t 2 t -1 t
4 5 6 0
1 2 3
codimension = 3 degree = 5 genus = 0
%
;this part of the Hilbert ;function gives the h-vector ;in case the ideal is aCM. ;Here it is not since it is ;not equidimensional . It ;consists of the union of a ;Gorenstein curve and a point.
90
CHAPTER 4
GORENSTEIN SUBSCHEMES OF pn
%
;produce the regular ;sequence J
% quotient J RE Q % quotient J Q IX
;IX is the top dimensional ; part
% hilb IX
1 t -5 t 5 t -1 t 1 t 3 t 1 t
0
2 3 5
0
1 2
codimension = 3 = 5 degree = 1 genus
;the genus has increased by 1 ;since the isolated point was ; removed
%
;verify that X contains V
Of course X must be the union of V with a suitable line. See also Example 5.2.26. 0
4.2.3
Linear Systems on aCM Schemes
The last construction which we will consider is potentially the most useful for liaison theory, and some of its consequences will be discussed in Chapters 5 and 6. It is based on the following theorem. Both the theorem and
4.2. CONSTRUCTIONS OF GORENSTEIN SCHEMES
91
the proof are taken from [121]. In order for the proof (and much of the theory built up in [121] and described in Chapter 5) to work, we have to use the machinery of Hartshorne's generalized divisors [90]. In particular, we will have to assume the condition G I , i.e. that we are working on a scheme which is Gorenstein in codimension 1. More precisely, we have the following definition. Definition 4.2.6 A Noetherian ring A (resp. a Noetherian scheme 5) satisfies the condition Gn "Gorenstein in codimension :S r," if every localization Ap (resp. every local ring Ox) of dimension :S r is a Gorenstein ring. 0 Example 4.2.7 This essentially says that the "bad" locus has codimension r + 1 or more. In the theorem below we will assume that our scheme satisfies condition G I , in order to be able to use Hartshorne's theory of generalized divisors. Unfortunately, this rules out some very simple and naturally occurring schemes. For example, the scheme 5 consisting of three lines in jp3 passing through a common vertex satisfies Go but it does not satisfy G I . See also Remark 4.2.9 and Example 4.2.10. The paper [121] discusses these problems. 0 Hartshorne also assumes Serre's condition 52, but we will assume a stronger condition, namely aCM. The following theorem is, in a sense, well-known. It comes from the fact that if an aCM ring R is generically Gorenstein (Go) then the canonical module is isomorphic to an ideal I, and Rj I is Gorenstein. One reference for this fact is [40] Proposition 3.3.18. Another reference, at least for the fact that the canonical module is an ideal, is [62] Exercise 21.18. An interesting result in the direction of a converse can be found in [24]. What is new in [121] is the geometric language, and the applications that are found from this point of view (for instance Theorem 5.2.27, Remark 5.2.28, Proposition 5.4.5 and Theorem 6.1.8) . Theorem 4.2.8 ([121]) Let 5 be an arithmetically Cohen-Macaulay subscheme of]pm satisfying condition GI , and let X be a subcanonical divisor on 5 (i.e. a subscheme of 5 defined by the vanishing of a regular section of ws(f) for some f E 7/.,). Let F E Ix be a homogeneous polynomial of degree d such that F does not vanish on any component of 5. Let Hp be the divisor cut out on 5 by F. Then the (effective) divisor Hp - X on 5, viewed as a subscheme of ]pm, is arithmetically Gorenstein. In fact, any effective divisor in the linear system IH p - X I is arithmetically Gorenstein.
CHAPTER 4.
92
GORENSTEIN SUBSCHEMES OF jpn
Proof: We are assuming that X is the divisor associated to a regular section of ws(t'). Let Y be the residual divisor, Y E IHF - XI = IdH - XI. Let I y be the homogeneous ideal of Y, viewed as a subscheme of F. Consider the ideal I y / Is in A = R/ Is and its sheafification IYI S in Os. We have
IYls(d) 2:! Os(dH - Y) 2:! Os(X) 2:! ws(€). Suppose that S has codimension r. Since S is arithmetically CohenMacaulay, the dual of the minimal free resolution of Is is a minimal free resolution for H~(ws) (cf. Remark 1.2.4) , and the last free module in this resolution has rank one. We thus have the following exact diagram (up to twist - what is important is the ranks): 0
.t.
0
R
Fr
F*1
.t.
.t.
.t.
F r-
.t. .t.
Fl
o
-+
.t.
Is
.t.
0
.t.
i
F.*2
.t.
(4.2)
.t.
F*r
.t.
-+ I y -+ H2(ws)(€ - d) -+ 0
.t.
0
The Horseshoe Lemma ([202J 2.2.8, p. 37) then shows that Y is arithmetically Gorenstein. Note that this shows that any element in the linear system of Y is arithmetically Gorenstein. 0
Remark 4.2.9 From the computational point of view, the construction of Theorem 4.2.8 is especially useful in producing arithmetically Gorenstein subschemes of codimension three, since S then has co dimension two, and the corresponding Hilbert-Burch matrix, A, of S is a homogeneous (t + 1) x t matrix. It turns out that under the above assumptions, the ideal of a subcanonical divisor X on S (viewed as a subscheme of F) can be obtained by deleting a column of A (possibly after a base change) and taking the ideal of maximal minors of the resulting (t + 1) x (t - 1) matrix.
4.2. CONSTRUCTIONS OF GORENSTEIN SCHEMES
93
See for instance [125] for details. Then the arithmetically Gorenstein ideal /y can be obtained by the ideal quotient [(Is + (F)) : Ix]. (This follows from liaison theory.) D Example 4.2.10 We would like to illustrate what can go wrong if we try to weaken the condition G 1 in Theorem 4.2 .8. Let us consider curves in 1P'3 of degree 3 and arithmetic genus 0, and look for Gorenstein zeroschemes on these curves. First, let S1 be a twisted cubic curve. This curve is smooth and aCM, hence it satisfies the hypotheses of Theorem 4.2.8, in particular G 1 . Since 2g - 2 = -2, we expect (and get) that a section of wS (1) will define a zeroscheme which has degree -2 + 3(1) = 1; i.e. it is a reduced point, X, on S1. A general hyperplane containing X yields an effective divisor in the linear system IH - X I which consists of two reduced points. This is a complete intersection, hence Gorenstein (as guaranteed by Theorem 4.2.8). Now consider the curve described in Example 4.2.7, which for convenience we now denote by S2 . This curve satisfies condition Go and the canonical module is an ideal. Any ideal that fits in the exact sequence (4.2) will clearly be Gorenstein. However, the geometric language of Theorem 4.2.8 is harder to apply to this situation. For example, a section of ws 2 (1) defines a zeroscheme, X, on S2 which is again a reduced point and in fact is the point of intersection of the three lines. A general hyperplane H through that point cuts out on S2 a zeroscheme Z lying on Hand having degree 3 (by Bezout's theorem) . What is this zeroscheme Z? One can check that it is simply the scheme corresponding to the square of the ideal of a point in H = ]p>2 . But since Z is not a local complete intersection, it turns out that trying to view H - X as a divisor on S2 does not give what one might expect from the example of S1 above - performing an ideal quotient as in Remark 4.2.9 gives a "residual" which is a single point rather than having degree 2. (This is of course still arithmetically Gorenstein, but one begins to see the problems that arise. In fact if one takes a homogeneous polynomial of degree 2 for H , the "residual" is not arithmetically Gorenstein.) This makes sense: a scheme of degree 2 supported at a point has a well-defined tangent direction, but the point X sitting inside Z does not "prefer" any direction over any other. See also Example 5.2.4. Finally, let S3 C ]p>3 be given by the square of the ideal of a line, say Is = (XJ, XOX1' X?). Then S is aCM with degree 3 and arithmetic genus 0, but it does not satisfy condition Go, hence the canonical module is not an ideal in the homogeneous coordinate ring. This is reflected in j
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the Hilbert-Burch matrix of S, which is the matrix
It is clear that any element of the column space of this matrix defines a curve rather than a zeroscheme, so the procedure described in Remark 4.2.9 fails. 0
4.3
Codimension Three Gorenstein Ideals
There is a surprisingly strong connection between the theory of codimension three arithmetically Gorenstein ideals and that of codimension two arithmetically Cohen-Macaulay ideals. In this section we describe some aspects of this connection. Nearly all results about codimension three arithmetically Gorenstein ideals ultimately rely on the structure theorem of Buchsbaum and Eisenbud [42], and we begin with that result. We first recall some terminology and facts (cf. [42]). If R is a commutative ring and F is a finitely generated free R-module, then a map f : FV --+ F is said to be alternating if, with respect to some (and therefore every) basis and dual basis of F and F V , the matrix of f is skew symmetric, and all its diagonal entries are O. If F has even rank, and f : F V --+ F is alternating, then det(f) is the square of a polynomial function of the entries of a matrix for f, called the Pfaffian of f, denoted Pf(f). That is, det(f) = (Pf(f))2. If F has odd rank, n, let A be the n x n matrix representing f. We define Pfn - 1 (f) to be the ideal generated by the Pfaffians of all submatrices of A obtained by deleting the ith row and the ith column, where 1 :S i :S n.
Theorem 4.3.1 ([42]) Let R be a noetherian local ring with maximal ideal m.
(a) Let n 2 3 be an odd integer, and let F be a free R-module of rank n. Let f : F V --+ F be an alternating map whose image is contained in mF. Suppose that Pfn-l(f) has grade 3. Then Pfn-l(f) is a Gorenstein ideal, minimally generated by n elements. (b) Every Gorenstein ideal of grade 3 arises as in (a). Notice that as a consequence, every arithmetically Gorenstein codimension three subscheme of r has a defining saturated ideal with an odd
4.3. CODIMENSION THREE GORENSTEIN IDEALS
95
number of minimal generators. One interpretation of Theorem 4.3.1 is that one matrix completely determines everything there is to know about the ideal and minimal free resolution of a co dimension three arithmetically Gorenstein ideal. This should be compared with the Hilbert-Burch theorem, Theorem 6.1.1, for the codimension two arithmetically CohenMacaulay situation. We now list a number of other facts from the literature, most of which follow from these matrix results, which highlight the amazing connection between these situations. We use the notation "CM2" for "arithmetically Cohen-Macaulay, co dimension two" and "G3" for "arithmetically Gorenstein, codimension three."
(1)
CM2
[130]. A necessary and sufficient condition for the h-vector of a CM2 subscheme of]pn (or indeed, a subscheme of any codimension) is given in terms of the maximal possible growth of the Hilbert function. See also (for instance) [22] for details.
G3
[192]. A symmetric sequence of integers is the hvector of a G3 subscheme of ]pn if and only if the first difference (cf. page 27) of the "first half" of the h-vector is the h-vector of a CM2 subscheme of]pn.
Example The h-vector (1,3,6,7,9,9,7,6,3,1) exists as the hvector of an arithmetically Cohen-Macaulay, codimension three subscheme of ]pn, but it cannot occur as the h-vector of a G3 subscheme since there is no CM2 subscheme of]pn with h-vector (1,2,3,1,2). (2)
(3)
CM2
[68]. The family of CM2 subschemes of]pn with the same Hilbert function is irreducible.
G3
[60] .The family of G3 subschemes of ]pn with the same Hilbert function is irreducible.
CM2
[37], [52]. The family of CM2 subschemes of]pn with the same Hilbert function and degrees of generating sets for the homogeneous ideal is irreducible.
96
(4)
(5)
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GORENSTEIN SUBSCHEMES OF Ipm
G3
[60]. The family of CM2 subschemes of]pn with the same Hilbert function and degrees of generating sets for the homogeneous ideal is irreducible.
CM2
[68] . Every CM2 subscheme of]pn is non-obstructed (i.e. the corresponding point of the Hilbert scheme is smooth).
G3
[158], [116], [43], [103] . Every G3 subscheme of]pn is non-obstructed.
Remark
The formula in [68] for the dimension of the Hilbert scheme (in terms of graded Betti numbers) in the CM2 case has an analog in the G3 case (again in terms of graded Betti numbers) which appears in [122].
CM2
[186], [77], [36], [49] . A necessary and sufficient condition, in terms of the degrees of the entries of the Hilbert-Burch matrix, is given for the existence of an integral CM2 subscheme of]pn.
G3
[97]. A precise analog is given for the existence of an integral G3 subscheme of]pn in terms of the degrees of the entries of the skew symmetric matrix arising from Theorem 4.3.1.
Remark
In [186], [36] and [49] conditions are in fact given for the existence of a smooth CM2 subscheme of ]pn. (The conditions depend also on n : for jp3 and r the conditions are the same, then in F they are slightly more stringent, and for n ?: 6 it is known that no smooth CM2 subschemes exist.) For the case of G3, it is proved in [97] that if n = 4 then whenever their condition is satisfied, there in fact exists a smooth G3 curve in r with the given degree matrix. So far there is no G3 analog for n > 4.
4.3. CODIMENSION THREE GORENSTEIN IDEALS
(6)
(7)
CM2
[83], [133], [77]. An integral CM2 subscheme of IF exists with given h-vector if and only if that h-vector has decreasing type (cf. page 32 and 180) .
G3
[58] An integral G3 subscheme of IF exists with given h-vector if and only if the "first half" of that h-vector has first difference of decreasing type.
Remark
In [83] and [133], the decreasing type condition is in fact shown to be necessary and sufficient for the existence of a smooth curve in p3. The analog seems to be still open for Gorenstein curves in p4.
CM2
[71], [52], [37]. Every arithmetically CohenMacaulay curve in p3 specializes to a stick figure , i.e. to a union of lines having only nodes as singularities. Furthermore, this can be done in such a way that the degrees of the minimal generators of the ideal are preserved. Finally, this notion can be extended to the general CM2 case by extending "stick figure" to "good linear configuration." (See also [35].)
G3
[79]. Every arithmetically Gorenstein curve in p4 specializes to a stick figure. Furthermore, this can be done in such a way that the degrees of the minimal generators of the ideal are preserved. Finally, this notion can be extended to the general G3 case by extending "stick figure" to "good linear configuration."
Remark
It has been known for some time which sequences of integers can be the set of graded Betti numbers of an Artinian co dimension two Cohen-Macaulay graded algebra, and that all of these in fact occur for a reduced set of points in ]P2, and hence for a CM2 subscheme of IF for any n ~ 2 (cf. for instance [186]). Diesel [60] described the set of all possible graded Betti numbers of an Artinian codimension three Gorenstein graded algebra. H. Kleppe [115]
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GORENSTEIN SUBSCHEMES OF Ipm
showed that any G3 zeroscheme in jp3 deforms to a reduced set of points. A consequence of the work in [79] is to show that all the possibilities allowed by Diesel for the Artinian case in fact occur for a reduced G3 subscheme of IF for any n ~ 3. See Theorem 4.3.2 below. To illustrate some of the ideas and connections described above, we will sketch the proof of the main result of [79], which uses CM2 results and techniques to prove a G3 result. It is based in a very simple idea: if C and C' are reduced with no common components, and if C u C' is a stick figure, then C n C' is also reduced. If C u C' is a complete intersection then C n C' is arithmetically Gorenstein. If we can control the graded Betti numbers of C, C' and C u C' then we control those of C n C' .
Theorem 4.3.2 ([79]) Any minimal free resolution which occurs for G3 Artinian ideals actually occurs for some reduced set of points in jp3 and for a stick figure in jp4. Every G3 curve in jp4 specializes to a stick figure with the same graded Betti numbers. Proof (Sketch): The fact that any G3 zeroscheme in jp3 deforms to a reduced set of points was also shown in [115]. Let I be a G3 ideal in a polynomial ring S. Consider a minimal free resolution
2k+1
o -+ S( -t) -+ EB i=l
S( -bi ) -+
2k+l
EB S( -ai) -+ I -+ 0, i=l
where a1 ~ ... ~ a2k+1 and b1 ~ ... ~ b2k+1. It follows from the selfduality of the resolution that bi = t - ai for all i, and a Chern class computation gives t + L ai = L bi · One can then check that k+l
L ai = i=l
(t - ak+2) + . .. +(t - a2k+1).
We are trying to construct a G3 ideal having the same graded Betti numbers as I, and which has certain nice geometric properties. The first step is to simply show the existence of some CM2 subscheme C of IF with minimal free resolution 0-+
2k+l
EB
j=k+2
S(-t+aj)
4
k+1 EBS(-ai) -+ Ie -+ O. i=l
(4.3)
4.3. CODIMENSION THREE GORENSTEIN IDEALS
99
This is done using the above numerical facts together with criteria from [77] and [60] on the existence of CM2 schemes and G3 schemes, respectively. For this step we do not yet try to achieve any geometric properties. Notice that we are not saying that for the original ideal I, the first k + 1 minimal generators necessarily define a CM2 scheme. This is unfortunately not true. (Iarrobino and Kanev [112] describe situations in terms of the h-vector where it is, in fact, necessarily true.) We are simply saying that a CM2 scheme exists with the graded Betti numbers we have just described. The next step is to show that we can find another CM2 scheme C' so that C' has no component in common with C, and the union of C and C' is a suitable complete intersection. This is done using liaison theory, and in fact the minimal free resolution of C' can be computed (by applying Proposition 5.2.10 in a suitable way). A mapping cone argument as in the proof of Theorem 4.2.1, followed by an analysis of the minimal generators, then gives that Ie + Ie' is G3 with the desired resolution . At this point Ie + Ic' is not even necessarily reduced. Now, to produce the desired ideal with the geometric properties we seek, we show how to arrange that the complete intersection C U C' is itself very nice. In particular, to produce a reduced G3 set of points in jp3 we arrange that C U C' is itself a stick figure. Since Ie + Ic' is the ideal of the intersection of C and C', we then get that this intersection is reduced. (The fact that C U C' has only nodes as singularities is important here.) To produce a G3 stick figure in p4, we similarly arrange that C U C' is a union of planes with good crossing properties, so that the intersection of C and C' is the desired G3 stick figure. (One can extend this by defining a "good linear configuration" in any dimension or codimension - cf. [35].) For convenience, let us consider the case where C and C' are curves in jp3 and we are seeking to construct a G3 zeroscheme. The idea for producing the suitable (stick figure) complete intersection C U C' is to first produce the curve C, and then build a suitable complete intersection around it. The curve that is "left over" (or residual, in the language of liaison theory) will be C'. The main step is producing C; notice that it must also be a stick figure, since C U C f is a stick figure. The construction is essentially an idea in [71] and [52]. From the resolution (4.3) we know the degrees of the entries of the Hilbert-Burch matrix, A, of C. One constructs A (and hence C - see Theorem 6.1.1) by putting along two diagonals homogeneous polynomials of suitable degree which are products
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of sufficiently general linear forms, and zeros everywhere else: PI,1
0 0 0 0
QI,2 P 2,2
0 0 0
0
0 0
0
Q3,4 P 4,4
0 0 0 0
0
0
Pk,k
Q2,3 P 3,3
0 0 0 0 Qk,k+l
The result of this IS that the two appropriate minimal generators of Ie , namely F = PI ,1 ... Pk,k and G = QI,2 ... Qk,k+l, are both unions of planes, with no common factor, such that the complete intersection they define is a stick figure. Then of course C, being a subscheme, is also a stick figure , and so is the residual C' . The last step is to show that any G3 subscheme of ~ specializes to one of these that we have constructed. (The statement of the theorem was for n = 3 and n = 4, but the same idea works in general.) This follows from a result of Diesel [60], that the scheme parameterizing G3 subschemes of ~ with the same graded Betti numbers is irreducible. D
Remark 4.3.3 As mentioned already, our purpose in this chapter, especially in the first two sections, is to give the facts about Gorenstein subschemes of projective space that we will need in the chapters on liaison theory, at the expense of omitting a wealth of interesting material. We have given some further references already, especially in this section. In this remark we would like to mention some additional references. Of special note is the manuscript of Iarrobino and Kanev [112], which gives many of the recent results on arithmetically Gorenstein ideals and describes much of the theory that is not treated here; it also gives many additional references. Other interesting references are [23], [24], [25], [26], [72], [110], [111], [114], [118], [120], [122]. D
Chapter 5 Liaison Theory in Arbitrary Codimension In this chapter we introduce liaison (or linkage) theory for subschemes of arbitrary codimension in projective space. The largest part of the theory has been developed around the notion of "linking" by complete intersections, but another large part has also been developed around "linking" by Gorenstein ideals. From now on we will refer to these as "complete intersection liaison" and "Gorenstein liaison" respectively, and we will abbreviate them by CI-liaison and G-liaison. The most complete picture (in terms of theory and also applications) has been in the case of codimension two, where we saw in Example 4.1.11 that the Gorenstein ideals are precisely the complete intersections. Even in higher co dimension, where much less is known, there is more work in the complete intersection setting than in the more general Gorenstein setting. For this reason, the first version of this book treated only the case of complete intersections. We will still treat the codimension two case in the next chapter. However, to pass to arbitrary codimension and view the "big picture," one has to decide whether to view the codimension two case as part of CIliaison theory or G-liaison theory. Recent research [121] strongly suggests (at least to the author) that G-liaison is, in many ways, a more natural approach to higher codimension. Many results which hold in codimension two, but which are false for CI-liaison in higher codimension, seem to be true for G-liaison in higher codimension. See, for example, Theorem 5.2.27, Remark 5.2.28 and Theorem 6.1.8. On the other hand, there are some disadvantages to G-liaison. One is the difficulty in finding a Gorenstein ideal, other than a complete intersection, with "nice" properties to link a given ideal to another one. This problem was addressed in
J. C. Migliore, Introduction to Liaison Theory and Deficiency Modules © Birkhäuser Boston 1998
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CHAPTER 5. LIAISON THEORY
Section 4.2. Another disadvantage is described in Example 5.2.26. As a result, this material has been extensively revised in this version of the book. It is the author's hope that these chapters will prove to be a useful source of information about liaison theory, both for someone wishing to learn the codimension two story, and for someone wishing to follow the developing picture of G-liaison for subschemes of projective space. (It should be noted that even this is a restriction - a great deal of work has been done in a more general algebraic context. We refer the interested reader to [199]' from which he or she can "get a foot in the door" and branch off to the vast literature in this direction.) There do not seem to be many introductory references for liaison theory. A very good first step is the short discussion in [62] and in [173], and the paper [92]. Another useful source, albeit only for curves in lP'3, is the lecture notes [181]. In this context, [137] is also a must-read. See also [188] and [195] for a discussion of liaison theory and its connections to Buchsbaum rings. A new approach to the subject, which develops the theory from the point of view of generalized divisors on a Gorenstein scheme, is given in [90]; we shall refer further to this paper. Other useful sources for G-liaison are [187], [36] and [163]. Historically, liaison began in the last century as a tool to study curves in projective space. The idea was to start with a curve, say in jp3, and look at its residual in a complete intersection. Since complete intersections are in some sense the simplest curves, it turns out that a lot of information can be carried over from a curve to its residual (or vice versa). So it was hoped that using this process of taking residuals, one could always pass to a "simpler" curve and so it would be easier to get information about the original curve. This idea in fact works nicely for aCM curves in jp3. This was essentially shown by Apery and Gaeta in the 1940's, and it was proved rigorously with modern machinery by Peskine and Szpiro [174] in 1974. A very exciting generalization of this result for G-liaison is mentioned below (Theorem 6.1.8). However, Joe Harris conjectured that this notion would not work in the non-aCM case even for curves in jp3 ([87] p. 80). This was proved by Lazarsfeld and Rao in 1982; the idea is that the "general smooth curve" is already the simplest curve in its liaison class, in many senses. This is a special case of (and indeed it inspired!) the structure theorem for codimension two even liaison classes (cf. [12], [137], [163], [167]) which we will discuss in the next chapter. Work of Huneke and Ulrich [104] can also be interpreted as showing that this idea also fails even for aCM curves in higher codimension.
5.1. DEFINITIONS AND FIRST EXAMPLES
103
Liaison has been used very extensively in the literature as a means of studying curves (or indeed higher dimensional varieties) and of producing interesting examples. Furthermore, beginning with the paper of Peskine and Szpiro, liaison has attracted a great deal of attention as a subject in and of itself, rather than simply a tool to study projective varieties. We will discuss several of these results in these chapters. As we will see, the deficiency modules play an important role also in this field, thanks especially to work of Rao [179], [180], Schenzel [187], Hartshorne [90], Nagel [163] and Nollet [167].
5.1
Definitions and First Examples
As we indicated above, liaison theory involves the study of properties that are shared by two schemes whose union is well-understood, namely either a complete intersection or simply arithmetically Gorenstein. These notions generate, respectively, the theory of liaison under complete intersections (CI-liaison) and the theory of liaison under Gorenstein ideals (G-liaison). Since complete intersections are arithmetically Gorenstein, many of the results in this chapter hold true in both cases. We will describe our results in the generality of G-liaison whenever possible, and also discuss the situations where the results are different. Actually, the point of view of looking at the union of two schemes is a bit too naive: when the two schemes have no common component then there are no problems, but otherwise "union" is too weak and we need to take a more algebraic approach. In any case we begin with the weaker notion. The terminology G-linked and CI-linked used below was introduced in [121]. Definition 5.1.1 Let VI, V2 be subschemes oflP'" such that no component of VI is contained in any component of V2 and conversely. Then VI is (geometrically) directly linked to V2 by an arithmetically Gorenstein scheme X if VI U V2 = X. We also say that VI is geometrically G-linked to V2 by X. If X is a complete intersection then we also say that VI is geometrically CI-linked to V2 • From the point of view of the saturated ideals, this says that Iv, n IV2 = Ix . The scheme VI is said to be residual to V2 in X. 0
Notice that deg VI +deg V2 = deg X. (See Corollary 5.2.13 for a more general statement.) The simplest example of two curves that are geometrically directly linked is the union of two lines, given as the intersection of a pair of planes with another plane (see Figure 5.1). The problem comes
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CHAPTER 5. LIAISON THEORY
x Figure 5.l: Geometric Link when we try to extend this notion to the case where the curves may have common components. For example, if the second surface (the plane) contains the line of intersection of the two planes comprising the first surface, this is still a perfectly good complete intersection (see Figure 5.2).
Figure 5.2: Algebraic Link The only natural way to interpret this would be to say that the line of intersection is linked to itself (since the complete intersection still has degree two), but using only unions of course we do not have the equality in the definition. The solution is to use ideal quotients.
Definition 5.1.2 Let VI, 112 be subschemes of JIM of codimension r, and let I x ~ I VI n I V2 be an arithmetically Gorenstein ideal. Then VI is (algebraically) directly linked to V2 by X (VI ~ V2 ) if and only if[Ix : IVI 1= IV2 and [Ix: IV2l = I vl · We also say that VI is directly G-linked to V2 by X. If Ix = (FI ' ... , Fr) is acorn plete intersection then we say that VI is directly CI-linked to V2 . The scheme VI is said to be residual to V2 in X. 0 We can check that this definition makes sense in the example described above in Figures 5.1 and 5.2:
5.1. DEFINITIONS AND FIRST EXAMPLES
105
Example 5.1.3 Assume n = 3 and let S = k[Xo, ... , X3]. Let Ix = (XOXl , X 2) and Iv! = (X o, X2). Then IV2 = [Ix : I vl ] = (Xl, X2). (So the first case is ok.) Now let Ix = (XOX l , Xo + Xl) and IVI = (Xo, Xl). Then IV2 = [Ix: I vl ] = (Xo, Xl) = IVI (so it is "self-linked"). D Remark 5.1.4 Our Definition 5.1.2 of algebraic linkage appears, at first, to not be quite the same as that used by Schenzel [187], Peskine and Szpiro [174] and Rao [179]. They say that VI and V2 are algebraically linked by X if and only if
(i) Ix c IVI n I v2 ; (ii) VI and
V2 are equidimensional, without embedded components;
(iii) IV2/Ix
= Homol?n(Ovl' Ox)({:} IV2/Ix = Homs(S/Ivll S/Ix))
(iv) IVI/Ix = Homopn (Ov2 ' Ox)({:} IvJIx = Homs(S/Iv2 ,S/Ix)) To see the equivalence, the main point is that there is a natural isomorphism
[Ix Ix : I vl ] '" = Horns (/ S lVI'S/Ix) . (There is a similar isomorphism obtained by reversing the roles of VI and
V2.) This equivalence is left to the reader as an exercise; it was first observed by Schenzel [187]. Notice that another way of saying this is that IV2 is the annihilator of IVI in S/ Ix. Peskine-Szpiro and Rao assume that the linking is done by a complete intersection, but Schenzel observes that it works equally well for arithmetically Gorenstein ideals. D
Remark 5.1.5 As one would expect from the way we have set things up, one can show that if VI and V2 are geometrically linked then they are algebraically linked. Indeed, assume that they are geometrically linked. This means that Iv! nlv2 = Ix and that neither IVI nor IV2 is contained in any associated prime of the other. We wish to show that [Ix: I vl ] = IV2 (and vice versa). The inclusion ;;2 is left as a quick exercise. For the inclusion ~, say F E [Ix: I vl ], so F· IVI C Ix. Choose C E IVI such that C is not in any associated prime of IV2 (i.e. C vanishes on no component of IV2). Then
FC E Ix = Ql n··· n Qr n Q~ n··· n Q:
~~
IV2
IVI
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CHAPTER 5. LIAISON THEORY
In particular FG E Qi for all i, but no power of G is in Qi. Therefore by the definition of "primary," FE Qi for all i so F E I v2 . Conversely, if Vl and V2 have no common components and are algebraically directly linked, then they are geometrically directly linked. See Proposition 5.2.2 for the proof. From now on we will assume that our links are algebraic links, and will write Vl~V2 without further comment. However, see Remark 5.1.9. 0 Note that the definition of direct linkage is symmetric but not usually reflexive (only in the case of "self-linkage") or transitive. So we need to extend it in order to get an equivalence relation: Definition 5.1.6 Liaison is the equivalence relation generated by direct (algebraic) linkage. If the links are by arithmetically Gorenstein ideals then the equivalence relation is called Gorenstein liaison, and if the links are by complete intersections then the equivalence relation is called complete intersection liaison. We will sometimes abbreviate these by G-liaison and CI-liaison, respectively. More precisely, we say that Vl is G-linked to V2 , denoted Vl £ V;, if there is a sequence of schemes W l , .. . , W k and a sequence of arithmetically Gorenstein schemes Xl , ... , X k + l such that
We say that Vl is CI-linked to V;, denoted Vl ~ V2 , if the links are by complete intersections. If there is no ambiguity (for instance in codimension two), we simply write Vl '" V;. The equivalence classes generated by this procedure are called, respectively, G-liaison classes (or G-linkage classes) and CI-liaison classes (or CI-linkage classes). If k + 1 is even, we say that Vi and V2 are evenly linked. Notice that even linkage also generates an equivalence relation, and the equivalence classes are called even liaison classes. An even liaison class will usually be denoted by C. The notion of even liaison is sometimes also called biliaison [137], [90], and evenly linked schemes are said to be bilinked (if necessary, we use the prefix "G-" or "CI-" to specify the kind of links being used). 0 Now that we have defined our equivalence classes, there are many natural questions that arise. We will discuss many of the answers in this book. We will see that, by far, the most complete picture is in the case of codimension two. However, we will also try to describe the known and the emerging picture in higher co dimension.
5.1. DEFINITIONS AND FIRST EXAMPLES
107
Questions /Pro blems 1. Find connections between directly linked schemes (e.g. degree, genus, and especially more subtle connections). Find properties that are preserved. For example, we will see that the property of being aCM is preserved, as is the property of being arithmetically Buchsbaum (Remark 5.3.2). Also the dimension is preserved (Proposition 5.2.2). As a consequence, find invariants of a liaison class or of an even liaison class (see for instance Section 5.3). 2. Is this a trivial equivalence relation? From what we said in item 1., we can already deduce that the answer is "no." Indeed, since the property of being aCM is preserved, as is the property of being arithmetically Buchsbaum, we see that there have to be at least three classes, since there exist non-aCM Buchsbaum curves and there exist non-Buchsbaum curves in JIM. This was already known to Gaeta in the 40's. (In fact , in codimension two the aCM subschemes form one liaison class; see Corollary 6.1.4. In higher co dimension it is known that there are infinitely many aCM CI-liaison classes.) 3. Parameterize the (even) liaison classes. Give necessary and sufficient conditions for two subschemes to be in the same (even) liaison class. 4. Describe anyone even liaison class. (We will see below that the even liaison class of a given scheme is the "right" object to study, rather than the entire liaison class.) Is there a standard way to describe "distinguished" elements in that class? What structure does the class have? In particular, is there a structure common to all even liaison classes? What consequences does this structure have? 5. Can liaison be studied in greater generality? For instance, can we talk about linkage of subschemes of something other than projective space? We will not talk about this much, but the answer is emphatically "Yes!" In this context important work has been done by Huneke, Kustin, Miller and Ulrich, among others. (For instance, see [104], where one can find further references. Also, [36] can be viewed as an attempt to bridge some of these ideas.) On the other hand, Hartshorne [90] has developed a more general theory of divisors. This approach allows him to give a new definition of linkage via complete intersections, which is equivalent to that of algebraic CI-linkage (Definition 5.1.2) "but closer in spirit to the intuition coming from the classical geometrical linkage." We will see that this
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approach extends also to Gorenstein liaison, from [121] . See also Example 5.1.7 (4.) and Theorem 5.2.27 below. Another direction is to ask whether we can perform links using something other than arithmetically Gorenstein schemes. The most natural guess would be to allow aCM schemes for our linking. However, Walter [200] has shown that if we replace "arithmetically Gorenstein" with "aCM" then that is too much - in that case there is only one equivalence class for any given codimension in projective space (or, more precisely, he shows that any locally Cohen-Macaulay equidimensional subscheme of lP'" can be aCM-linked to the empty set in a finite number of steps). Similar results have been obtained by Martin [134]. And finally, liaison has been generalized to the notion of residual intersections (where even the dimension is not preserved); this was introduced by Artin and Nagata [6] and studied for instance by Huneke and Ulrich in [109]. 6. What are some applications of liaison theory?
Example 5.1.7 1. We saw that in p3 we can directly link a line to another line (provided that the lines either meet in one point or coincide). In fact, if one considers p3 as a complete intersection linear subvariety of ]pn, one can show that the same holds in ]pn (but now the codimension is n - 1 so the number of generators of the complete intersection performing the link is n - 1 rather than two). 2. Any curve of type (n, n) on a smooth quadric in 1P'3 is a complete intersection, so a twisted cubic is directly linked to a line, two skew lines are directly linked to two skew lines (in the opposite ruling), etc. Furthermore, one can directly link two skew lines to a double line (i.e. a scheme of degree 2 supported on a line) in the opposite ruling by the same reasoning. More generally, one can link any curve of type (a, b) to any curve of type (c, d) if and only if a + c = b + d. 3. Schwartau [189J gives a simple proof that any two complete intersections of the same codimension are linked. The proof follows from -the following lemma: If IXl = (Fl , ... , Fd- l , F), IX2 = (Fl , ... , Fd- l , G) and Ix = (Fl , ... , Fd - l , FG) then Xl~X2. So now starting with arbitrary complete intersections Yl and Y2 one can produce a sequence of links from one to the other in d steps: just apply the lemma to change one generator at a time.
5.1. DEFINITIONS AND FIRST EXAMPLES
109
In particular, any two hypersurfaces are linked. (For example, any two plane curves are linked. Note that a plane curve is still a complete intersection even if it is embedded in a higher projective space, so even in this context any two plane curves are linked.) As remarked earlier, we will see in Chapter 6 that for co dimension two, being in the linkage class of a complete intersection is equivalent to being aCM. In higher codimension, this is not true. A great deal of work has been done on subschemes of projective space that are lieci; that is, subschemes that are in the linkage class of a complete intersection. See for instance [104], [106], [107], [108], [156], [196], [197] . The result indicated in this remark justifies the use of the term "linkage class of a complete intersection," since it shows that there is just one such class (for a given codimension). In [121], the notion of subschemes that are glieei (i.e. in the Gorenstein linkage class of a complete intersection) was introduced. We will further discuss this notion below, for instance in Section 6.1. 4. If two curves on a smooth surface F in 1P'3 are linearly equivalent then they are evenly linked. Here is an intuitive proof. Say C is linearly equivalent to C' on F . Then C - C' is the divisor of a rational function ~, so there is some divisor D on F such that G j cuts out C + D (i.e. C is linked to D) and G 2 cuts out C' + D (i.e. D is linked to C'). Hence C is evenly linked (in two steps) to C'. More generally, we may replace F by a smooth complete intersection subscheme of IF of any co dimension and obtain a similar result for divisors on a complete intersection (cf. [174] Exemple 2.4). But in fact, in the context of Gorenstein liaison, it turns out that the complete intersection can be replaced by any aCM scheme satisfying property G j (in particular a smooth one)! This will be described in Theorem 5.2.27 below. Using his notion of generalized divisors on Gorenstein schemes mentioned above, Hartshorne [90] showed that the above idea using linear equivalence is in a sense the central idea behind liaison, and does not require the hypothesis of smoothness used above. He derives from this approach all of the basic theorems of liaison found in the first three sections of this chapter (in the case of complete intersection links). Theorem 5.2.27 below will describe a similar set-up for Gorenstein liaison. 5. Let Z be a set of four points in p3 in linear general position. This is the simplest example of a scheme which is not licci (see Example 6.1.6). For instance, if one tries to make a link using the smallest possible complete intersection containing Z, namely the complete intersection of three quadrics, one gets back a set of four points (possibly non-reduced). It is,
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however, glicci: simply adding a general point to Z gives a Gorenstein zeroscheme (cf. Example 4.1.11 (b)), and so Z is directly G-linked to a complete intersection. This shows that the G-linkage class of a complete intersection is not the same as the CI-linkage class of a complete intersection, except in codimension two. Much less work has been done to date on the former than on the latter. Remark 5.2.28 and Theorem 6.1.8, both from [121], are worth noting in this regard. 6. Let] = (F, G) be a complete intersection in codimension two. Then it is not hard to show that the complete intersection ideal Jk = (Fk, Gk) gives a direct CI-link between ]k and ]k-l. In particular, these powers are the saturated ideals of aCM schemes. See Example 6.1.5 for the proof. For instance, the scheme defined by the square of the ideal of a line in ]p3 (which has degree 3) is directly linked to the line itself. This is not true for CI-liaison in higher codimension. There are a few results along these lines for G-liaison. It can be shown that if ] is the ideal of a single point in ]p3 then ]k is directly G-linked to ]k-l by a suitable Gorenstein ideal. We omit the details. I am not aware of other results in this direction for direct linkage, but see also Corollary 6.1.10 for even G-liaison. 7. There are two very useful programs for making computations in Commutative Algebra, which in particular are very nice sources of examples for liaison theory. They are Macaulay [15] and CoCoA [183] . See also Remark 1.1.7. 0
Example 5.1.8 We mentioned above that liaison has also been used extensively as a tool, especially for constructing interesting varieties. Here is a very short list of examples of ways in which it has been applied. (Some of these will be described in much better detail in Section 6.4.) Note that they are almost all in the context of codimension two. 1. In [86] Harris gives sharp bounds for the genus of an integral curve lying on an irreducible surface S of degree k in ]p3 (in terms of k and the degree d of the curve). The curves which achieve his bounds are exactly the integral curves residual to a plane curve in a complete intersection of S with a surface of degree [dkl] + 1.
2. In [133], Maggioni and Ragusa use liaison to produce smooth aCM curves in ]p3 for every Hilbert function of "decreasing type." 3. In [19], Beltrametti, Schneider and Sommese use liaison to construct smooth aCM threefolds of degree 9 and 10 in F and to show that they are the unique examples with the given invariants. In particular, for degree
5.1. DEFINITIONS AND FIRST EXAMPLES
III
10, every smooth threefold in ]p'5 is aCM. On the other hand, in [157] MiroRoig uses liaison to construct smooth non-aCM threefolds in ]p'5 of degree lOn, n > 1, completing the work begun by Banica [14] of determining the dpgrees of non-aCM smooth threefolds of ]p'5. Beltrametti, Schneider and Sommese also do a similar analysis of the case of degree 11 in [20]. See also [57] for an overview of work on this problem and that of surfaces in
IfD4.
4. In [35], the structure of an even liaison class (see Question 4 above) is used to show that every Buchsbaum curve in ]p'3 specializes to a stick figure (see Section 6.4 for more details). 5. In [177], Popescu and Ranestad give a very powerful use of both deficiency modules and liaison to complete the description of the components of the Hilbert scheme of surfaces of degree 10 in IfD4 . 6. Liaison has also been used to study unobstructedness and obstructedness, both of subschemes of JP'1l and of flags of subschemes of JP'1l (cf. [43] for aCM subschemes and [116], [30] more generally). This is closely connected to the question of studying families of subschemes of]p'n. It turns out that if each member of a given family X of subschemes of JP'1l is contained in a corresponding member of a family Y of complete intersections of fixed type , we can talk about Y "linking" X to another family X'. In such a setting, assuming the vanishing of some of the deficiency modules in certain degrees, one may prove that unobstructedness (as well as the property of being general in the Hilbert scheme) is preserved under CI-linkage. See for instance [116], [30], [121], [137], [94]. 7. Liaison supplies a simple and efficient way of finding the top dimensional part of a subscheme of JP'1l. See Remark 4.2.4, Example 4.2.5 and Remark
5.2.5.
0
Remark 5.1.9 In Remark 5.1.5 we observed that two schemes are geometrically linked if and only if they are algebraically linked and have no common component, and we said that we will thus focus our attention on algebraic links. There is an important point that is left open by this approach, and we briefly discuss it here. We restrict ourselves in this remark to CI-liaison. It is not hard to see that if V is geometrically CI-linked to V' then both V and V' are generic complete intersections (and of course equidimensional). Hence if one restricts to the set H (c, n) of codimension c, generic complete intersection, equidimensional subschemes of JP'1l, one can study the equivalence relation generated by geometric links. One can also
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study the equivalence relation on H(c, n) generated by algebraic links, where at each step we assume that we remain in H (c, n), and ask if these two are the same. It was proved in [179] for codimension two subschemes of IF that these equivalence relations are in fact the same if we also assume that our schemes are locally Cohen-Macaulay. This was generalized in [121], allowing the co dimension to be arbitrary and removing the locally Cohen-Macaulay assumption. There is an important variant of this question. Suppose that V and V' are both elements of H(c, n), and that one can get from V to V' in a series of algebraic links not all of which are necessarily in H(c, n) (so in particular, some of the links are not geometric and some of the schemes obtained in the process are not necessarily generic complete intersections). Is it still possible to get from V to V' by a series of geometric links? This was answered affirmatively by Schwartau [189] for the case of curves in p3, and for codimension two in general (assuming only equidimensional) in [121]. The case of higher co dimension seems to be open. Similarly, there seems to be little work on the question for G-liaison. In [179] Remark 1.5, Rao makes the following observation about what is required to make a geometric link, and what is required to make an algebraic link where the residual is again a generic complete intersection: Let V be a generic complete intersection of codimension two in IF. Let Ix = (Fl ' F2) ~ Iv be a complete intersection and suppose Ix : Iv = IVI. Then (a) V'is also a generic complete intersection if and only if at each generic point 17 of V, either Fl or F2 yields a minimal generator ofIv,t). If Fl E HO(IF, Iv(vl)) is given arbitrarily, then for V2 » 0 there is an open set in HO (IF , Iv (V2)), elements F2 of which satisfy this condition. (b) The link is geometric if and only if at each 'f/, Fl and F2 give a minimal basis of IV,t). If VI, V2 » 0 then there is an open set in (IV)Vl x (Iv)v2' elements of which satisfy this condition. In fact, it is always possible to find FI and F2 forming part of a minimal basis for Iv and giving a geometric link. Note that while Rao assumed that his schemes are locally CohenMacaulay, they hold equally well in our more general context. These observations form the local version of Remark 6.2.2. 0
5.2. RELATIONS BETWEEN LINKED SCHEMES
5.2
113
Relations Between Linked Schemes
In this section we will give some answers to the first question raised in Section 5.1. That is, we will find some connections between linked schemes. The main goal here is to provide some basic tools, which will be used later. We shall see in Section 5.3 that there are some subtle invariants of a liaison class, but for now we begin with some more natural observations. We will assume some knowledge about primary decomposition and associated primes; see for instance [7J for details.
Lemma 5.2.1 Let IVI be a saturated ideal and Ix C IVI an arithmetically Gorenstein ideal defining a scheme of the same dimension. Consider J = [Ix: IvJ Then J is saturated.
Proof: This follows from the fact that Ix is saturated; we leave the details as an exercise. 0 Our next observation is that the dimension is preserved under liaison. In fact, a scheme cannot participate in a link unless it is "unmixed" (i.e. its associated primes all have the same height):
Proposition 5.2.2 Assume that VI~V2 (cf. Definition 5.1.2). Then
(a) As sets, VI U V2 = X. (b) VI and V2 are equidimensional of the same dimension, and have no embedded components. (c) If VI and V2 have no common component then they are geometrically linked. Proof: We follow the proof in [189J. Recall that by definition, Ix C IVI n I v2, [Ix: IvlJ = IV2 and [Ix: I v2 J = I vi . For part (a) we have to show that the radicals vi n IV2 = VJX. From the above facts it follows that IVI ·Iv2 C Ix ~ IVI n I v2 . Then (a) will follow once we prove that in fact vi · I v2 = vi n I v2 . This is
Jl
Jl
Jl
left as an exercise. For (b), we will show that every associated prime of either IVI or IV2 is an associated prime of Ix. Then we will be done, since all associated primes of Ix have the same height: its deficiency modules are zero so it is in particular locally Cohen-Macaulay and equidimensional, by Theorem 1.2.5, and this implies that all associated primes have the same height.
CHAPTER 5. LIAISON THEORY
114
Let P be an associated prime of Iv!' so P = [Iv! : x] for some homogeneous XES, x ~ Iv!. In particular, xP C Iv!. But Iv! = [Ix : I v2 ], so xP·Iv2 CIx . (5.1) Now, x ~ Iv! so xIv2 C/. Ix· Thus there exists y E IV2 with xy tJ. Ix. But by (5.1) , xyP c Ix. Therefore, P ~ [Ix: xy] . We now claim that in fact P = [Ix: xy]. Once we prove this we will be done since the associated primes of Ix are exactly the prime ideals among all ideals of the form [Ix : z] with z E S, z ~ Ix. To prove the claim consider the set { ideals [Ix:
zll
[Ix: z] 2 [Ix : xyl }.
Let Q be a maximal element of this set (possibly [Ix : xy] itself). In particular Q is an associated prime of Ix. Now, P c Q and both are prime ideals. If they are not equal, then height P < height Q; that is, codim V(P) < co dim V(Q) (where for an ideal I, V(1) is the vanishing locus of 1). This says that Vi has a component of dimension strictly larger than any component of X (which all have the same dimension). But X = Vi U V2 so this is impossible. This proves (b) . To prove (c), we have to show that Ix = Iv! n I v2 . The inclusion ~ is part of the definition of algebraic linkage, so we have only to prove the reverse inclusion. Since Vi, V2 and X are all equidimensional of the same dimension by (b), the associated primes all have the same height, and correspond to components of Vi, V2 and/or X of maximal dimension. Hence it follows from (a) (independently of the hypothesis of (c)) that any associated prime of Ix is an associated prime of either Iv! or I v2 . We have by hypothesis that [Ix: Iv!] = IV2 and [Ix: IV2l = Iv!. Let F E Iv! n I v2 . By choosing polynomials of large degree, we can choose G 1 E I v! such that no power of G 1 is in any associated prime of I V2 (since Iv! and IV2 do not share any associated primes). (It helps to think geometrically.) Similarly we can choose G 2 E IV2 such that no power of G 2 is in any associated prime of Iv! . Let F E Iv! nIv2 . We want to show that FE Ix. If we write a primary decomposition Ix = Qi n .. . n Qt, then we need to show that F E Qi for all i. Suppose that for some i, F ~ Qi. Let Pi be the corresponding associated prime. We have seen that Pi is then an associated prime of either Iv! or IV2 but not both (by hypothesis). Suppose, without loss of generality, that Pi is an associated prime of Iv!. Then F . G 2 E Ix by
5.2. RELATIONS BETWEEN LINKED SCHEMES
115
definition of the ideal quotient (since F E Iv! and [Ix: IV2l = IvJ. So F . G 2 E Qi. But no power of G 2 is in Pi, so F E Qi. 0 A useful application of liaison is to produce new schemes from given ones. To do this, one typically starts with Vi and chooses an arbitrary arithmetically Gorenstein scheme X containing VI and looks for a residual V2 using the ideal quotient. In practice it has often been the case that X was chosen to be a complete intersection, but Section 4.2 shows how one can also choose other arithmetically Gorenstein schemes containing VI. In any case, one needs to know that, under reasonable hypotheses, this construction yields V2~ Vi. Proposition 5.2.3 Let VI be a closed subscheme of pn with saturated ideal Iv!. Let Ix C Iv! be an arithmetically Gorenstein ideal of the same dimension. Consider J = [Ix : Iv!l (which is automatically saturated) . Let V2 be the scheme defined by J, so J = I v2 . Then (a) VI U V2
=X
as sets.
(b) V2 has no embedded components and is equidimensional of the same dimension as X. (c) If VI has embedded components or is not equidimensional then VI is not linked to V2 .
x
(d) Let WI be the top dimensional part of VI. Then WI'" V2. In particular, if VI has no embedded components and is equidimensional then V1 ?:f-V2 • Proof: For (a), note that [Ix: Iv!l = IV2 implies in particular that IvJv2 ~ Ix. It also implies that Ix C I v2 , so we have that Ix C Iv! n I v2 . Then
the same proof as in Proposition 5.2.2 (a) works. For (b), note that in Proposition 5.2.2 (b) what we really proved was that if Iv! = [Ix: IV2l and if P is an associated prime of Iv!, then P is an associated prime of Ix. Here we have exactly the same situation but with VI and V2 interchanged. Hence we conclude that every associated prime of IV2 is an associated prime of Ix so we have (b). For (c) we need to consider [Ix : [Ix : Iv!]] and show that this is not equal to Iv!. But the proof of (b) gives that [Ix : [Ix: IvJl has no embedded components and is equidimensional, so we immediately get (c). Part (d) is somewhat deeper, although notice that it is immediate if VI is equidimensional with no embedded components and having no
116
CHAPTER 5. LIAISON THEORY
component in common with V2 (all components then have the same codimension by (b) and by hypothesis) since then one can show that they are geometrically linked. For the general case, we have to show that [Ix: I vl ] = [Ix: IwJ = I v2 , and then we again have to look at a double ideal quotient and show that [Ix : [Ix : I wl ] = I wl . The result (at least the second statement) of (d) is stated without proof by Peskine and Szpiro ([174] Proposition 2.1). A proof is given by Schenzel ([187] Proposition 2.2), Eisenbud ([62] Theorem 21.23 (a)) and Vasconcelos ([199] Theorem 4.1.1). We will give a proof shortly (see page 123).0 Example 5.2.4 We will show that (d) above is not necessarily true if X is not arithmetically Gorenstein. This example was produced using the computer program Macaulay [15]. Let S = k[Xo,···, X 3 ] and let Ix = (Xo , Xd 2 = (X6 , XOXI,Xl) . X has degree 3 and is not arithmetically Gorenstein. Let Iv) = (X6,XOX I , Xr, XOX2 - X IX 3 ). VI has degree 2. Let J = [Ix : IvJ One calculates that J = (X o, Xd = I v2 , which is the ideal of a line Y2 (degree 1) supported on the same set as X and VI. But [Ix: I v2 ] = I v2 , so we do not recover I vl . 0 Remark 5.2.5 A nice application of Proposition 5.2.3 is to find the top dimensional part, W , of a projective scheme V. It follows immediately from what we said in the proof of Proposition 5.2.3 that if X is arithmetically Gorenstein of the same dimension as V and with Ix ~ Iv, then
[Ix: [Ix: Ivll
= Iw·
That is, in a link the only components that matter are those of maximal dimension. If VI has embedded components or is not equidimensional, X is arithmetically Gorenstein of the same dimension containing VI, and we twice apply the ideal quotient with Ix as above, then the components of IVI of non-minimal height disappear (they are swallowed up by V2 ) . See also Remark 4.2.4 and Example 4.2.5. The proof of this fact will be given on page 123. 0 We now describe a very useful exact sequence from [174], relating the ideal sheaves Ix, Iv! of X and Vl and the dualizing sheaf WV2 of V2 • Proposition 5.2.6 Let vI?f., Y2 where VI, V2 C IF have codimension rand X is arithmetically Gorenstein with minimal free resolution
o -t S( -t) -t Fr - I -t ... -t FI
-t
Ix -t O.
5.2. RELATIONS BETWEEN LINKED SCHEMES
117
Then we have an exact sequence
o -+ Ix
-+ IVJ -+ WV2 (n
+1-
t) -+
o.
(5.2)
Proof: From Remark 5.1.4, and sheafifying, we get
IVJ IIx
2:!
1iomopn (OV2' Ox).
We need to show that this last term is isomorphic to WV2 (n + 1 - t). This follows immediately from [96] (5.20) by sheafifying. However, we will give another proof which is a modification of a proof in [181]. Consider the exact sequence of sheaves
o -+ Ix
-+ OlP'n -+ Ox -+
o.
Applying 1iomopn (OV2' -) we get
1iomOpn (OV2,OlP'n ) -+ -+
1iomopn (Ov2, Ox) £ xthpJ OV2 , OlP'n)
But by [88] III.7.3, £xtbpJOV2' OlP'n) = 0 for 0
1iomo pn (OV2' Ox)
2:!
£ xth pn(Ov2,Ix ) -+
-+ -+
....
:s: i < r.
Hence
£xthpn (OV2' Ix).
But in fact we can carry this idea even further. Consider the sheafification of the minimal free resolution for the arithmetically Gorenstein scheme X: ----7
\. ?
Fr- 2
Kr- 2
?\.
o ----7
\. ? 0
?
K2
\.
F2
0
----7
\. ?
?
Kl
ED OlP'n (-ai)
\.
o
0
0
----7
\. ?
?
Ix
\.
We get that
Iv)Ix
C:>! C:>!
C:>! C:>!
£ xt 1l'n (OV2' Ix) £ xt~pn (OV2' Kd £ xto~~ (OV2 ' Kr - 3 ) £xto~~(OV2' Kr - 2 )
OlP'n -+ Ox -+ 0
0
CHAPTER 5. LIAISON THEORY
118
since the Fi are free. What is this last term? We use the left-most short exact sequence in the diagram above and get an exact sequence
o
-+ £xt~~~(Ov2,Kr-2) -+ £xtOnm(OV2,OlPn(-t))
~
£ xtOll'n(Ov2,Fr-d -+ ... But we know exactly what a is, since X is arithmetically Gorenstein: its entries are the generators of Ix (cf. Corollary 4.1.3). Hence a annihilates every Ox-module. Therefore the image of a is zero and we get
IvJIx
OlP n ( -n - l))(n + 1- t)
~
£xt~pn(OV2'
~
£xt~lI'n(OV2,WlPn)(n+ 1-
t)
wV2(n + 1 - t) as desired. 0
Remark 5.2.7 The exact sequence in Proposition 5.2.6 gives us that IVI/Ix ~ wV2(n + 1 - t). But this is the kernel of the natural map Ox -+ OVI' Thus we also have an exact sequence
o -+ WV2 (n + 1 -
t) -+ Ox -+ Ov! -+ O.
We have a similar exact sequence if we reverse the roles of Vl and V2 . 0
Remark 5.2.8 We want to explicitly give the translation of Proposition 5.2.6 in the case where X is a complete intersection. Say that Ix = (Fl , ... , Fr) where deg Fi = di . Then the Koszul resolution gives t = L di (cf. Example 1.5.1) and we have the exact sequence
0-+ Ix -+ Iv! -+ wV2(n + 1- 2:di) -+ O. This is in fact what is contained in [174]. 0
Remark 5.2.9 The exact sequence of Proposition 5.2.6 has a very geometric interpretation which we would like to mention in passing. Suppose, for convenience, that Vl and V2 are smooth and geometrically linked by X. Let Y be the scheme-theoretic intersection of Vl and V2 , with defining ideal Iv! + I v2 . We saw in Theorem 4.2.1 that Y is a divisor on either Vl or V2 , and that as a subscheme of IF, Y is arithmetically Gorenstein. Consider the linear system IIv! (k) I of hypersurfaces of degree k in IF containing Vl . It can be shown from Proposition 5.2.6 that if FE IIv! (k)1 then F cuts out on V2 the divisor Y + D, where D E IWV2(n + 1 - t + k)l; i.e. D is a subcanonical divisor. Conversely, any such divisor is cut out by some hypersurface of degree k containing Vl . Compare this with Theorem 4.2.8. 0
5.2. RELATIONS BETWEEN LINKED SCHEMES
119
Notice that so far we have not needed to assume that our schemes are locally Cohen-Macaulay. A useful consequence of Proposition 5.2.6 is that in the locally Cohen-Macaulay case, one can produce a locally free resolution of the linked scheme from a locally free resolution of the original scheme plus the resolution of the arithmetically Gorenstein scheme. This is by way of the Mapping Cone procedure.
Proposition 5.2.10 ([174]) Let Vl~V2 where VI, V2 C IF of codimension r and X is arithmetically Gorenstein with sheafified minimal free resolution
----+
0-+ OlP'n( -t) -+ F r - 1 -+ . . . -+ Fl
~
Ix
/'
o
OlP'n -+ Ox -+ 0
/' ~
o
Assume that VI is locally Cohen-Macaulay, and suppose that we are given a locally free resolution for Iv! of the form
0-+ 9 -+ 9r-l -+ ... -+ 91
----+ ~
/'
/'
~
Iv!
o
Then there is a locally free resolution for
0-+ 9i(-t) -+ Fi(-t) EB
9~(-t)
IV2
OlP'n -+ Ov! -+ 0
o of the form
-+ F~(-t) EEl 9~(-t) -+ . . .
. . . -+ ;="'/-1 (-t) EEl 9 v (-t) -+
IV2
-+ O.
Proof: Such a locally free resolution for Iv! can be obtained, for example, as in Section 1.2: all that is needed for this construction is that all the sheaves 9 and 9i be locally free, but in particular we can arrange that the 9i be free. By Proposition 5.2.6 we have an exact sequence
The sheafified minimal free resolution for Ix and the locally free resolution for Iv, combine with this short exact sequence to form a commutative
CHAPTER 5. LIAISON THEORY
120 diagram
0
0
.!.
-+
.!. 9 .!.
F1
-+
91
Ix
-+
IVI
O]pn( -t)
.!. .!.
0-+
.!. .!.
.!. .!. .!.
0
-+ WV2 (n
+ 1 - t)
-+ 0
0
Then the mapping cone of this diagram gives a locally free resolution of
WV2 (where we write 0 for O]pn):
Now we apply Homo( -,0), and get
0-+ 9( -+
Fi' EEl 9i -+ ... -+ 9 v EEl F:_ 1 -+ O(t) -+ £xtS(wv2(n
+1-
t), 0) -+ O.
The proposition will be proved once we show (a) this is exact, i.e. £xt~(WV2 ' 0) = 0 for 0 ~ i ~ r - 1; (b) £xtSlI'n(WV2(n + 1- t), O]pn) = OV2(t). To see this, consider a locally free resolution for OV2' namely,
Applying Hom( - , 0) and using [88] Lemma III. 7.3, we get a locally free resolution
0-+ 0 -+ H~ -+ ... -+ H: -+ wV2(n + 1) -+ O. Applying Hom( -,0) again we get
0-+ Hr -+ ... -+ HI -+ 0 -+ £xtO(wV2(n
+ 1) , 0)
-+ O.
This is not exact a priori, but we know it is by comparison with the original resolution. Therefore we get £xt~(WV2' 0) = 0 for 0 ~ i ~ r - 1 and we have shown (a).
5.2. RELATIONS BETWEEN LINKED SCHEMES
121
For (b), we again compare the two resolutions. We get that t'xtO(WV2(n + 1), O)(t) OV2(t)
as desired. D
Remark 5.2.11 It was noted at the beginning of the proof of Proposition 5.2.10 that all that is needed is that the sheaves 9 and gi be locally free, but that in particular we can arrange that the gi be free. In this latter case where the gi are free , the result in fact holds without the assumption that VI is locally Cohen-Macaulay, although the proof is more delicate. See [163] for details. D As an immediate corollary (considering the projective dimension and using the local version of the Auslander-Buchsbaum theorem; see page 10), we get the important fact that the property of being locally CohenMacaulay is preserved under liaison:
Corollary 5.2.12 Let VI~V2 where VI, V2 c IF and X is arithmetically Gorenstein. Then Cohen-Macaulay.
Vi
is locally Cohen-Macaulay if and only if V2 is locally
Notice that the corresponding statement for local complete intersections is false. For instance, if VI c p3 is a line and V2 is the scheme defined by I~l (the first infinitesimal neighborhood) then VI is directly linked to V2. See also Example 5.1.7 (6.), Remark 5.1.9 and Example 6.4.5. These results are very useful in general. We will give an important consequence of Proposition 5.2.10, namely the Hartshorne-Schenzel Theorem, in Section 5.3. For now we give some more elementary consequences of Proposition 5.2.6.
Corollary 5.2.13 If VI~V2' and if one (hence both) of Vi and V2 are locally Cohen-Macaulay, then deg VI
+ deg V2 = deg
X.
Proof: This is clear in the case of geometric linkage. We will sketch the proof in the case of algebraic linkage. For convenience say that dim VI = dim V2 = dim X = p. Note that for t » 0, P(VI' t) (the Hilbert polynomial of Vd deg VI - - I-tP + (lower terms). p.
CHAPTER 5. LIAISON THEORY
122
Hence for t» 0, since hI (IVI (t))
hO(Ivl (t)) = ( t + n
n) -
hO(Ix(t)) = ( t + n
n)
= 0, we have g VI [de ~tP
+ (lower terms) ]
.
gX - [de ~tP
+ (lower terms) ]
.
Similarly,
Now, for t
»
0 use [88] pp. 230, 243 and 244 to deduce that
P(V2' -t)
=
That is, for t
»
hO(OV2(-t)) - h1 (Ov2( -t)) (-l)Ph P(OV2 (-t)) (-l)P hO(WV2 (t)) .
+ ... + (-1)Ph P(Ov2( -t))
0 we have
hO(WV2(t)) =
(-l)P [de!, V2 (-t)P [ de!, 1;2 tP
+ (lower terms)]
+ (lower terms)]
.
Finally, use the exact sequence of Proposition 5.2 .6, twist by t » 0, take cohomology, compute dimensions using the above facts, and compare leading terms. D. One can also give formulas relating the Hilbert polynomials of the linked schemes in any dimension. See for instance [90], Proposition 4.7 for the complete intersection case and [163] Corollary 3.6 for the general Gorenstein case. We restrict ourselves to the case of curves, where a nice formula emerges. Let 91 and 92 be the arithmetic genera of C1 and C2 respectively.
Corollary 5.2.14 (cf. [121]) Assume C 1ii.,C2 , where X is an arithmetically Gorenstein curve in ]pn with minimal free resolution
o-t S( -d) -t Fr - 1 -t ... -t Fl -t Ix -t O. Then
5.2. RELATIONS BETWEEN LINKED SCHEMES
123
Proof: Recall that for a curve C of arithmetic genus g, the Hilbert polynomial is given by P(C, t) = (deg C)t - 9 + 1. Then the computation of Hilbert polynomials in the proof of Corollary 5.2.13 gives, in this case,
Pa(X) - 1 - (deg C2)(d - n - 1)
= gl -
g2.
But by Proposition 4.1.12, 1
Pa(X) = 2(de g X)(d - n - 1) + 1. Combining these and using the observation that deg X - 2deg C2 deg C1 - deg C 2 (by Corollary 5.2.13) gives the result . 0
=
Remark 5.2.15 The formula given in Corollary 5.2.14 appeared in this generality in [121], although it is equivalent to a formula in [163]. However, in the case where X is a complete intersection and C 1 and C 2 are aCM, it appears in [174]. The formula for complete intersection liaison in the non-aCM case appears in [90]. 0 Corollary 5.2.16 Let C1 and C2 be curves. If C/f.C2, and if deg C1 = deg C2, then gl = g2 ·
It is an amusing exercise to study the converse of Corollary 5.2.16 and see what it would take to find a counterexample. For instance, suppose n = 3.
As promised, we now give the proof of Proposition 5.2.3 (d).
Proof of 5.2.3 (d): Our proof is essentially the same as that of Eisenbud [62]. We have a scheme VI, possibly with embedded or isolated components of higher codimension, with top dimensional part WI. We want to show that [Ix; I vl ] = [Ix ; I wl ] and [Ix; [Ix; IWlll = I wl · In either case, the main observation is this. From the proof of Proposition 5.2.6 one can see that instead of assuming that Vl~V2' it is enough to assume that [Ix; I vl ] = IV2 in order to get the exact sequence 0-+ Ix -+ IV2 -+ WVl (n
+ 1 - t)
-+ 0,
and consequently get by Corollary 5.2.13 that deg VI + deg V; = deg X. Notice that deg VI = deg WI. We first check that [Ix; I vl ] = [Ix; IwJ It is clear that we have the inclusion 2. On the other hand, both are saturated ideals of subschemes of IP'n of the same dimension and degree (= deg X - deg VI)'
124
CHAPTER 5. LIAISON THEORY
which are equidimensional and without embedded components. Hence they are equal. In fact, both are equal to I v2 . The proof that [Ix: [Ix: I wl ]] = [Ix: I v2 ] = IWI is very similar. We have seen that [Ix : I v2 ] is unmixed, and by definition so is I wl . Both are saturated ideals defining subschemes of IF of the same dimension and degree, which are equidimensional and have no embedded components. It is easy to check that IWI C [Ix : [Ix: IwJ]. Therefore they are equal. 0 This trick of showing that two unmixed ideals are equal by proving one inclusion and showing that they define two schemes of the same dimension and degree is very useful. We use it again to prove another important fact, namely that linkage is preserved under hyperplane or hypersurface section (see Section 1.3). This is intuitively obvious, but there is something to prove.
Proposition 5.2.17 Let VI~V2 in IF, where 2 ~ codim X = r < n. Let F be a general hypersurface of degree d. Then (VI n F) ~ (V2 n F) in IF .
Proof: Notice that X
nF
is again arithmetically Gorenstein (Proposition 1.3.7). By Proposition 5.2.3 we need to show that the saturated ideals of the hypersurface sections satisfy (a) IxnF C Iv,nF n Iv2nF , and (b) [IxnF : Iv,nF] = Iv2nF . The proof of (a) is quick:
IxnF
= c c
Ix + (F) Iv, + (F) Iv, + (F) Iv,nF
and similarly for V2 • For (b), notice that both sides of the desired equality represent saturated, unmixed ideals of schemes of the same codimension. We first show that these schemes have the same degree; after that it suffices to show one of the two inclusions to get equality. We know that deg VI + deg V2 = deg X. Hence the left-hand side of the desired equality is the saturated ideal of a scheme of degree (deg X) . d - (deg VI) . d = (deg V2) .d, which is the same as the degree of the scheme represented by the right-hand side.
5.2. RELATIONS BETWEEN LINKED SCHEMES
125
Now we just show that [IxnF : Iv!nF] :J I v2nF . We leave it to the reader to verify the fact that if I and J are ideals with I = J, then [I : J] = [I : J]. Having this fact, we only need to show that
But clearly [Ix + (F) : Iv! + (F)] :J IV2 saturated, so we are done. 0
+ (F),
and the left-hand side is
Remark 5.2.18 The assumption in Proposition 5.2.17 was that V was not necessarily aCM, but that it had dimension at least 1. Now suppose that VI and V2 are zeroschemes in rn which are directly linked by an arithmetically Gorenstein zeroscheme X having minimal free resolution 0-+ S( -t) -+ Fn -
1
-+ ... -+ Fl -+ Ix -+
o.
In particular, they are all aCM . Let F be a general hypersurface of degree d (i.e. one not vanishing on any component of X). The ideals Iv! + (F), IV2 + (F) and Ix + (F) all define the empty set, so geometrically it does not make sense to talk about linkage in this case. Indeed, if we saturate these ideals we get the whole ring. However, algebraically it makes a good deal of sense and still conveys geometric information. We discuss this here. We first claim that the ideal Ix + (F) is still arithmetically Gorenstein, of co dimension n + 1. This follows from the exact sequence
o -+ Ix n (F) -+ Ix 81 (F) -+ Ix + (F)
-+ 0,
the observation that (F) S:! S (-d) and Ix n (F) S:! Ix (-d), and a mapping cone. (Compare with Proposition 1.3.7.) This proof also shows that the twist in the last free module (of rank 1) in the minimal free resolution of Ix + (F) is -(t + d). Now let J 1 = Iv! + (F), h = IV2 + (F) and J = Ix + (F). Notice that the algebraic definition oflinkage (Definition 5.1.2) still makes sense in this context; all we need are ideal quotients and Gorenstein ideals. We claim now that J links J 1 to J2 • Clearly J ~ J 1 n J 2 and [J : J1 ] 2 J2 . We have to show that these latter two ideals are equal. We now use essentially the same trick as above, comparing degrees. Let 12 = [J : Jd and let K2 be the canonical module of S /12 . From the first three sentences of Proposition 5.2.6 it follows that there is an exact sequence
(5.3)
CHAPTER 5. LIAISON THEORY
126
Remembering that dimkSj 12 is finite, we have as in Remark 1.4.8 that K2 is the k-dual of Sj 12 • It follows from the exact sequence above that
dimkSjJ - dimkSjJI
= dimkSjh
But dimkSj J = d· deg X and dimkSj J I = d· deg VI, so using Corollary 5.2.13 we get dimkSj 12 = dimkSj J2. Combining this with the above inclusion, we get that J2 = [J : J I], and J links J I to J2. In a similar way we can show that if VI is directly linked to V2, where both are aCM, then the Artinian reductions are directly linked. In this case, of course, we always take F to be a linear form, and in reducing modulo F we preserve the socle degree (and graded Betti numbers) of our Gorenstein ideal - the argument is similar to that of Theorem 1.3.6. 0 We can use the ideas in this last remark to prove the result of Davis, Geramita and Orecchia [56] on the behavior of the Hilbert function under liaison.
Corollary 5.2.19 Let VI ~ V2 where VI and V; are aCM of codimension c in IF and X is arithmetically Gorenstein with sode degree r = t-c. Let VI, V2 and X have h-vectors, respectively, (1, aI, a2,· . . ,ak), (1, bI , b2, ... , be) and (1,c,c2, ... ,cr _2,c,1). Then
for all i. Proof: We first pass to the Artinian reduction by modding out n + 1 - c linear forms. The last paragraph of Remark 5.2.18 says that we then get ideals J I , J2 and J in R = k[X I , ... , Xc] such that J is arithmetically Gorenstein linking J I to J2 , and J has the same socle degree and graded Betti numbers as Ix. Recall that the h-vectors given in the statement of the corollary are simply the Hilbert functions of Rj J I , Rj J2 and Rj J respectively. Since we are now in the ring k[X I , . .. , Xc], the exact sequence (5.3) becomes Then we get Ci -
ai = dimk[K2(c - t)]i dimdK2]c-t+i dimk[Homk(Sj J2, k)]c-t+i = dimk[SjJ2]t-c-i = bt- c- i
=
(by (1.10))
o
5.2. RELATIONS BETWEEN LINKED SCHEMES
127
Example 5.2.20 Suppose that Z is a zeroscheme in ]P'2 linked to a zeroscheme Z' by a complete intersection X of type (aI , a2)' Notice that given al and a2, we know precisely what the Hilbert function of X is. Now t = al + a2 and c = 2. Let N = al + a2 - 2. Then Corollary 5.2.19 says that
t:..H(Z' , N - i) = bN - i =
Ci -
ai
=
t:..H(X, i) - t:..H(Z, i).
For example, the following configuration of points is a complete intersection of type (3, 4) linking the "open points" Z to the "solid" points Z' : o o
0
0
•
0
•
•
• • • • Think of the complete intersection X as consisting of three "horizontal" lines and four "vertical" lines. Corollary 5.2.19 says that to compute the Hilbert function of Z' we subtract the h-vector of Z from that of X and then "read backwards." In this case, we have that Z lies on a unique conic, so one quickly computes that the h-vector of Z is given by (1,2,2). The h-vector of X is (1,2, 3, 3, 2, 1). Then from i h-vector of X -h-vector of Z
I0
1 1 2 1 2 0 0
2 3 4 5 3 3 2 1 2 1 321
we get (by reading backwards) that the h-vector of Z' is (1 , 2,3,1). 0 Example 5.2.21 Corollary 5.2.19 is very useful for constructing examples and counterexamples. For instance, in liaison theory and its applications it is often the case that we are given a scheme and need to know what kind of arithmetically Gorenstein schemes contain it, especially "small" ones. Corollary 5.2.19 is a good first tool to try to do this. For example, if Z is a zeroscheme in jp3 with h-vector (1 , 3, 6,7,4), does there exist an arithmetically Gorenstein scheme X containing Z and having h-vector (1 , 3,6, 10, 6, 3, 1)? Setting up a table as before, we get i h-vector of X -h-vector of Z
I0
1 1 3 1 3 o 0
2 3 4 5 6 6 10 6 3 1 6 7 4 3 2 3 1 0
CHAPTER 5. LIAISON THEORY
128
If such an arithmetically Gorenstein scheme exists, it links Z to a zeroscheme with h-vector (1,3,2,3). The growth of this Hilbert function from 2 to 3 violates Macaulay's growth condition [130], so no such X exists. 0
We can use Proposition 5.2.17 to remove the hypothesis of being locally Cohen-Macaulay from Corollary 5.2.13:
x Corollary 5.2.22 If VI'" V2 then deg VI
+ deg V2 = deg X.
Proof: If the schemes are zero-dimensional then Corollary 5.2.13 applies. Otherwise take enough hyperplane sections to reduce to this case, and note that Proposition 5.2.17 guarantees that linkage is preserved, while Bezout's theorem guarantees that the degrees are preserved. 0
The next several results and remarks concern the general question of "lifting" links. First, a natural question is whether the converse of Proposition 5.2.17 is true. The answer is affirmative, at least in the case of geometric, complete intersection liaison. Clearly with suitable assumptions this could be generalized to Gorenstein liaison, and even the assumption of geometric linkage could be removed. However, we will limit ourselves to the simple case, giving an interesting application of Theorem 2.3.1.
Proposition 5.2.23 Let VI and V2 be locally Cohen-Macaulay equidimensional subschemes of IF. Assume that for a general homogeneous polynomial F of degree d, VI n F is directly CI-linked to V2 n F and that VI n F and V2 n F have no common components. Then Vi is directly CI-linked to V2 , unless Vl and V2 are curves and one of the following holds:
= 1, n = 3 and Vl U V2 lies on a quadric surface; d = 1, n = 3 and deg Vi + deg V2 = 4.
(i) d (ii)
Proof: Since VinF and V2nF have no common component and are directly CI-linked, they are geometrically CI-linked (by Proposition 5.2.2). That is, their union is a complete intersection. But since they have no common component, we also have that Vl and V2 have no common component (and no component of one is contained in a component of the other since the components are all of the same dimension). So the scheme-theoretic union VI U V2 has the property that its general degree d hypersurface section is a complete intersection. Then by Theorem 2.3.1 (for curves) or Theorem 1.3.3 (for higher dimension), VI U V2 is a complete intersection, so since they have no common component they are directly linked. 0
5.2. RELATIONS BETWEEN LINKED SCHEMES
129
Again, something similar should be true for the case of algebraic linkage, although not quite as strong (i.e. the list of exceptions should be slightly bigger). This is still an open problem. The following example illustrates some of the possible considerations and obstacles. Example 5.2.24 In jp3, let VI be a double line (i.e. a scheme of degree 2 supported on a line - cf. [143]) and let V; be a scheme of degree 4 supported on the same line as VI. The general hyperplane section of VI is a complete intersection. Assume that the general hyperplane section of V; is a complete intersection, say (x 2, y2). By considering links of the form (x 2, y2 . e) (where eis a linear form) one can verify that the general hyperplane section of V; is directly linked to the general hyperplane section of VI. However, by changing the linkage class of VI if necessary (cf. [143]) we have that VI is not directly linked to V;. 0 The following result is closely related to Proposition 5.2.23. However, in addition to the obvious difference concerning the assumption that the schemes have vanishing first cohomology (for instance aCM schemes), there is a somewhat subtle difference: in Proposition 5.2.23 we began with a particular V2 , while here we ask whether a suitable V2 can exist. In addition, here we do not assume that the link is geometric, but we do assume that it is a complete intersection link. Then in Example 5.2.26 we show that this hypothesis is important. Proposition 5.2.25 Let V c IF be a subscheme of codimension c ~ 2 satisfying H;('Iv) = O. Let H be a general hyperplane, defined by a linear form L, and let 11 = V n H (as a subscheme of H = IF-I), i.e. Iv = IVcijL) C S / (L). Let X be a complete intersection in H linking 11 to some subscheme W of H. Then this link lifts; that is, there exists a complete intersection X containing V and a subscheme W c IF such that X links V to W, X = X n Hand W = W n H. Proof: We have an exact sequence of modules
o-t Iv(-1) -t Iv -4 Iv -t 0 where the homomorphisms all have degree zero (the surjection comes from the vanishing of the first cohomology). Let PI, ... , Pe E I v be a regular sequence defining X. Make a sufficiently general choice of FI , .. . , Fe E Iv such that p(F;.} = Pi for all i. Notice that all the components of the scheme X defined by the ideal (FI , ... , Fe) (which is, a priori, not necessarily
130
CHAPTER 5. LIAISON THEORY
saturated) must have codimension ~ c since there are only c generators in the ideal. But the hyperplane section X n H has pure co dimension c in H, so X cannot contain a component of codimension < c. It follows that X is a complete intersection containing V, and X = X n H. Let W be the scheme residual to V in X. We only have to show that W = W n H. But this is obvious: Proposition 5.2.17 says that V = V n H is linked by X = X n H to W n H since V is linked by X to W, while the hypothesis says that V is linked by X to W. Hence W = W n H. 0
Example 5.2.26 We now show that the situation with Gorenstein liaison is not so simple, leading to one advantage of complete intersection liaison over Gorenstein liaison. This advantage concerns lifting a link from the general hyperplane section of a given aCM scheme to the scheme itself. We just saw in Proposition 5.2.25 that this can always be done for complete intersection liaison. Now let PI, . . . ,Pk E Iv generate an ideal which is arithmetically Gorenstein. We are interested in knowing if the general choice of FI , . .. , Fk E Iv as above defines an ideal which is again arithmetically Gorenstein, or even if there necessarily exists such an ideal at all. We show that this is not necessarily the case, with a geometric example. Let V c JIM be a rational normal curve, so that V = V n H is a set of four points in H = jp3 in linear general position. Let P be a fifth point, such that V U P remains in linear general position. Then V U P is arithmetically Gorenstein (Example 4.1.11 (b)), having five minimal generators PI, ... , P5 of degree 2. When does this ideal lift to that of an arithmetically Gorenstein curve X in JIM containing V? If such a curve X exists, its minimal free resolution must be the same as that of its hyperplane section, namely o -t S( -5) -t 5S( -3) -t 5S( -2) -t Ix -t O. Then clearly the degree of X is 5, and Proposition 4.1.12 says that the arithmetic genus of X must be 1. The degree condition says that X must be the union of V with a line, and the genus condition says that this line must in fact be a secant line (since the genus of V is zero). Now, the intersection of this secant line with H is precisely the point P. Hence if P is chosen off the secant variety of V, it is impossible for the Gorenstein ideal (FI , •. • , F5 ) to lift to the ideal of an arithmetically Gorenstein curve containing V! In fact , in this case the ideal (FI , ••• , F5 ) obtained as above is the ideal of V U P; the fact that there are more generators than the codimension allows for components of higher codimension. 0
5.2. RELATIONS BETWEEN LINKED SCHEMES
131
We conclude this section with a very geometric result from [121], as promised in Example 5.1.7 (4.). For our purposes it is enough to assume smoothness, for simplicity of proof. However, in [121] it was only assumed that S is G 1 , but as a result the proof was more complicated. Theorem 5.2.27 ([121]) Let S be a smooth, arithmetically CohenMacaulay subscheme of IF. Let C c S be a divisor, i. e. a pure codimension one subscheme with no embedded components. Let C' be any element in the linear system IC + kHI, where H is the hyperplane section class and k E Z. (Notice that k = 0 if and only if C and C' are linearly equivalent.) Then C' is G-bilinked to C in two steps.
Proof: Throughout this proof, equality means "as subschemes," not as linear equivalence classes. For a homogeneous polynomial F not vanishing on any component of S, we denote by Hp the subscheme of S cut out on S by F, i.e. the scheme defined by the saturated ideal Is+(F) (see Section 1.3). Let K be an effective subcanonical divisor on S (cf. Theorem 4.2.8). For a » 0 we can find a form A of degree a such that HA - K is effective; it is enough to choose A E IK not vanishing on any component of S. By interchanging C and C' if necessary, we may assume that k ~ O. Because S is aCM, the condition that C' E IC + kHI is equivalent to the assertion that there exist forms Band B' with deg B' - deg B = k, and an effective divisor D, such that HB - C = D and H B, - C' = D. (Again, the case k = 0 is precisely the condition that C and C' are linearly equivalent.) Now we link. Observe that HAB - K is arithmetically Gorenstein, by Theorem 4.2.8. On the other hand, HAB-K -C = (HA -K)+(HB-C) = (HA - K) + D. This means that the arithmetically Gorenstein scheme HAB - K links C to the curve (HA - K) + D (which is just D together with an arithmetically Gorenstein "tail") . In a similar way, H AB , - K is arithmetically Gorenstein and links C' also to (HA - K) + D. D Remark 5.2.28 Using Theorem 5.2.27, it is shown in [121] that on a smooth, aCM rational surface S in ]p>4, all aCM curves are glicci (cf. page 109); in particular all are in the same Gorenstein liaison class. It is also shown that the same result is far from true in the setting of complete intersection liaison. Indeed, if C is an aCM curve on such a surface (but not a complete intersection) and if Ca E IC + aHI and Cb E IC + bHI with a ¥- b, then Ca and Cb are aCM curves which "most of the time" belong to different CI-liaison classes. This is accomplished by studying certain liaison invariants. In fact, it is shown that it is even possible for C
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CHAPTER 5. LIAISON THEORY
and C' to be in the same linear system and still not be CI-linked in any number of steps! This is done using arguments similar to those of [198J. Of course if S is a complete intersection then these curves are all in the same even CI-liaison class (combine Remark 3.2.5 and Example 5.1.7 (4.).) See also Remark 5.3.8. 0
Remark 5.2.29 It was mentioned before Theorem 5.2.27 that rather than assume S to be smooth, it is enough to assume that S satisfies property C 1 . In fact, the only reason to assume even C 1 is to be able to talk about linearly equivalent divisors (cf. [90]). If one only considers the explicit divisor Z = C + HA for some homogeneous polynomial A of degree k not vanishing on any component of S, rather than its linear equivalence class, then assuming only Co it is still true that Z is G-bilinked to C. See Proposition 5.4.5 and the discussion following it. 0
5.3
The Hartshorne-Schenzel Theorem
In this section we give a more subtle relation between linked schemes, in the form of a necessary condition for two schemes to be linked. That is, we will show the invariance (up to shifts and duals and re-indexing) of the deficiency modules. The fact that the dimensions of the components (suitably re-indexed) is preserved was known already by Gaeta [71], at least for curves, but the invariance of the module structure of course requires knowledge of schemes and sheaf cohomology. Throughout this section we will assume that the schemes we consider are locally Cohen-Macaulay, so that we obtain a locally free sheaf from the resolution of the ideal sheaf. This assumption has been removed by Hartshorne in the case of even liaison; he proved Corollary 5.3.3 below without this hypothesis by using reflexive sheaves and his notion of generalized divisors ([90J Proposition 4.5). Similar results have been obtained by Nollet [167J and by Nagel [163J. Notice that Theorem 5.3.1 itself cannot be extended to the non locally Cohen-Macaulay case since for any such V the module (Mi) (V) will then fail to have finite length, but it is always zero in large degree, so the conclusion of the theorem cannot hold. So in a sense, Corollary 5.3.3 is more basic than Theorem 5.3.1. See also Example 5.1.7 (4.). The theorem was proved for curves in jp3 by Hartshorne (cf. [179]), for curves in ]pm by Chiarli [50], and for arbitrary dimension by Schenzel [187J and subsequently by Migliore [145J. A different proof has recently been given by Hartshorne [90J . We follow the proof given in [145J. The original
5.3. THE HARTSHORNE-SCHENZEL THEOREM
133
proof of [145] was for complete intersection liaison, but the same proof works in general. We will use the following notation:
(see page 3).
Theorem 5.3.1 Let Vi, V2 c IF have dimension r and assume that Vi ~ V2 where X is arithmetically Gorenstein with sheafified minimal free resolution
o -+ Gpn( -t) -+ F n- r- 1 -+ ... -+ Fl -+ Ix -+ o. Then
Proof: Notice that we are making a small change in notation from Proposition 5.2.10 in that now r represents the dimension rather than the codimension. This is more in line with the material in Chapter 1, and it simplifies the notation. Consider a locally free resolution for IVl as in Proposition 5.2.10:
o -+ 9 -+ 9n-r-l
... -+ 92
----7
----7
)'I
'\t
'\t
li n - r - 1
)'I
'\t
)'I
0
0
0
)'I
11.2
'\t
91 -+ IVl -+ 0
0
(5.4) where 9 is locally free and the 9i are free. Then by Proposition 5.2.10 we have a locally free resolution
o -+ 9t (-t) -+ Ft (-t) EEl 9{ (-t) -+ ... '\t
o
)'I
K2
)'I
'\t
0
... -+ 9v (-t) EEl F~-r-l (-t) -+
'\t
)'I
)'I
'\t
Kn - r -
o
1
0
(5.5) IV2
-+ O.
CHAPTER 5. LIAISON THEORY
134
From the short exact sequences obtained from this latter resolution, beginning from the left, one sees that
o= H~-r-2(IC2) ~ ... ~ H;(IC n- r- 2) ~ H:(IC n- r- 1) o=
H~-2(IC2) ~
...
~
H;+2(ICn_r_2) ~ H;+1(IC n_r_1)
Therefore we have
(Mr-i+l)(V2) ':::! ':::!
':::! ':::!
=
H;-i+l(Iv2 ) H;-i+l (Qv (-t)) H~-r+i-l(Q(t - n - 1))* H~-r+i-2(lln_r_l (t - n - 1))* H!+1(1l2(t - n - 1))* H!(Iv,(t-n-1))* (Mi)V(Vl)(n + 1 - t)
(by (5.5)) (by Serre duality) (by (5.4)) (by (5.4)) (by (5.4))
o
Remark 5.3.2 With this theorem we can now answer some of the questions raised in Section 5.1. First, the equivalence relation of liaison is not trivial, at least in dimension> O. Indeed, this theorem says that the deficiency modules are invariant, up to duals and shifts and re-indexing, in a liaison class. But as we remarked in Section 1.2, a theorem of Evans and Griffith [69] states that any collection of n - 2 graded S-modules of finite length can be realized, up to shift, as the deficiency modules of a co dimension two subscheme of ]P'n, and this was generalized to arbitrary co dimension in [153]. Huneke and Ulrich have shown that it is not trivial in dimension 0 either, apart from sets of points in ]P'2. However, this does not follow from Theorem 5.3.1. Their approach is to study the linkage class of a complete intersection, and show that not every set of points is in this class. In fact, they study arithmetically Cohen-Macaulay schemes of any dimension from this point of view. This already says that there are at least as many liaison classes (roughly) as there are collections of graded S-modules. However, the question remains as to whether to each collection of modules (up to shifts and duals) there is associated just one liaison class, or many. We will discuss this later. (This is related to the problem of finding sufficient conditions for two schemes to be linked.) Another immediate consequence is that the property of being arithmetically Cohen-Macaulay is invariant in a liaison class, since this property depends only on the collection of modules being zero.
5.3. THE HARTSHORNE-SCHENZEL THEOREM
135
We can also show that the property of being arithmetically Buchsbaum is preserved under liaison. For curves it is an immediate consequence of the definition (Definition 1.5.7) since it depends only the fact that the deficiency module is annihilated by the maximal ideal m = (Xo,"', X n ), and this does not change with shifts or duals. For higher dimension, the definition of being arithmetically Buchsbaum is given after Definition 1.5.7. What is different in higher dimension is that we require that m should annihilate the deficiency modules not only of the original scheme V, but also of the general hyperplane section V n HI, the general hyperplane section V n HI n H 2 , etc. That is, we require that the scheme obtained by intersecting V with a general linear space of dimension 2 n - dim V + 1 also have its deficiency module(s) annihilated by m. But then the fact that the Buchsbaum property is preserved under liaison follows from Proposition 5.2.17. A useful application of Theorem 5.3.1 is to show that two schemes are not linked, by showing that their collection of modules are not the same even after dualizing and shifting. We will apply this idea in a geometric way in Section 5.5. D We have been a little careless in saying that Theorem 5.3.1 implies that the deficiency modules are invariant up to shifts and duals and reindexing. For curves there is no problem, but for higher dimension one has to be a little careful. It is very important to realize that the twist n + 1 - d is the same for each module. Hence the "positioning" of the modules (Mi)(V) with respect to each other stays the same in even liaison and "flips" in odd liaison. In particular we have Corollary 5.3.3 Let VI and V2 be evenly linked schemes of dimension r. Then there is an integer p such that (Mi)(VI)(p) ~ (Mi)(V2) for each l:S;i:S;r. The point of this corollary is that the same p is used for each i. That is, in even liaison the configuration of modules is preserved up to shift. This will play an important role in the structure theory introduced in Section 5.4, and the main structure theorem in the next chapter (Theorem 6.3.1). We will discuss it further below. This is our first indication that even liaison is considerably simpler than odd liaison, since dual modules do not have to be considered. It will turn out that an even liaison class has a "nice" structure, while a liaison class does not (apart from being a union of two even liaison classes in general).
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CHAPTER 5. LIAISON THEORY
The following two corollaries assume smoothness for convenience. As mentioned before Theorem 5.2.27, the hypothesis that S is smooth can be relaxed somewhat.
Corollary 5.3.4 Let S be a smooth, aCM subscheme of]pn. Let V be a divisor on S, i. e. a pure codimension one subscheme with no embedded components. Let V' be any element of the linear system IV + kHI, where H is the hyperplane section class and k E Z. Then for 1 :::; i :::; dim V,
In particular, if V'is linearly equivalent to V then
Proof: This follows immediately from Theorem 5.2.27 and the HartshorneSchenzel theorem. One has to follow the explicit links given to determine the shifts. 0
Corollary 5.3.5 Let S be a smooth, aCM subscheme of]pn. The property that a divisor on S is aCM (resp. arithmetically Buchsbaum) is preserved under linear equivalence and adding hyperplane sections. Proof: This follows from Theorem 5.2.27 and Remark 5.3.2. 0
Remark 5.3.6 An example is given in [121] to show that in the statement of Corollary 5.3.5, if the assumption that S is aCM is removed then at least the aCM property is not necessarily preserved under linear equivalence. It seems likely that the Buchsbaum property should similarly fail to be preserved. 0 Note that we have assumed throughout that our schemes are locally Cohen-Macaulay and equidimensional. While this latter condition cannot be avoided, much of the theory can be carried out (with a good deal more work) without the assumption of locally Cohen-Macaulay, especially for even liaison. Corollary 5.3.3 is a good example. In particular, this was carried out for complete intersection liaison by Hartshorne [90] and by Nollet [167]' and for Gorenstein liaison by Nagel [163]. The basic idea is to study stable equivalence classes of certain reflexive sheaves, in analogy with Rao's study of stable equivalence classes of vector bundles (cf. [180] and also Chapter 6).
5.4. THE STRUCTURE OF AN EVEN LIAISON CLASS
137
Remark 5.3.7 One of the questions raised in Section 5.1 was whether we could perform links using something other than arithmetically Gorenstein ideals and still get a good linkage theory. We mentioned that Walter has shown that if we replace "arithmetically Gorenstein" by "arithmetically Cohen-Macaulay" then there is just one equivalence class (for fixed dimension) , at least if we allow the empty set to be an element of a liaison class. It is easy to see that at the very least we cannot expect the deficiency module to be invariant in this case: let VI be a set of two skew lines in 1P'3 and let \!2 be a line meeting both components of Vi, Then Vi U V2 is arithmetically Cohen-Macaulay, but V2 has a trivial deficiency module while VI has a one-dimensional module (cf. Example 1.5.2). 0 Remark 5.3.8 While the Hartshorne-Schenzel theorem gives an important invariant of a liaison class in any codimension, it is by no means the only invariant. Other invariants are known, especially for arithmetically Cohen-Macaulay liaison classes, thanks to work of Buchweitz, Huneke, Kleppe and Ulrich, among others. For example, let X C IP'N and X' C IP'N be aCM subschemes of dimension n. Assume that they are algebraically linked by some complete intersection. If n > 1 then, as graded S-modules, we have H!(Nx ) ~ H!(Nxt) for 1:::; i :::; n - 1, where N x = Homo" (1:x, Ox) is the normal sheaf of X in IP' = IP'N. In addition, assume that X = Proj(S/1) and X' = Proj(S/ J') are local complete intersections, let c = N - n be the codimension, and let KA = Ext~(S/ I, S) and KAt = Ext~(S/ I', S) be the canonical modules. Then we have H?n(KA 0s 1) ~ H?n(KA' 0s 1') for 0 :::; j :::; n, as graded k-modules. While these and related invariants are beyond the scope of this book, we refer the reader to (for example) [43], [44], [116], [30], [121) and [103). See Remark 5.2.28 for an application. 0
5.4
The Structure of an Even Liaison Class
We can now begin to describe the structure of an even liaison class, although the main theorem will come in Chapter 6 (Theorem 6.3.1). We will have to be careful to distinguish those aspects of the theory that hold for Gorenstein liaison in general, and those that work only for complete intersection liaison.
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The first ingredient is the notion of Basic Double Linkage (see Remark 3.2.4), and we will now see why it is so called. This is a feature which is, for the most part, special to complete intersection liaison. We will again change the notation slightly to be consistent with that of the current chapter: r shall continue to denote the dimension rather than the codimension. Recall that we start with a scheme VI =j:. (/) of dimension r and we choose general homogeneous polynomials FI E S of degree dl and F2 , ... ,Fn - r E IVI of degrees d2 , .. . ,dn - r respectively, such that (FI' F2 , ... , Fn - r ) form a regular sequence. We then consider the scheme Z with defining (saturated) ideallz = FIIvl + (F2 , •• • , Fn-r). Recall (Remark 3.2.5) that this amounts to studying the divisor VI + HFI on the complete intersection scheme S defined by Is = (F2 , ... , Fn-r). Recall also that for 1 ::; i ::; r we have (Mi)(Z) ~ (Mi)(VI)( -d l ). The key observation now is that we in fact have something stronger: Z is CI-linked to VI in two steps. (This was observed in [128J for codimension two and in [34J and [78J in general.) The basic idea is to choose a general homogeneous polynomial G E IVI of sufficiently large degree so that (G, F2 , .•. , Fn - r ) forms a regular sequence, and hence so does (GFI' F2 , ... , Fn-r). Then one can check that if one links VI to a scheme W using the complete intersection (G, F2 , ..• , Fn - r ), and then links W using the complete intersection (G F I , F2 , ... , Fn - r ), the resulting scheme is exactly Z. Now fix an even liaison class C of dimension r subschemes of IP". We have associated to C a configuration of modules {(MI ), ... , (Mr )}, unique up to shift. (See Corollary 5.3.3 and the sentence following it.) By Proposition 1.2.8, a sufficiently large leftward shift of this configuration of modules (i.e. {(Md(d), (M2 )(d), ... , (Mr)(d)} for d » 0) is not the configuration of modules of any scheme V E C, and so there is a leftmost shift (i.e. maximum d) for which there exists some V E C with (Mi)(V) = Mi(d) for some d. This observation does not rely on the choice of complete intersection or Gorenstein liaison, so we have the following definition.
Definition 5.4.1 Let C be an even liaison class, either with complete intersection or Gorenstein links. We will denote by £0 the set of schemes V E £ whose configuration of modules coincides with the leftmost possible one among elements of £. The elements of £0 are the minimal elements of the even liaison class £ and the corresponding configuration of modules is said to be in the minimal shift. Let Va E £0. We will denote by Ch the set of schemes V E £ satisfying Mi(V) ~ Mi(VO)( -h) for 1 ::; i ::; r. By Basic Double Linkage, £h =j:. (/) for all h ~ O. (Recall that a complete
5.4. THE STRUCTURE OF AN EVEN LIAISON CLASS
139
intersection link is also a Gorenstein link.) 0 As a consequence, we have partitioned £ = £0 U £1 U £2 ... according to the shift of the configuration of modules. The key to answering question 4 of Section 5.1 (i.e. to describe the structure of an even liaison class) is to study the relations between the elements of these subsets. This structure has been called the LazarsJeld-Rao Property (cf. [34], [12]). This is known to hold for co dimension two (cf. [12], [137]), and it will be described carefully in Chapter 6. However, for complete intersection liaison the structure can be phrased in arbitrary codimension, even if it is unknown to hold in co dimension 2 3, so we turn now to a description of that property. Definition 5.4.2 (cf. [34]) Let £ be an even complete intersection liaison class of dimension r subschemes of IF. We say that £ has the LazarsJeldRao Property if the following conditions hold: (a) If V1 , V2 E £0 then there is a deformation from one to the other through subschemes all in £0; in particular, all subschemes in the deformation are in the same even liaison class. (b) Given Va E £0 and V E £h (h 2 1) , there exists a sequence of subschemes Yo, V1, ... , 1ft such that for all i, 1 ::; i ::; t, Vi is a basic double link of Vi-1 and V is a deformation of 1ft through subschemes all in £h. This is also sometimes referred to as the LR-Property. 0 Remark 5.4.3 It follows immediately from Definition 5.4.2 that the minimal elements of £ all have the same degree, and they are strictly minimal in £ with respect to the degree. A similar result can be proven for the arithmetic genus. See also Section 6.4.3. If one combines this observation with Example 3.3.2, one might be tempted to conclude that the LR-Property does not hold in codimension 2 3, at least for Buchsbaum curves. Indeed, in Example 3.3.2 we saw that among curves in whose deficiency module is one-dimensional, the leftmost shift occurs in degree 0, but there are curves of degree 2 and curves of degree 3 occurring with this module. It is worth stressing that we do not know that these curves are all in the same CI-liaison class! (This contrasts sharply with Example 5.4.8 for Gorenstein liaison.) The converse to the Hartshorne-Schenzel theorem is false for complete intersection liaison in co dimension 2 3.
r
140
CHAPTER 5. LIAISON THEORY
For instance, there are non-licci aCM curves in JP4; the rational normal curve provides the simplest example. (The reason is identical to that given in Example 6.1.6, where the analogous zeroschemes in jp3 are studied.) In the situation of Example 3.3.2, it was conjectured in [145] that if C and C' each consist of a set of two skew lines, and if H and H' respectively are the hyperplanes containing C and C', then C and C' are in the same CI-liaison class if and only if H = H' . In fact, it seems very reasonable to conjecture that the LR-Property is true for complete intersection liaison in co dimension ~ 3. At the moment this is an open problem. 0 We conclude with some remarks about how this can be extended to Gorenstein liaison and where the problems lie. We begin with Basic Double Linkage, which is somehow at the heart of the problem. It was observed by Martin-Deschamps and Perrin [137] and by Hartshorne [90] that what is really at the heart of Basic Double Linkage and the LR-Property is the following (see also Remark 3.2.5 and Example 5.1.7 (4)).
Definition 5.4.4 (cf. [90]) Let Vi and V2 be schemes of equidimension r without embedded components in ]pn. We say that V2 is obtained from VI by an elementary biliaison of height h, for some h E Z, if there is a complete intersection scheme S of dimension r + 1 containing VI and l/2, such that V2 is linearly equivalent to VI + hH as generalized divisors on S , where H is a hyperplane section of S. 0 The difference between Basic Double Linkage and elementary biliaison is the following. In either case, V2 is obtained from VI by first adding a suitable "complete intersection tail" to VI; as mentioned in Remark 3.2.5, this is achieved by viewing VI as a divisor on a complete intersection Sand considering the divisor VI + HF for suitable homogeneous polynomial F of degree h. This is the Basic Double Link. Then in elementary biliaison we also allow ourselves to move in the linear equivalence class of VI + HF on S, to obtain V2 . In view of this approach, the natural way to attack Gorenstein liaison is clearly via Theorem 5.2.27 (see also the sentence preceding the statement of Theorem 5.2.27). This is the approach taken in [121], where the following notion of Basic Double G-Linkage is introduced. (Compare also with Theorem 3.2.3, Remark 3.2.4 and Remark 3.2.5.) We first give a result from [121], and we stress the weakness of the hypothesis Go rather than G I .
5.4. THE STRUCTURE OF AN EVEN LIAISON CLASS
Proposition 5.4.5 Let S C
141
be an aCM subscheme satisfying property Go (cf. Definition 4.2.6). Let C be an equidimensional divisor on S without embedded components, and let F be a homogeneous polynomial of degree d not vanishing on any component of S. Let Z = C + HF be the divisor on S defined (as a subscheme of]pm ) by the homogeneous ideal Is + F· Ie. (a) As sets, Z
]pm
= C UHF.
(b) We have an exact sequence of ideals
o -t Is( -d) -t Ie( -d) EB Is -t F· Ie + Is -t O. (c) F· Ie
+ Is
is a saturated ideal, i.e.
Iz
= F . Ie
+ Is.
(d) The Hilbert function of Z satisfies H(Z, t)
= =
H(S, t) - H(S, t - d) + H(C, t - d) H(HF' t) + H(C, t - d)
(e) H!(]pm,Iz ) ~ H! (Jpm , Ie)(-d) for all 1 ~ i ~ dim C = dim Z. (f) Z is in the same even G-liaison class as C, and is G-linked to C in two steps. Proof: Part (a) is straightforward, and almost identical to the proof of
part (a) of Theorem 1.3 of [78J. Parts (c), (d) and (e) follow easily from (b); notice that the equality H(HF' t) = H(S, t) - H(S, t - d) comes from the exact sequence
o -t Is( -d) -t Is EB R( -d) -t Is + (F)
-t
O.
Part (f) is rather technical, and we refer the reader to [121J where the links are given explicitly; however, if we assume that S satisfies G t then (f) follows simply from Theorem 5.2.27 (stated in suitable generality). We have only to prove (b). Notice that the last map is given by (A, B) f-t A - F B. It is clearly surjective, and its kernel is isomorphic to
[(Is: F) n Ie]
~
Is( -t)
since F does not vanish on any component of S. 0 This motivates the following definition, generalizing the notion of Basic Double Linkage.
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CHAPTER 5. LIAISON THEORY
Definition 5.4.6 Let S c pn be an arithmetically Cohen-Macaulay subscheme of dimension r + 1 satisfying property Go. Let VI C S be an equidimensional subscheme of codimension 1 with no embedded components. Let F be a homogeneous polynomial of degree h. The scheme Z defined by Iz = F . Iv! + Is is a Basic Double G-Link of VI. 0 It should be noted that the case where S is at least arithmetically Gorenstein was done in [36] and [163]. Unfortunately, we need that S be G I in order to talk about divisors being linearly equivalent (cf. [90]) and move to elementary biliaison in the Gorenstein setting. In view of this, the natural way to proceed is to replace "complete intersection" in Definition 5.4.4 by "arithmetically Cohen-Macaulay and G I :"
Definition 5.4.7 Let VI and V2 be schemes of equidimension r without embedded components in pn. We say that V2 is obtained from VI by an elementary G-biliaison of height h, for some h E Z, if there is a aCM scheme S of dimension r + 1, having property G I and containing VI and V2 , such that V2 is linearly equivalent to VI + hH as generalized divisors on S, where H is a hyperplane section of S. 0 This does seem to be the natural definition of elementary G-biliaison. However, the problem is that it does not seem to lead to as nice a picture for Gorenstein liaison as that given by the Lazarsfeld-Rao Property for complete intersection liaison. (On the other hand, the Lazarsfeld-Rao Property has only been proved in codimension two, where there is no difference between CI-liaison and G-liaison.) A simple example to illustrate the difficulties is the following. Example 5.4.8 Let X be the curve in p4 given in Figure 5.3. Here, the
Figure 5.3: Gorenstein link lines are chosen as generally as possible subject to the condition that they
5.5. GEOMETRIC INVARIANTS OF A LIAISON CLASS
143
intersect as indicated. It is not too hard to see that X is arithmetically Gorenstein. As such, we can view X as giving a geometric G-link of two curves, C 1 and C2 , where C 1 consists of two skew lines and C2 consists of the disjoint union of a line and a plane curve of degree 2. We know the deficiency module of C 1 (Example 1.5.2) and one can check that the minimal free resolution of X is
o ~ 5(-5) ~ 55(-3) ~ 55(-2) ~ Ix
~
0,
so by the Hartshorne-Schenzel theorem (Theorem 5.3.1) or by a direct computation (Proposition 1.3.10 (c)), we see that both C 1 and C2 have a one-dimensional deficiency module, concentrated in degree O. (Notice that the curve C2 is non-degenerate. This can be contrasted with the curve of this form lying in p3, which was given in Example 1.5.4. In particular, we see that C 2 is arithmetically Buchsbaum, while Example 1.5.4 showed that the corresponding curve in p3 is not.) On the other hand, C 1 is directly linked by a complete intersection of type (1,2,2) to another union of two skew lines, which is then evenly linked to C2 . What then should the set of minimal curves in the even liaison class look like? It is no longer true that all curves with the leftmost shift have the same degree, which is a consequence of condition (a) of the Lazarsfeld-Rao Property (Definition 5.4.2). And if we take as our minimal curves those with least degree (cf. Remark 5.4.3), namely degree 2, there is no natural way using Definition 5.4.7 to obtain from a minimal curve the curve C2 . 0
5.5
Geometric Invariants of a Liaison Class
In Section 5.3 we saw that the graded modules (Mi)(V), with 1 ::; i ::; dim V, are invariant, up to shifts and duals (resp. shifts), in the liaison class (resp. even liaison class) of V. This implies that the degeneracy loci Vb described in Section 1.1, are invariant (after suitable re-indexing). These loci essentially parameterize the set of linear forms that have "unusual" rank when viewed as homomorphisms from a component (Mi)(vh to the next component (Mi)(Vh+l' In this section we will show hJw this fact can be used to study the liaison class of V when V is a curve (locally Cohen-Macaulay and equidimensional, as always), and hence i = 1. Our main references are [142] and [145]. However, we remark that it would be interesting to understand the connection between the Vk and V in the case where dim V is larger and especially when i is larger. The general philosophy that emerges, which is
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CHAPTER 5. LIAISON THEORY
illustrated in several examples below, is that the points of Vk correspond to those hyperplanes meeting V in "unusual" ways, either by containing a component of the curve or by having unusual postulation for the hyperplane section. This often yields useful necessary conditions for curves to be evenly linked, as we will see. We remark that this general description should still be true for the higher cohomology modules, but it needs to be made precise. (The meaning of "unusual" may be entirely different for larger i.) We will use the letter C to denote our curves rather than V, to avoid confusion with the degeneracy loci Vk . We now illustrate the ideas with several examples.
Example 5.5.1 Let C c JP3 be the disjoint union of a line A and a plane curve Y of degree d ~ 1. This situation was analyzed in Example 1.5.4. It was shown that M(C)i is one-dimensional for 0 ~ i ~ d - 1, and 0 elsewhere. Furthermore, the degeneracy locus Vo C (JP3)* is just the plane dual to the point P of intersection of A with the plane of Y. Since this is an isomorphism invariant, we have that any curve evenly linked to C has the same degeneracy locus in the first non-trivial module multiplication. Furthermore, by Remark 1.3.11 (c), if we denote by C the even liaison class of C, then C E Co. We now describe a simple link that can be done with C. (We will use the same names for surfaces and for polynomials defining those surfaces.) Let FI = Al U A2 be a surface of degree 2 consisting of the plane Al of Y and a plane A2 containing A. Let F2 be a surface of degree d+ 1 containing C but not containing any component in common with Fl. (FI' F 2 ) is a complete intersection, and hence gives a residual curve ct. What is C'? Looking first at the intersection of F2 with Al we see that the residual to Y is a line A' on Al passing through P (since Y avoids P but both Al and F2 contain P). The residual to A in the complete intersection of F2 with A2 is a plane curve Y' of degree d. Hence C' = Y' U ,XI. If Y' and ,XI were to meet, then by Example 1.5.5 C' would be aCM. But we know that C is not aCM and that the property of being or not being aCM is preserved under liaison. Therefore C' is also the disjoint union of a line and a plane curve of degree d, with P again the point of intersection of A' and the plane of Y'. We now ask when two such curves can be in the same liaison class. The following can be shown from the above discussion (the details are left as an exercise). Let C be the disjoint union of a line A and a plane curve Y of degree d, with P the point of intersection of A with the plane of Y.
5.5. GEOMETRIC INVARIANTS OF A LIAISON CLASS
145
Let C' , >..', yl, pI and d' be similarly defined. Then C and C' are in the same liaison class if and only if d = d' and P = P' . To show that the conditions d = d' and P = P' imply that C and C' are linked, for now one has to proceed by brute force, constructing a sequence of links in the manner outlined. However, once we have Rao's theorem in the next chapter, this will follow from the observation that these two conditions force M(C) and M(C') to be isomorphic. 0
Example 5.5.2 Let C be a nondegenerate set of d skew lines in IF, with 2d > n + 1. Assume that the general hyperplane section Z = C n H of C is nondegenerate in the hyperplane H. We first compute that dim M(C)o = d - 1 and dim M(Ch = 2d - (n + 1). These come from the exact sequence
o --+ Ie --+ Opn --+ Oe --+ 0,
twisting by either 0 or 1, and taking cohomology. Now, for a linear form L defining a hyperplane H that does not contain any component of C, there is an exact sequence
This is from the exact sequence in cohomology associated to the short exact sequence (1.6) of Section 1.1. The first 0 is because C is nondegenerate. The hypothesis that the general hyperplane section is a set of points which is nondegenerate in H guarantees that CPo (L) is injective for general L. In particular, we must have that dim M(C)o :S dim M(Ch, i.e. n:S d. In fact, clearly if L does not vanish on any component of C then Lean be viewed as a point of Vo if and only if the points of H n C are degenerate in H. (For example, if C is a curve in jp3 then L is a point of Vo if and only if H contains a d-secant line of C.) On the other hand, we still have to consider the linear forms L which do vanish on a component A of C. Let Y be the remaining set of lines. Then for any k we have the following two exact sequences of sheaves:
0--+ Ie(k) --+ Iy(k) --+ O)..(k) --+ 0 and Here C
0--+ Iy(k) ~ Ie(k + 1) --+ IenH(k + 1) --+ O.
n H is one-dimensional. Notice that the composition 0--+ Ie(k) --+ Iy(k)
xL)
Ic(k
+ 1)
CHAPTER 5. LIAISON THEORY
146
induces the map
and
o ~ HO(IcnH(l))
~ M(Y)o ~ M(Ch-
Since hO(O>.) = 1, we get that n + 1, but now we assume that the general hyperplane section is degenerate in the hyperplane. We will show that C must lie on a quadric hypersurface. We use the "Socle Lemma" of [105J (see Lemma 2.3.3 and take d = 1) and apply it to M(C) . Since hO(IcnH (l)) =I 0 by hypothesis, this means that b = 1. Hence Lemma 2.3.3 guarantees that i([O :
5.5. GEOMETRIC INVARIANTS OF A LIAISON CLASS
147
that the degeneracy locus Va identifies the hyperplanes in JP'l that contain either a d-secant line or else a component of C. There are two possibilities (from the point of view of what we want to discuss) : either C lies on a quadric surface or it does not. If C lies on a quadric surface Q then clearly it must be smooth in order to contain skew lines. One checks that the hyperplanes tangent to Q are precisely the hyperplanes that we are looking for. One can thus recover Q from the module, but no more. And indeed, C is linked via Q and an appropriate union of d hyperplanes to a set of d skew lines in the other ruling of Q. It follows (since dim M(C)o = dim M(C')o = d - 1) that any set of skew lines in the liaison class (either even or odd) of C also has d components, and also lies on Q, but there is no uniqueness; any such set of lines is in the liaison class. In fact , it can be shown that if C is a set of d ~ 3 skew lines lying on a quadric surface Q, and if C' is set of d' skew lines, then C is evenly linked to C' if and only if d = d' and C' also lies on the same ruling of Q. They are oddly linked if and only if d = d' and C' lies on the other ruling of Q. On the other hand, if C does not lie on a quadric surface then by Example 5.5.3, the general hyperplane section does not consist of collinear points. That is, the general hyperplane section gives an injection of M(C)o ---+ M(Ch. Now, since C does not lie on a quadric surface, it must consist of at least four lines. And for the same reason, one can check that it has at most two d-secants. (If it had three d-secants, Bezout's theorem would force C to lie on the quadric surface spanned by the three d-secants.) Then Vo consists of a finite set of lines in (JP'l) *, and so from the configuration of lines one can recover C from M(C) . That is, C is the only set of skew lines in its even liaison class. (See [128) for another example, more algebraic, of how to recover a curve from the deficiency module, under certain circumstances.) However, there may be a set of skew lines (at most one) oddly linked to C . Asimple example is provided by a so-called "double-six" configuration on a cubic surface (cf. [81) ; see also Example 5.5.6), in which a set of six skew lines is directly linked to another set of six skew lines. See [142) for more details. 0 Example 5.5.5 In this example and the next we consider general smooth rational curves. We will consider the case d = 5 in this example and d = 6 in the next example, and we will ask the question of which other smooth rational curves can be in the liaison class of C . This was first studied in [142). To begin, let C be any smooth rational curve of degree d. We can
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use the Riemann-Roch theorem and the exact sequence
o-+ Ie -+ OlP'3 -+ Oe -+ 0 to compute dim M( C)i for any i if we know the Hilbert function of C. In particular, we see immediately that dim M(Ch = d - 3. If C is a general rational curve of degree d, we have from [98] and [13] that C has maximal rank, and this is also enough to compute dim M(C)i for any i. Now let C be a general smooth rational quintic curve in JP3. In this case we have 2 if i = 1; dim M(C)i = { 1 ifi = 2; o otherwise, and furthermore C lies on no quadric surface and hO(JP3,Ie(3)) = 4. In particular, there is a (projective) 3-dimensional family of cubic surfaces containing C, all of which are irreducible. The intersection of any two of these is a complete intersection containing not only C but any 4-secant line of C. Hence C has a finite number of 4-secants. The general hyperplane section of C consists of five points lying on a unique conic in ]p2. Hence the degeneracy locus VI consists of those hyperplanes whose corresponding hyperplane section lies on a pencil of conics. This happens if and only if it contains four collinear points, i.e. it contains a 4-secant of C . Note that for any 4-secant line, there is a pencil of hyperplanes containing it. Since there is a finite number of 4-secants, and since the expected dimension of VI is 1 (from the dimensions of the components computed above), we can apply Lemma 1.1.1 to conclude that deg VI = 1, i.e. there is a unique 4-secant line, which is dual to the line VI C (JP3)*.
Suppose that C' is another smooth rational curve, and we now ask if C' can be linked to C in any number of steps. It is not hard to show from the above computations that C' cannot be oddly linked to C at all, and that if C' is evenly linked to C then C' also has degree 5, lies on no quadric surface, and shares the same 4-secant line. Furthermore, one can check that in this case VI determines M(C) up to isomorphism. Hence we can conclude from Rao's theorem, Theorem 6.2.7, that another rational curve C' is linked to C if and only if C' also has degree 5, lies on no quadric surface, has the same (unique) 4-secant line that C has, and is evenly linked to C. In [168], Nollet shows that in fact one can choose two cubics to link C to a curve supported on the 4-secant, which has degree 4 and generic embedding dimension 3. 0
5.5. GEOMETRIC INVARIANTS OF A LIAISON CLASS
149
Example 5.5.6 Let C be a general smooth rational sextic curve in JP3. We will briefly describe the lovely geometry relating the geometry of C to the geometry of the degeneracy locus VI; for details see [142]. The problem we would like to solve is to determine which other curves C' of arithmetic genus 0 are in the liaison class of C. Because C has maximal rank (cf. [98], [13]), C lies on a unique cubic surface 5 . Since C is general, 5 is smooth. Hence we know all the divisors on 5, and in particular there are 27 lines (cf. [88], [81]). The geometry of 5 will play an important role. One immediate observation, from Example 5.1.7, is that any curve which is linearly equivalent to C on 5 is evenly linked to C. Also, any curve linked to C by 5 and a quartic surface has degree 6 as well, and hence also has arithmetic genus 0 (cf. Corollary 5.2.14). These are precisely the curves in the linear system 14H -CI (where H is the class of the hyperplane section). We will see that these are the only curves of degree 6 and/or arithmetic genus 0 in the liaison class of C. On the other hand, maximal rank (together with the Riemann-Roch theorem) also allows us to compute that the deficiency module M(C) is 3-dimensional in degrees 1 and 2, and zero elsewhere. This shows, in particular, that if C' is in the liaison class of C with arithmetic genus 0 then C' also has degree 6. (One can also use the Lazarsfeld-Rao Property to deduce this.) It is not hard to show that the general hyperplane section of C consists of 6 points not on a conic. (For example, the general hyperplane section does not have any collinear points, so if it lay on a conic it would be a smooth conic rather than a union of two lines. But then it would be a complete intersection, and so by Strano's theorem [193] (see Theorem 2.3.1) C would have to be a complete intersection.) Hence the degeneracy locus VI consists of those linear forms whose corresponding hyperplanes intersect C in six points on a (possibly singular) conic. By Porteous' formula (cf. Lemma 1.1.1) we see that VI is a cubic hypersurface in (JP3)*. (This is also clear since it is obtained by taking the determinant of a 3 x 3 matrix of linear forms, but it is nice to see Porteous' formula at work.) We would like to identify this cubic surface and to describe its relationship to the cubic surface 5. We can only give the basic idea of the answer here. There is a special type of configuration of lines on 5 called a "double-six," and there are only 36 such configurations. These configurations consist of two separate sets of 6 skew lines, meeting in a special way. Of these 36, there is exactly one with the property that each of one of the sets of 6 skew lines is a 4-secant
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CHAPTER 5. LIAISON THEORY
to C, and each of the other 6 lines is disjoint from C. But then it is clear that the lines dual to each of these lines all lie on V1 . One has to check that a plane containing any of these 12 lines meets C in 6 points which lie on a conic in that plane: in the former case the conics are all unions of lines, and in the latter case they are generically smooth. We leave the verification as an exercise, or see [142] for details. But these 12 dual lines also form a double-six configuration in (p3)*, and hence V1 is the unique cubic surface containing this configuration. Knowing that this is V1 limits the possible linear systems of curves of degree 6 and arithmetic genus 0 on cubic surfaces in p3 (by looking at all the double-sixes on V1 and going back to their duals in p3). The last step is to consider a certain line bundle on V1 and use it to "weed out" the sextic curves which do not lie on S. D The examples in this section serve as a useful illustration of the fact that the curves in an even liaison class which are minimal (cf. Remark 1.3.11 (b) and Definition 5.4.1) really do fall into nice families, including the possibility that they are unique. This is part of the Lazarsfeld-Rao property described in Definition 5.4.2 and proved in the next chapter for codimension two.
Chapter 6 Liaison Theory in Co dimension Two As we have mentioned before, the strongest results to date on liaison theory are for the case where the schemes have co dimension two. We have seen that in this situation, the arithmetically Gorenstein ideals are precisely the complete intersections. Until recently, it was also necessary to restrict to schemes that are locally Cohen-Macaulay and equidimensional. However, Nollet [167] has modified the original proofs of these theorems to remove the hypothesis of locally Cohen-Macaulay, now assuming only that all components have the same dimension. Similar generalizations were obtained by Nagel [163]. This is an important contribution, but for simplicity in this chapter we maintain this hypothesis. The main results which are known in co dimension two, but which are missing in the case of arbitrary codimension, are a sufficient condition for two subschemes to be linked (or evenly linked), and a structure theorem describing an even liaison class. These will be discussed in this chapter. See also [146] for an expository description of some of the main results in codimension two. Liaison theory (as a subject rather than simply a tool) began with Apery and Gaeta in the 1940's (cf. [3], [4], [70], [71]). In these papers, it was shown (essentially) that a smooth curve C in ]p3 is in the linkage class of a complete intersection (i.e. lieci) if and only if it is arithmetically Cohen-Macaulay (i.e. its deficiency module is zero). This result was extended to arbitrary co dimension two subschemes of projective space by Peskine and Szpiro [174], and they put the whole theory of liaison into the framework of modern scheme theory. Some of their work was described in the last chapter. In Section 6.1 we discuss the co dimension two part of their work.
J. C. Migliore, Introduction to Liaison Theory and Deficiency Modules © Birkhäuser Boston 1998
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CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
After the landmark paper of Peskine and Szpiro, the next major contribution to the theory of liaison was by Rao in his two papers [179] and [180]. The latter paper contains the main result of the former as a special case, but the former can be viewed as the starting point of the burst of activity for codimension two subschemes that has occurred in liaison theory in the last two decades. The main result ofRao which we will describe here is to give a necessary and sufficient condition for two co dimension two subschemes of projective space to be evenly linked. (He also observes that his result holds in a more general context.) In the last chapter we have seen a necessary condition: the Hartshorne-Schenzel theorem. In the case of curves in p3 he notes that this is also sufficient. However, for F it is necessary to approach the problem from the point of view of locally free sheaves, as we will see. The main result is given in terms of stable equivalence classes of locally free sheaves. In this chapter we will also give the structure theorem of [12] (in F) and [137] (in p3) , the so-called Lazarsfeld-Rao Property, which describes an even liaison class and how the elements of the class are related to each other. This can also be done for the more general context mentioned by Rao (cf. [36]). Passing to curves in p3, we have even stronger results from [137] (and, as mentioned above, we have Rao's converse to the HartshorneSchenzel theorem). Finally, in Section 6.4 we give some applications of liaison. For example, we will describe how it has been used in the classification of surfaces in ]p4 and threefolds in F. We will also show how the Lazarsfeld-Rao Property has been used to prove that every Buchsbaum curve specializes to a stick figure, proving a special case of the classical question (often called Zeuthen's problem) of whether every smooth curve specializes to a stick figure. Throughout this chapter, we will assume that all schemes in question are locally Cohen-Macaulay and equidimensional. It should be remarked, again, that Nollet [167] and Nagel [163] have generalized the main results of codimension two liaison theory by removing the assumption of locally Cohen-Macaulay and working with reflexive sheaves.
6.1
The aCM Situation and Generalizations
The simplest situation in Liaison Theory is the case of arithmetically Cohen-Macaulay (aCM), codimension two subschemes ofF. In this short
6.1.
THE aCM SITUATION AND GENERALIZATIONS
153
section we will discuss what is true in this situation. As indicated below, most of the proofs can be found in [62]. We saw on page 11 that if V is aCM of codimension two with Iv = (F1 , . .. , Fr ) (minimal generators) then it has a minimal free resolution
Iv
o
/'
(6 .1)
'\I
0
where B = (F1 ,· ·· , Fr) and A is called the Hilbert-Burch matrix of Iv . The first amazing fact is that the saturated homogeneous ideal Iv of V is given precisely by the ideal of maximal minors of A . This is called the Hilbert-Burch theorem. Here we give the version from [62].
Theorem 6.1.1 (Hilbert-Burch)
(a) If a complex
r-1
r
j=l
i=l
o -+ EBS(-bj)~EBS(-ai)~S -+ SII -+ 0 is exact, then there exists a non-zerodivisor a such that I = a· I r - 1(A) (the ideal of maximal minors of A). In fact , the i-th entry of the matrix B is (-1) i a times the minor obtained from A by leaving out the ith row. The ideal I r - 1(A) is the saturated ideal of an aCM codimension two subscheme of pn, and in particular it has depth exactly 2. (b) Conversely, given any r x (r-l) matrix A such that depth I r - 1(A) ~ 2 and a non-zerodivisor a, the map B obtained as in part (a) makes the complex into a free resolution of SI I with I = a· Ir-1(A) . Proof: This is given in [62], Theorem 20.15. 0
Remark 6.1.2 Theorem 6.1.1 can be used to give a different proof of Corollary 2.2.3. If V is defined by the maximal minors of an r x (r - 1) homogeneous matrix A, then clearly v {Iv ) = rand a {Iv ) ~ r - 1 (with equality if and only if there is an (r -1) x (r - 1) submatrix of A consisting entirely of forms of degree ~ 1). For further discussion on when the equality is reached, see [16], [17], [18] . 0
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CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
Next we address the question of what happens to the Hilbert-Burch matrix when we link. Again the result is amazingly simple and elegant, and is due primarily to Gaeta. We give the main special case.
Theorem 6.1.3 Let V be an aCM codimension two subscheme of m. Assume that the minimal generators of Iv are chosen generally. Consider a minimal free resolution of Iv given by (6.1), so that Iv is given by the maximal minors of an r x (r - 1) homogeneous matrix A. Let Fi be the (r - 1) x (r - 1) minor of A obtained by omitting the i-th row. (a) There is some choice of i and j so that (Fi' Fj) is a regular sequence. (b) Assume that (Fi' Fj ) is a regular sequence, and let A' be the matrix obtained from A by deleting the ith and jth rows. Then the complete intersection (Fi ' Fj ) links V to the aCM scheme V' defined by the ideal of maximal minors of A'. Proof: Since the generators of Iv were chosen generally, the generator of largest degree must have no component in common with any of the other generators, or otherwise the ideal would not define a scheme of codimension two. This proves part (a). Part (b) is proved in Proposition 21.24 of [62).0
What this theorem says is that if we start with an aCM codimension two subscheme, V, ofm whose homogeneous ideal is given by the maximal minors of a homogeneous r x (r - 1) matrix A, and if we link V using two forms which are minimal generators of Iv, then the residual scheme V'is again aCM of codimension two, and up to multiplication by a scalar, the Hilbert-Burch matrix of V'is obtained from A by deleting two rows and transposing. Gaeta gave a more detailed treatment, considering also the case where the link is not by two minimal generators, and giving the Hilbert-Burch matrix of the residual. This leads to the following famous theorem; for the case of curves in p3 it is due to Apery [3] and Gaeta [70], and in codimension two in the modern language it is due to Peskine and Szpiro [174].
Corollary 6.1.4 There is only one liaison class consisting of aCM codimension two schemes in any projective space. In other words, any aCM codimension two scheme is licci .
6.1.
THE a,OM SITUATION AND GENERALIZATIONS
155
Proof: Starting with any aCM co dimension two scheme V with HilbertBurch matrix A, we can find a sequence of links so that at each step the number of minimal generators goes down by one from the step before, by Theorem 6.1.3. Finally we arrive at a complete intersection. The fact that all complete intersections of fixed codimension (not just codimension two) are in the same liaison class is given in Example 5.1.7 (3.) . 0 Another proof of Corollary 6.1.4 can be found in Remark 6.2.2, using a more general set-up (the mapping cone procedure described in Section 5.2) and specializing to the aCM case. Example 6.1.5 We mentioned in Example 5.1.7 that if j = (F, G) is a complete intersection and Jk = (Fk, Gk ), then Jk directly links jk to jk-l, and in particular these are the saturated ideals of aCM schemes. We can give a simple proof of this fact now. Indeed, we have that jk is precisely the ideal of maximal minors of the matrix
F 0
k+l
0
0
0 0 0 0
0 0
0 0
G
G F 0 G
F .., k
Then removing the first and last rows and applying Theorem 6.1.3 (and transposing) gives the result. Notice that the degree of the scheme defined by jk is (k!l) . deg F· deg G (one can check this inductively). 0 The results in this section are essentially about ideals defined by the maximal minors of a r x (r - 1) homogeneous matrix, i.e. about determinantal ideals of codimension two. In higher codimension it is not true that any aCM scheme, or even any scheme defined by the maximal minors of a homogeneous matrix (and these are no longer equivalent), behaves so nicely under complete intersection links. In particular it is no longer true that they are licci. A very surprising result is that they are nevertheless glicci (cf. page 109). We now briefly describe the setup and result. We first give an example of a determinantal scheme of codimension three which is not licci.
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CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
Example 6.1.6 Let S
= k[Xo, Xl, X 2 , X 3] and let A be a 2 x 4 matrix of
linear forms in S whose ideal of maximal minors defines a zeroscheme Z in jp3. The degree of Z is 4 (cf. for instance Porteous' formula, Lemma 1.1.1). Two examples are
0]
[ Xl X 2 X3 o Xl X 2 X3
whose ideal of maximal minors, II, is the square of the ideal of a point, (Xl, X 2 , X 3 ), and
[ Xo Xo
o
+ Xl Xo + 2X2 Xo + 3X3
Xl
X2
]
X3
whose ideal of maximal minors, 12 , is the ideal of the four coordinate points in p3. In either case, the minimal free resolution of the ideal has the form 0-+ 3S( -4) -+ SS( -3) -+ 6S( -2) -+ Ii -+ 0 (where i = 1,2), which is a so-called pure resolution. Theorem 2.5 of [126] says that in codimension three, if the resolution of I is pure then I is lieci if and only if it is arithmetically Gorenstein. Hence neither of these ideals is lieei. 0
Remark 6.1.7 A classical result of Watanabe [201] gives that a codimension three, arithmetically Gorenstein subscheme of F is licci. Contrasting this, it was shown in [19S] that there are infinitely many CI-liaison classes of arithmetically Gorenstein curves in F, and infinitely many CI-liaison classes of aCM curves in IF. See also [121], and Remark 5.2.2S. 0 For many results on determinantal ideals (cf. for instance [62] and [41]), one has to assume that the ideal of maximal minors of the given homogeneous matrix defines a scheme of the "expected codimension." More precisely, the ideal of maximal minors of a homogeneous t x (t + r) matrix, A, is assumed to have co dimension r + 1. Such an ideal is called a standard determinantal ideal in [125]. Many results require a further assumption. One way of saying this condition very succinctly is that the scheme defined by the ideal of maximal minors of A should be a generic complete intersection. Another way of saying it is that after (possibly) a base change, the matrix obtained by removing some row of A again has the expected codimension, this time r + 2. Such ideals are called good determinantal ideals in [125]. The following unexpected result was proved in [121], and it provides a very nice generalization in arbitrary codimension to Corollary 6.1.4. It
6.1.
THE aCM SITUATION AND GENERALIZATIONS
157
shows the power of passing from complete intersection liaison to Gorenstein liaison. Theorem 6.1.8 A standard determinantal ideal of any codimension is glicci. Proof: The proof is rather long, but we give the essential idea. Given the t x (t + r) homogeneous matrix A, it was shown that after possibly
making a base change, the matrix A' obtained by removing a row and a column from A is again standard determinantal. Let I and I' be the ideals defined by the maximal minors of A and A', respectively. Then an ideal J was produced such that both I and l' are directly G-linked to J. Hence the ideal of maximal minors of A is evenly G-linked to the ideal of maximal minors of a submatrix of A, and just as in the codimension two case described above, one proceeds step by step until one arrives at a complete intersection. 0 Remark 6.1.9 The fact that the second ideal, 12 , in Example 6.1.6 is glicci was already observed. Indeed, we saw in Example 5.1.7 that the four coordinate points are directly G-linked to a single point, which is a complete intersection. The first ideal, 11 , is less clear since it is supported at a point and is not a local complete intersection. However, it was shown in [154] Remark 5.1 that there is an arithmetically Gorenstein ideal J c It defining a scheme of degree 5, so by Corollary 5.2.13, the square of the ideal of a point is G-linked to a single point. The result of Theorem 6.1.8 is slightly different. It says that both 11 and 12 are evenly G-linked (in two steps) to the point with defining ideal (Xl, X 2 , X 3 ), because in either case we can obtain this ideal by removing a suitable row and column from the original matrix. The following result generalizes this observation. 0 Corollary 6.1.10 Let I = (F1 , •. . , Ft ) be a complete intersection in IF. Then for any k, Ik is G-linked to I k- 1 in two steps. Proof: The proof is an obvious modification of Example 6.1.5, writing Ik
as the maximal minors of a suitable matrix and removing instead the last row and the last column as in the description of the proof of Theorem 6.1.8. 0
158
6.2
CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
Rao's Results
From now on we assume that our scheme V is locally Cohen-Macaulay and equidimensional, but not necessarily aCM. If V c IF has co dimension two, then we have seen that there is a short exact sequence m
o-+ 9 -+ EB Ojp>n( -ai) -+ Iv -+ o.
(6.2)
i=l
This is obtained, as in Section 1.2, from a minimal free resolution of the saturated homogeneous ideal Iv. (The integers ai are the degrees of the minimal generators of Iv.) It follows that 9 is locally free of rank m - 1 , and that H; (9) = o. (The latter fact follows because we began with a minimal free resolution, so the map EB~l Ojp>n( -ai) -+ Iv is already surjective on global sections.) The fact that 9 is locally free , especially, was important in our proof of Proposition 5.2.10 (but see also Remark 5.2.11) . It follows from Proposition 5.2.10 that if V and V' are evenly linked, and if Iv has the locally free resolution above, then Iv' has a locally free resolution
0-+ g' -+ :F' -+ Iv' -+ 0 where :P is free and g' = 9 (c) EB A, with A free and c E Z. Note that here the direct summands of :P do not necessarily correspond to a minimal generating set for Iv'. For many purposes, for instance Rao's proof of injectivity and surjectivity of the correspondence in Theorem 6.2.5 below, and the proof in [12] of the Lazarsfeld-Rao Property, it is convenient to have a short exact sequence of the form 0-+ (free) -+ K -+ Iv -+ 0
(6.3)
where K is locally free. This is obtained using the above ideas as follows. Choose any complete intersection X containing V, and let W be the residual. Suppose that Ix is generated by forms of degrees b1 and b2 , and let b = b1 + b2 . If W has a locally free resolution
o -+ £ -+ EB Ojp>n( -ai) -+ Iw -+ 0 as above, where £ is locally free, then Proposition 5.2.10 gives that V has a locally free resolution 0-+
EB Ojp>n(ai -
b) -+ £ v (-b) EB Ojp>n( -b2 ) EB Ojp>n( -b 1 ) -+ Iv -+
Then simply take K = £v (-b) EB Ojp>n( -b2 ) EB Ojp>n( -b 1).
o.
6.2. RAG'S RESULTS
159
Remark 6.2 .1 To remove the assumption of locally Cohen-Macaulay (in any codimension) , Nollet [167] and Nagel [163] develop a set-up very similar to this one. However, what is lost is the property that 9 and K. are locally free . They have to develop the corresponding machinery (in the case of [163] this is done even for Gorenstein liaison) using reflexive sheaves. See also Remark 6.2.6. 0 Remark 6.2.2 As Rao observes, if the generators of Ix are taken from a minimal generating set of Iw then the summands OJPn( -b2 ) EB OJPn( -b 1 ) of K. are redundant and can be canceled. This idea can be used, for example, to show that any arithmetically Cohen-Macaulay codimension two subscheme V of IF is in the linkage class of a complete intersection (i.e. licci). We briefly describe the idea. Note first that if V is arithmetically Cohen-Macaulay then 9 is in fact free of rank m -1, not just locally free. When we perform a link, the ideal Ix of the complete intersection X that we choose has two generators, and it can happen that 0, 1 or both of these generators are part of a minimal generating set of Iv. (In particular, for any V, arithmetically CohenMacaulay or not , it is always possible to find a complete intersection X satisfying any of these three possibilities.) If we link V by a complete intersection X, both of whose generators are minimal generators of Iv, one can check that the canceling referred to at the beginning of this remark gives us that the residual to V in the link has one fewer minimal generator (m - 1) than does V. (Use the fact that 9 is free.) So one can proceed by induction. This is essentially the approach of Gaeta and Peskine-Szpiro. However, this result also follows from Rao's more general theorem below. Notice that this idea, that in this case the residual has one fewer generator, is made precise in Theorem 6.1.3 where the Hilbert-Burch matrix of the residual is described. If we link V by a complete intersection X , only one of whose generators is a minimal generator of Iv, a similar consideration gives that the residual has the same number of minimal generators as does V . For example, if V is a complete intersection then in this case so is the residual. 0
Definition 6.2.3 Two reflexive sheaves 91 and 92 on IF are stably equivalent if there exist free sheaves £1 and £2 and an integer c such that 91 EB £1 ~ 92(C) EB £2. This notion can also be restricted to locally free sheaves. (Recall that by a "free sheaf" £ we mean a direct sum of line bundles, £ = EBi O(ai)') 0
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CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
The following result, due to Nollet [167] in the generality given, clarifies this definition somewhat. It is central to the approach of MartinDeschamps and Perrin to prove the structure theorem described in the next section. What it says is that a stable equivalence class is built up from a unique (up to isomorphism) minimal element; this is almost exactly what the Lazarsfeld-Rao property (Definition 5.4.2) says for an even liaison class! Proposition 6.2.4 Let 0 be a stable equivalence class of reflexive sheaves on jpn . Let F o be a sheaf in 0 of minimal rank. Then for each g E 0, there exists h E Z and a free sheaf L such that g ~ Fo(h) EB L . In particular, the isomorphism class of F o is uniquely determined up to twist. Proof: We repeat the proof given in [167], Proposition 2.3. Let
g E 0 be
arbitrary. Since F o and g are in the same stable equivalence class, there exists h E Z and free sheaves L1 and L2 such that (6.4)
The idea is to split off line bundles one by one and show that all of L2 can be removed in this way. Let O( -a) be a line bundle summand of L2 , so L2 = O( -a) EBL; with .q free. Twist the isomorphism (6.4) by a and consider the map on global sections. We get an induced surjection
If the induced map HO(jpn , Fo(h + a)) ~ k is surjective, then the corresponding map Fo(h + a) ~ 0 is a split surjection. If £ is the kernel then £ EB 0 ~ Fo(h + a) shows that F o was not of minimal rank in 0, a contradiction. Hence the induced map must be zero, and so the map HO(jpn,L1(a)) ~ k is surjective. This means that the induced map L1 ~ O( -a) is split surjective, and if L~ is the kernel then we can split off O( -a) from (6.4) to get Fo(h) EB L~ ~
g EB L;.
Continuing in this manner gives the desired splitting off of L2. 0 Rao's main result on liaison in codimension two (cf. [180]) is the following:
6.2. RAO'S RESULTS
161
Theorem 6.2.5 (Rao) In JP", n ~ 2, the even liaison classes in codimension two are in bijective correspondence with the stable equivalence classes of locally free sheaves 9 on JP" with Hl(JP", 9(p)) = 0 for all p E Z. The correspondence is given as above: first, to any locally CohenMacaulay, equidimensional scheme V we associate a locally free sheaf 9 obtained in the exact sequence (6.2). Proposition 5.2.10 then says that if V and V' are evenly linked, then the locally free sheaves one obtains are stably equivalent. This means that we have a well-defined map 1> from the set of even liaison classes to the set of stable equivalence classes of locally free sheaves with the indicated vanishing of cohomology. The hard part is to show that this map is injective and surjective. We will only give the basic idea of Rao's proof, and refer the reader to [180] for the details. To show injectivity, suppose that Vi and \12 are codimension two subschemes of JP" which yield locally free sheaves 91 and 92, as in (6.2) , which are stably equivalent. Rao shows that then Vi and V2 are evenly linked. His idea is that we can use the above type of arguments to obtain two locally free resolutions r
0-+
EB OlP'n( -aj) ~ !C(a) -+ Iv!
-+ 0
j=l
The maps a and {3 have the form (Sl"'" BT ) and (t 1 , •.. , tT ), where the Si and ti are global sections of various twists of !C. He then shows how, by a specific series of pairs of links, we may begin with the exact sequence involving a and (essentially) end up with the exact sequence involving (3. That is, he shows how a prescribed even number of links will take us from Vi to \12. For surjectivity, Rao begins with a locally free sheaf 9 as in the theorem, and considers the dual sheaf 9 v . His goal is to set up an exact sequence of the form (6.2) with the locally free sheaf 9. He will do this by first finding one of the form (6.3). By considering global sections of9 V (p) for p » 0, he obtains a locally Cohen-Macaulay scheme Y with resolution 0-+ (free) -+ 9 V (c) -+ I
y
-+ 0
for some c E Z. Then any direct link Y rv V gives a codimension two locally Cohen-Macaulay equidimensional scheme V with resolution of the desired form. This completes our description of Rao's proof of the bijection.
162
CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
Remark 6.2.6 This theorem, of course, implies Corollary 5.3.3 of the Hartshorne-Schenzel theorem, in the case of co dimension two even liaison. In fact, the cohomology of the sheaf 9 gives us back the deficiency modules: H!+1 (Jpn , Q) ~ (Mi)(V) for 1 ~ i ~ n - 2, and this cohomology is invariant under stable equivalence (up to shifts). However, to obtain this corollary we do not need the full weight of Rao's bijective correspondence between co dimension two even liaison classes and stable equivalence classes of locally free sheaves 9 with H; (Jpn ,9 (p)) = O. All that is needed is the fact that the map ¢, which we mentioned in the paragraph following Theorem 6.2.5, is well-defined. In fact, this works in arbitrary codimension. Given a locally CohenMacaulay, equidimensional subscheme V of Jpn of codimension c, we get in a similar wayan exact sequence
o--+ 9 --+ 9c-1 --+ ... --+ 91 --+ Iv --+ 0, where the 9i are free and 9 is locally free. To prove Corollary 5.3.3 we need only that this induces a well-defined map ¢ from the set of even liaison classes in co dimension c to the set of stable equivalence classes of locally free sheaves 9. This is guaranteed by applying Proposition 5.2.10 an even number of times, as was done at the beginning of the proof of Theorem 6.2.5. What is open (and probably not true) is that this map ¢ is both injective and surjective. It should be remarked, too, that in the non locally Cohen-Macaulay case, Nollet [167] and Nagel [163] showed the existence of a bijective correspondence between the set of even liaison classes in co dimension two and certain stable equivalence classes, but in place of locally free sheaves they required reflexive sheaves with more technical conditions. The two papers worked in slightly different contexts, and we do not quote the precise conditions here. Theorem 6.2.5 also implies that co dimension two arithmetically CohenMacaulay subschemes ofJpn comprise an entire liaison class (the licci class). Indeed, the arithmetically Cohen-Macaulay codimension two subschemes of Jpn are exactly those for which the sheaf 9 in (6.2) is free, and any two free sheaves are obviously stably equivalent. As a corollary, among zeroschemes in 1P'2 there is just one liaison class. D Thanks to the classification, due to Horrocks [102]' of stable equivalence classes of vector bundles 9 on IP'n with H; (9) = 0, we have a very nice restatement of Rao's result for the case of curves in p3. This result actually preceded Rao's main result described above (cf. [179]).
6.2. RAG'S RESULTS
163
Theorem 6.2.7 (Rao) Let C and C' be curves in ]p3 with deficiency modules M(C) and M(C'). Then C is evenly linked to C' if and only if M(C) is isomorphic to some shift of M(C') . That is, in the language of Theorem 6.2.5, the even liaison classes of (locally Cohen-Macaulay, equidimensional) curves in ]p3 are in bijective correspondence with the isomorphism classes of graded S-modules of finite length, after identifying those that differ only by shift. Hence this provides the converse to the Hartshorne-Schenzel theorem for curves in ]p3. Of course it then follows immediately from the Hartshorne-Schenzel theorem and its corollary (Theorem 5.3.1, Corollary 5.3.3) that C is oddly linked to C' if and only if M(C) is isomorphic to some shift of M(C't k. Rao also mentions in [180] that the Horrocks classification also yields a (somewhat more complicated) classification of the even liaison classes of surfaces in ]p4 in terms of graded modules:
Corollary 6.2.8 For graded S-modules M 1 , M2 of finite length, consider equivalence classes of triples (M1 ,M2, TJ), where TJ E Exfs(M(k , M~k) and (M1 , M 2, TJ) '" (N1 , N 2, () iff we have Ml !+ Nl(a) and M2 .4 N 2(a) isomorphisms for some integer a which maps TJ to (. Then the even liaison classes of surfaces in]p4 are in bijective correspondence with the equivalence classes of such triples. This shows that the converse to the Hartshorne-Schenzel theorem is false for surfaces in ]p4. Indeed, we need not only the preservation of the modules (Ml )(V) and (M2)(V) up to shift, but also an element of Ext1(M(k, M~k). Bolondi [27] has used this classification to give a construction of the minimal elements in any even liaison class of surfaces in ]p4, much as Martin-Deschamps and Perrin [137] did for curves in ]p3. It is also worth noting that under Liaison Addition (cf. Section 3.2) , the behavior of stable equivalence classes of vector bundles in the case of co dimension two subschemes of IF works as one would expect (and so we really do have an "addition" in the context of liaison, even in codimension two). Specifically, we have
Theorem 6.2.9 ([34]) Let VI and V2 be codimension two subschemes of IF, and choose Fl E IV2 and F2 E Iv! of degrees d1 and d2 respectively, such that Fl and F2 form a regular sequence, defining a complete intersection X. Let Z be the scheme defined by the saturated ideal Fl . Iv! + F2 . I v2 . Assume that we have locally free resolutions
° 91 -t
-t
;:1
-t
Iv!
-t
°
164
CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
o -t 92 -t F2 -t IV2
-t 0
where the 9i are locally free, the Fi are free and H!(9i) Then
= 0 for i = 1,2.
(1) Z has a locally free resolution
o -t 91(-dl)EB92(-d2)EBOlP'n(-dl-d2) -t F 1 (-d 1 )EBF2(-d 2) -t Iz -t O. (2) The vector bundle corresponding to Z is in the same stable equivalence class of vector bundles as the direct sum of the vector bundles corresponding to VI and V2 , suitably shifted. (3) There is an exact sequence
Proof: (1) and (3) can be proved easily using the methods of [78J described in Section 3.2 (and in fact an analogous statement can be obtained in higher codimension). We omit the details. It was originally proved in [34J in a different way. (2) follows immediately from (1). 0 Remark 6.2.10 Rao also remarks that his approach in [180J works in greater generality: one can replace IF by any complete, connected arithmetically Gorenstein variety P (cf. page 11), of dimension at least two, possessing a very ample line bundle.c. One then performs links using only complete intersections coming from powers of .c. That is, if we view P as embedded in IF, these complete intersections on P are the restrictions to P of codimension two complete intersections in IF which meet P in a subscheme of codimension two in P. For example, if P is a smooth quadric surface in ]p'3, then linking is done only with divisors of type (a, a). For instance, let Zl and Z2 each consist of one point on P. If the line A joining ZI and Z2 lies on Q, then ZI and Z2 are not directly linked even though their union is the intersection of a divisor of type (0, 1) with one of type (2, 0). On the other hand, if the line A joining Zl and Z2 is not on Q, then Zl and Z2 are directly linked by a complete intersection of two divisors of type (1,1). This is because ZI U Z2 is the intersection of P with A, which is a complete intersection in ]p'3. 0
In higher codimension there is no known analogue to Rao's converse of the Hartshorne-Schenzel theorem for curves in jp3, or to his restatement of the problem in terms of stable equivalence classes of locally free sheaves. Any such result would be a very welcome addition to the theory.
6.3. THE LAZARSFELD-RAO PROPERTY
6.3
165
The Lazarsfeld-Rao Property
As we mentioned on pages 107-108, there are several natural questions to answer about the equivalence relation of even liaison. The first two (connections between linked schemes and whether all schemes are linked) were answered in Chapter 5. The third and fourth are solved only in codimension two. The third, to parameterize the even liaison classes, was answered by Rao, and his work was described in Section 6.2. In this section we consider the fourth question, to describe the structure of an even liaison class. This structure has been called the Lazarsfeld-Rao Property. It was introduced in full generality, for complete intersection liaison in arbitrary codimension, in Definition 5.4.2. Some comments can be found in the last part of Section 5.4 about extending this notion to Gorenstein liaison. In this section we will focus on codimension two, where there is no distinction between complete intersections and arithmetically Gorenstein ideals. We first recall the set-up in codimension two. (See Section 5.4 for the set-up in arbitrary codimension.) We will then outline the proof of the structure theorem from [12J, and we will give full proofs of those parts where the details are not too involved. Recall, from Definition 5.4.1 and the discussion following it, that any non-arithmetically Cohen-Macaulay even liaison class .c may be partitioned .c = .cO u .c1u .. . u .ch... according to the shift of the deficiency modules (M 1 ), •.• , (Mn-2) . At the heart of the Lazarsfeld-Rao Property is the set of minimal elements of the even liaison class, i.e. the elements of .co. The rest of .c will be "built up" from these elements. The first step is Basic Double Linkage. We now record the definition and important properties in the case of codimension two (but see Remark 3.2.4 and page 138 for more details about these facts in the general case of arbitrary codimension). Let VI be a codimension two subscheme of ]pn , and choose F2 E IVI of degree d2. Let Fl be a form of degree d1 such that Fl and F2 form a regular sequence, defining a complete intersection X. Then the ideal Fl . I VI + (F2) is the saturated ideal of a scheme Z, with the following properties: (1) As sets, Z
= VI U X
(in particular co dim Z
= 2).
(3) Z is linked to VI in two steps. In particular, if VI E
.ch+dl.
.ch
then Z E
166
CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
(4) If VI has a locally free resolution m
o-+ 9 -+ EB OlP'n( -ai) -+ Iv!
-+ 0
i=1
where H; (9) = 0 (cf. Section 6.2), then I z has a locally free resolution m
0-+ 9( -dd EBOlP'n( -d 1 - d2 ) -+
EB OlP'n( -d
1-
ai) EBOlP'n( -d2 ) -+ I z -+
o.
i=1
The only fact which we have not already seen is (4). This can be proved either by applying the mapping cone procedure (Proposition 5.2.10) twice and canceling redundant terms (as was done in Remark 6.2.2), or by applying Theorem 6.2.9 and using the fact that Basic Double Linkage is really an application of Liaison Addition, taking one of the schemes to be trivial. Either (1) or (3) of Theorem 6.2.9 can then be used to give a proof. For instance, it follows by taking a mapping cone (see page 4) of the diagram
0
J.
9(-dd EB 0
0
J. O( -d1
o
-
J.
0
EB O( -d
1 -
ai) EB O( -d2 )
i =1
J.
-+ O( -d1
d2 ) -+
J.
m
-
d2 ) -+
IVl (-d 1 )
J.
EB O( -d2 ) -+ I z -+ 0
J.
0
where 0 = OlP'n. We remark that for curves in p3, the fact that Z is evenly linked to VI would follow from (2) by Rao's theorem (Theorem 6.2.7) . But in higher dimension or codimension we do not have these results, and in any case we have the further information that the linking can be done in two steps. Recall from Definition 5.4.2 that an even liaison class £ of codimensiontwo subschemes of IF has the LazarsJeld-Rao Property (or simply the LRProperty) if the following conditions hold: (a) If VI, V2 E £0 then there is a deformation from one to the other through subschemes all in £0; (in particular, all subschemes in the deformation are in the same even liaison class) .
6.3. THE LAZARSFELD-RAO PROPERTY
167
(b) Given Va E .co and V E .ch (h ~ 1), there exists a sequence of subschemes Va, VI , ·· ., Vi such that for all i, 1 :S i :S t, Vi is a basic double link of Vi-I and V is a deformation of Vi through subschemes all in .ch . The main theorem which we will describe in this section is the following:
Theorem 6.3.1 ([12]) Every even liaison class oj non-arithmetically Cohen-Macaulay codimension two subschemes oj Rao Property.
rn
has the LazarsJeld-
Remark 6.3.2 In this remark we give the history of work on this problem. The LR-Property was first proved for special even liaison classes of curves in jp3 by Lazarsfeld and Rao [128] . Their main theorem was that if C is a curve in jp3 with index of speciality e = e( C) = max{ t I hI (Oc(t)) =1= O}, and if C lies on no surface of degree e + 3, then C is minimal in its even liaison class, and this even liaison class has the property described above. Furthermore, if C lies on no surface of degree e + 4 then .cO = {C}; that is, C is the only curve in .ca. This paper inspired the main structure theorem, both in [12] and in [137]. See also Remark 6.3.10. Lazarsfeld and Rao's paper was motivated by the conjecture of Harris mentioned on page 102. That is, the general curve of fixed degree and genus cannot be "linked down" to another curve of smaller degree and/or genus. Indeed, they recall that when d ~ 2g - 1 the family of all smooth irreducible curves in ]p3 of degree d and genus 9 is irreducible. (Note also that if a smooth curve X of degree d and genus 9 satisfies d ~ 2g - 1 then e(X) :S 0.) Then they show that for a fixed integer 9 ~ 0 there exists a constant C(g) 2: 2g -1 such that a sufficiently general curve X of genus 9 and degree d 2: C (g) lies on no surface of degree V5d or less. Hence their structure theorem applies to the even liaison class of X. The next case proved was in [33], where it was shown that the even liaison class of every arithmetically Buchsbaum curve (see Definition 1.5.7) in jp3 has the LR-Property. If the deficiency module M( C) of an arithmetically Buchsbaum curve C has diameter one, then the minimal curves all fail to lie on surfaces of degree e(C)+3 (cf. [31]), so the theorem of [128] applies. In fact, the LR-Property was important in obtaining the results in [31]. But if the diameter is greater than one, then it was shown in [32] that no curve in the liaison class satisfies the hypothesis of Lazarsfeld and Rao. Hence [33] gave a large class of curves not covered by [128].
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CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
The proof of the main result of [33) for Buchsbaum curves used the paper of Bolondi [28], which provided the approach to get the desired deformations. (This paper in turn was based on the original paper of Lazarsfeld and Rao.) It was inspired by the papers [1), [31), [32], [75) and [76), which provided technical tools as well as very compelling evidence that the LR-Property should also hold at least for Buchsbaum curves. The paper [34) suggested a general approach to the problem and gave several useful technical results. It also proved the first case of the LRProperty in co dimension two in any projective space, but it was a very limited case. The main structure theorem for codimension two subschemes of F (Theorem 6.3.1 above) was proved in [12J. The proof will be described below. A similar proof was subsequently given in [36J for even liaison classes of codimension two sub schemes of a smooth arithmetically Gorenstein variety in F. (See also page 164.) The main structure theorem was given independently for curves in p3 by Martin-Deschamps and Perrin [137). Although their results hold only for curves in p3, they give a much more detailed description of the structure and the connections between the curves and the deficiency module (such as an improvement of Theorem 1.2.7 of Rao, and an analysis of the connections with the locally free sheaves mentioned in Section 6.2). They also show how to construct the minimal elements of the even liaison class. A similar analysis was subsequently done by Bolondi [27) for surfaces in
F.
As has been noted already, the main structure theorem of [12) and [137] assumes that the schemes studied are, in particular, locally CohenMacaulay and equidimensional. This assumption has also been made on several other occasions in this book, and in most work on liaison theory to date. The equidimensional assumption cannot be removed; however, the machinery and main results of liaison theory have been extended to avoid the locally Cohen-Macaulay assumption (but with a good deal of extra work) in the papers of Nagel [163J and Nollet [167]. In particular, both papers show that the Lazarsfeld-Rao Property holds true without this assumption. The proof of the Lazarsfeld-Rao Property given by Nagel in [163J more closely follows that of [12), giving fresh ideas and approaches at the places where "locally free" needs to be replaced by the weaker "reflexive." We do not elaborate here since a good deal of new machinery is required to make it work, but in any case a comparison of [12J and [163J is very useful in seeing what is involved in passing to the non locally CohenMacaulay case. Similarly, the approach of Nollet more closely follows
169
6.3. THE LAZARSFELD-RAO PROPERTY
that of Martin-Deschamps and Perrin [137] and a comparison of these two works is very useful. It should also be noted that in a more algebraic context, many results in this direction have been obtained by Huneke and Ulrich. See especially [104], Section 6. 0 There are two main technical tools used in the proof of Theorem 6.3.1 found in [12]. The first tool is a sufficient condition for the existence of a deformation of the desired form. Let 12 be an even liaison class of locally Cohen-Macaulay, codimension two subschemes of F. It turns out [34] that if two elements of 12 occur in the same shift C h and have the same Hilbert function, then such a deformation exists. This result (essentially) was known for aCM subschemes of codimension two in F [68], and proved for non-aCM curves in p3 by Bolondi [28], inspired by the paper [128]. The proof in [34] followed the same idea, and we will now describe it .
Proposition 6.3.3 ([34]) Let 12 be an even liaison class of locally CohenMacaulay, codimension two subschemes of IP'n , and let VI, V2 E Ch . Assume that VI and V2 have the same Hilbert function (i.e. hO(F , IVl (t)) = hO(F,Iv2 (t)) for all t) . Then (a) it also holds that hn-I(F,Ivl (t)) = h n- I (F,Iv2 (t)) for all t; (b) there exists an irreducible fiat family {Vs}sES of codimension two subschemes of F to which both VI and V2 belong; and (c) 5 can be chosen so that for all s E 5, Vs E C h, and so that furthermore Vs has the same Hilbert function as that of VI and V2 . Proof: Consider a locally free resolution
o-t g -'-t EB OJP'n( -ai) -t IVl f
m
-t
0
(6.5)
i=I
where H;(Y) = 0 (cf. Section 6.2). Since V2 is evenly linked to VI, a repeated use (an even number of times!) of the mapping cone procedure (cf. Proposition 5.2.10 and also page 158) gives that V2 has a locally free resolution o -t g (d) E& A ~ B -t IV2 -t 0 (6.6) where A and B are direct sums of line bundles. Since Vi and V2 are in the same shift, it follows by taking cohomology that d = O.
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CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
By adding a trivial addendum A to the first two terms in the locally free resolution (6.5), we get
0-7 9 ~ A
m
EB QJPn( -ai) ~ A -7 IVI
/$IA)
-7
o.
(6.7)
i=1
We rewrite (6.6) and (6.7) as
o-7 £
r
~
EB QJPn( -hi) -7 IV2 -7 0, i=1 r
0-7£ ~
EB OJPn( -kd -7 IVI -70, i=1
where £ = 9~A (and hence H;(£) = 0) and r = rk£+l. Because VI and V2 are assumed to have the same Hilbert function, a calculation (taking cohomology) shows that hi = ki for all i. Hence we have u, v E Hom (£, $ OJPn( -hi)) . Now, following [128J, we can produce the desired deformation. Given s E k (the base field), let
For general s E k, the cokernel of Ws is the ideal sheaf Ivs of a codimension two subscheme of IF, and these subschemes fit together in an irreducible flat family (see also [128J and [113]). By Theorem 6.2 .5, Vs is in the same even liaison class as VI and l!2. By taking cohomology one can check that they are all in the same shift and that they all have the same Hilbert function. 0 The second technical tool, in some sense the heart of the proof of Theorem 6.3.1, is the following key lemma.
Lemma 6.3.4 Let £ be a rank (r + 1) vector bundle on IF and let
$i=1 QJPn( -ai) -7 £
al :::; a2 :::; ... :::; ar , b1 :::; b2 :::; ... :::; br , be morphisms whose degeneracy loci have codimension two. Then there exists a morphism r
( :EB QJPn( -Ci) -7 £ i=1
with
Ci = mini ai,
bJ Vi,
whose degeneracy locus has codimension two.
171
6.3. THE LAZARSFELD-RAO PROPERTY
The reason that this lemma is central to the proof of the Lazarsfeld-Rao Property is the following (but see also Remark 6.3.8). It is rather clear that the main step in proving the LR-Property is to find the minimal elements and their relationship to the other elements in the even liaison class. We will see shortly that if V and V' are in the same even liaison class £ then we can find exact sequences r
o --t EB OlP'n( -ai) --t £, --t Iv --t 0 i=l
and
r
o --t EB OlP'n( -bi) --t £, --t IVI(o)
--t 0
i=l
where £, is locally free and L: bi - L: ai = O. The condition that V and V' occur in the same shift £h is equivalent to saying that L: bi = L: ai. The power of this lemma is to say that if h = 0 (i.e. V and V' are minimal elements) then in fact bi = ai for all i. The proof of the lemma is rather involved. Nagel [163] gives the generalization to the non locally Cohen-Macaulay case by replacing "rank (r + 1) vector bundle" by "reflexive module of rank (r + 1) which is locally free in codimension 1." The idea, either in [12] or [163], is to prove the following claim and apply it repeatedly until the lemma is proved. (The proofs of the claim in [12] and [163] are different.)
Claim: Assume that al ~ .. . ~ ar and bl ~ .. . ~ br . Assume further that ai = bi for i ~ k (with 0 ~ k < r) and ak+l < bk + 1 . Then there exists a morphism r
T :
EB OlP'n( -Ci) --t £' , i=l
with
Ci
= ai if i
~
k + 1 and
Ci
= bi if i
> k + 1.
We refer the reader to [12] and [163] for details. A useful analogy is the following. Let I be an ideal defining a subscheme of projective space. Assume that there is a regular sequence in I offorms of degrees aI, a2, ... , ar and a regular sequence in I offorms of degrees bl , b2 , ... , br . Let Ci = min {ai, b;} for 1 ~ i ~ r. Then there exists a regular sequence in I of forms of degrees Cl, C2,···, Cr·
With these two tools, there are essentially three steps in the proof of Theorem 6.3.1:
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CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
(1) Take care of the minimal elements (i.e. prove condition (a) of the LR-Property); (2) Relate a locally free resolution for a minimal element Vo E £,0 to one for an arbitrary element V E £,\ and (3) Find the right sequence of basic double links to complete condition (b) of the LR-Property. The first step is accomplished by the following: Proposition 6.3.5 If VI, V2 E £0 then there is an irreducible fiat family {Vs}sES of codimension two subschemes of IF to which both VI and V2 belong. Proof: By Proposition 6.3.3 it is enough to show that VI and V2 have the same Hilbert function. Consider a locally free resolution
o -+ P -+ F
(6.8)
-+ IVI -+ 0
with P a direct sum of line bundles and F a vector bundle with H:- I (F) = o (see (6.3) of Section 6.2). Since V2 is evenly linked to VI, by repeatedly using the mapping cone procedure (Proposition 5.2.10), one obtains a locally free resolution
o -+ P EB T3 -+ F EB A
-+
IV2
(6.9)
-+ 0
where A and T3 are direct sums of line bundles. On the other hand, we can trivially add A to sequence (6.8) to get
o -+ P EB A -+ FEB A -+ IV2 -+ o.
(6.10)
Clearly we will be done if we can show that P EB A ~ P EB T3 (just take cohomology on (6.9) and (6.10)). Suppose we write PEBA = $~=I Grn( -ai) and PEBT3 = $~=I Qrn( -bi ). If ai =I bi for some i then by Lemma 6.3.4 there exists a morphism r
EB Qrn( -Ci) -+ FEB A i=1
with Ci = min{ ai, bi } for all i, whose degeneracy locus Z has codimension two. Hence its ideal sheaf has a locally free resolution 0-+
r
EB Gr i=1
n (
-Ci) -+ F $ A -+ Iz(b) -+
o.
6.3. THE LAZARSFELD-RAO PROPERTY
173
By considering Chern classes, one can check that 2:r=I ai = 2:r=I bi , and that hence 8 = 2:r=I (Ci - ai) < O. But this means that the deficiency modules of Z are shifted to the left of those of VI, contradicting the hypothesis that VI E CO. 0 A similar proof takes care of step (2) of the proof. Specifically, one proves
Proposition 6.3.6 Let
Va
E CO have the locally free resolution
o -t P
-t :F -t Ivo -t 0
(with P and:F as above) and let V E Ch have the locally free resolution
o -t P (J) B -t :F (J) A
-t Iv(h) -t 0
(with A and B direct sums of line bundles). If P (J) A = E9r=I GJPn( -ai) and P (J) B = E9r=I GJPn( -bi ) with al ::; ... ::; ar and bi ::; ... ::; brl then bi ~ ai for all i. The proof is a straightforward application of Lemma 6.3.4, and is very similar to the proof of Proposition 6.3.5. It remains only to complete step (3) of the proof. Again we omit the details, but the proof is similar to that given originally by Lazarsfeld and Rao in [128] and we will give the basic idea. We consider locally free resolutions of the type we have been using above. If Vo E Co and V E C h then we have the two resolutions mentioned in Proposition 6.3.6, with bi 2: ai for all i. We have seen that whenever we get a short exact sequence of this form, with the first two terms of the sequence the same, it follows that the corresponding schemes are in the same shift (8 = 0) and have the same Hilbert function, and so there is the desired deformation. So we have to focus on those i for which ai < bi , and perform basic double links to produce new schemes for which "fewer" of these degrees are different. The idea of the proof is to start with the maximum of such i (essentially) and perform a basic double link starting from Vo , using d l = bi - ai and d2 = bi , to produce a scheme Vi, (See the set-up for basic double links on page 165. VI here plays the role of Z in that set-up.) This is not quite correct, because of a technicality, but it gives the right idea. One then uses fact (4) in the description of basic double links, on page 166, to produce the appropriate locally free resolution for VI. One compares this to the locally free resolution of the scheme V (suitably altered), and finds that
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CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
"fewer" of the degrees differ. So after a finite number of steps the degrees are all the same, and we are done. This completes our description of the proof of Theorem 6.3.1. We conclude this section with a number of remarks related to the LazarsfeldRao Property and its proof. Remark 6.3.7 Notice that it is possible to construct (with, say, the computer program Macaulay [15]) a minimal element in the even liaison class of a given scheme V . From V one obtains the vector bundle F, and then one finds the "smallest" set of rk F - 1 sections of F (chosen generally) which drop rank in codimension two. As in Lemma 6.3.4 and the discussion after it, this will be the minimal element. C. Peterson has implemented this idea for curves in jp3. A more careful algorithm can be found in [84] . 0 Remark 6.3.8 An important idea lurking in the background in this section, and made explicit in the approach of Nollet [167]' is that of domination. (See also [137] Chapter V for a closely related notion.) Let C be an even liaison class of co dimension two subschemes of IF and let Vl , 112 E C. For the sake of this remark we will not assume locally Cohen-Macaulay, since Nagel [163] and Nollet [167] have shown that this assumption is not necessary. We say that V2 dominates Vl if V2 can be obtained from V l by first performing a sequence of basic double links (if necessary) and then deforming within the even liaison class, preserving the cohomology, as we have seen in this section. Nollet shows that domination is transitive and hence defines a partial ordering on an even liaison class. In this language, Lemma 6.3.4, together with some extra work (namely the ideas in the proof of Theorem 6.3.1 described above), says the following. Under the hypotheses given, if Vl and V2 are the degeneracy loci of the given morphisms ¢ and 'IjJ, then there is an element of the even liaison class, namely the degeneracy locus of (, which is dominated by both VI and V2 • The Lazarsfeld-Rao Property simply says that in any co dimension two even liaison class C, there is an element that is dominated by every element in the class, i.e. C contains a minimal element with respect to domination. This follows easily from the above interpretation of Lemma 6.3.4, since we already know that there is a minimal shift (Definition 5.4.1). Domination allows for the study of interesting subsets of an even liaison class. For example, consider curves in C c jp3 and recall the notation ra(C) and ro(C) (page 15) for the degrees in which M(C) begins and ends, respectively. In [136], curves C were studied having equal cohomology,
6.3. THE LAZARSFELD-RAO PROPERTY
175
i.e. curves for which e(C) = ro(C) and h1(Ic(t)) = h2 (Ic(t)) for t = ra(C), .. . ,ro(C) . The homogeneous ideals of these curves have interesting properties (they have generators of large degree). What was amusing was that the set of all such curves in a given even liaison class satisfies a Lazarsfeld-Rao Property, namely it has a minimal element with respect to domination. Similarly, a result of Nollet ([169] Corollary 2.6, Remark 2.7) describes another subset of an even liaison class with this kind of Lazarsfeld-Rao Property. For any scheme X c IP'n, recall that a(X) denotes the least degree of a hypersurface containing X (page 29 and Definition 2.2.1). Let
Sl(X) = min{t 2': a(X)
I (Ix):s,t
is not principal}
t1(X) = min{t 2': a(X) I V((Ix)t) contains no hypersurfaces} For example, let X C jp>3 consist of the union of a general plane curve of degree 5 and a general plane curve of degree 10 (on a different plane). Then a(X) = 2, Sl(X) = 6 (= 5 + 1) and t1(X) = 11. (This can be checked on the computer, of course, but it is amusing to convince yourself of this from scratch, especially the last one. Hint: the component of Ix in degree 11 defines a curve of degree 16.) Then Nollet shows that in any even liaison class C, if we denote by M the set of elements X E C for which Sl(X) = t1(X), then M has a minimal element with respect to domination. 0
Remark 6.3.9 We would like to make a remark about the paper [36]. The main contribution of this paper was to show how the theory of codimension two liaison described in this chapter goes over very nicely when F is replaced by a smooth, arithmetically Gorenstein variety X, following a remark by Rao in [180] (see Remark 6.2.10). It amounts to saying that the theory works equally well if we work over a Gorenstein ring instead of a polynomial ring, but it is given a geometric interpretation. More precisely, for X smooth it was shown in [180] that the even liaison classes of codimension two subschemes of X (with the usual assumptions of locally Cohen-Macaulay and equidimensional) are in bijective correspondence with the stable equivalence classes of vector bundles, just as it was in F. It was then shown in [36] that the non-licci codimension two even liaison classes on X also satisfy the Lazarsfeld-Rao Property. This includes classes that are aCM but not licci, as described below. The licci class was also studied separately. In both cases, this was generalized in [163] to the case where X is not necessarily smooth and, as mentioned, the subschemes are not necessarily locally Cohen-Macaulay.
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CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
Notice that in this setting, codimension two aCM subschemes are no longer necessarily licci, in sharp contrast to Corollary 6.1.4. For example, consider four general points, Z , in p3 lying on a smooth quadric surface Q. Z has co dimension two on Q and is aCM, but if it were licci on Q then it would be licci in p3 since Q is itself a complete intersection. But we saw in Example 6.1.6 that this is not the case. Nevertheless, it was shown in [36] how to define the minimal elements even in an aCM, non-licci, even liaison class. This is done by looking at a locally free resolution of the form
o -+ F
-+ N -+ IVlx -+ 0
where F is free and N is locally free as Ox-modules, and H;(X,N) = o. One then considers which twists of N can actually appear in the stable equivalence class of N corresponding to elements of the even liaison class, and shows that this is bounded above. From this, one can prove the Lazarsfeld-Rao Property also for these classes. 0 Remark 6.3.10 As mentioned on page 167, the original result of Lazarsfeld and Rao was essentially that if C c p3 is a curve not lying on any surface of degree e( C) + 3, then the even liaison class of C satisfies the LR-Property, and C is a minimal element. If, furthermore, C lies on no surface of degree e( C) + 4, then C is the unique minimal element. We would like to mention two generalizations of this result. First, Popescu and Ranestad [177] observed that the following more general version of this result is true with the same proof. Let Z be a codimension two, locally Cohen-Macaulay subscheme of pn, and define the index of speciality of Z as e(Z) = max { t I hn - 2 (Oz(t)) -=I 0 } .
(a) If hO(Iz(e(Z) liaison class;
+ n))
= 0 then
(b) If HO(Iz(e(Z) + n + 1)) even liaison class.
Z is a minimal element in its even
= 0 then
Z is the unique element in its
The second generalization is due to Nollet [169]. Let [, be an even liaison class of codimension two subschemes of pn . Let X o E [,0 and let e = e(Xo). Let X E [,h. If hO(Ix(e + n + 1 + h)) = 0, then X o is the unique minimal element of [" and furthermore X o eX . We illustrate Nollet's result with a simple example. Let X o be a set of 4 skew lines in p3 not lying on a quadric surface and let [, be the corresponding even liaison class. Let X be any curve obtained from X o by a basic double link (see page 165) using d1 = 1 and d2 ~ 4. We t hen
6.4. APPLICATIONS
177
have e = -2, h = d1 = 1 and hO (Ix (e + n + 1 + h)) = hO(Ix(3)) = o. Of course X contains X o, by construction. The interesting fact here is that any curve in 1: with the same cohomology as X in fact must also have X o as a subscheme. For instance, if we link X o to a curve Y using any two surfaces of degree 4 (say), and then link Y to a curve X' using surfaces of degrees 4 and 5, then X' must contain X o as a subscheme. 0
6.4
Applications
In this section we will describe a number of ways in which liaison theory has been applied in the literature. The topics chosen here are by no means close to being a complete list. Furthermore, because of space limitations we will obviously not be able to give more than an overview of the techniques and ideas involved in these topics. We just want to give a flavor of some of the ways in which liaison theory can and has been used. The first topic deals with trying to find curves in JP3 having certain nice properties related to the Hilbert function of the general hyperplane section. Liaison is used to find these curves. We will describe work from [86] and [133]. The second topic passes to surfaces in jp4 and three folds in p5. We would like to use liaison to find smooth examples, working toward a classification theorem. We describe work of several authors. The next application describes some consequences of the LazarsfeldRao Property, giving some ways in which properties of the whole even liaison class (say of a curve in JP3) can be given just from knowledge of a minimal element. In particular, we will show how in principle we can give a complete list of all possible pairs (d, g) (degrees and genera) of Buchsbaum curves in JP3. We then show how the Lazarsfeld-Rao Property can give information about how codimension two subschemes of]pn can specialize to "nice" subschemes. In particular, we describe the proof from [35] that every Buchsbaum curve in JP3 specializes to a stick figure. This generalizes the corresponding fact for arithmetically Cohen-Macaulay curves, due to Gaeta. To do this, we will also have to use some of the results and ideas from Chapter 3. Finally, we give some connections between low rank vector bundles on ]pn and subschemes of ]pn defined by a small number of equations. Our discussion is based on Chapter 3 of [175].
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CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
6.4.1
Smooth Curves in p3
We would first like to describe two problems concerning curves in jp3. These problems both turn out to involve the Hilbert function in a key way, and they both use liaison to find the desired curves. Furthermore, in both cases although the problem is given in general, for the solution it turns out to be enough to consider arithmetically Cohen-Macaulay curves. The first problem is to find a bound for the (arithmetic) genus of a non-degenerate, integral curve lying on an irreducible surface of given degree k. This was solved in [86]. The second problem is to describe all possible Hilbert functions for the general hyperplane section of an integral curve. This was solved in [133]. (See [83] and [186] for similar results, and [77] for a discussion of the relations between these approaches.) Proposition 1.4.2 gave a formula for the arithmetic genus of an aCM curve. We now show how to extend this to the non-aCM situation. Let C C ]pm be any curve of degree d and arithmetic genus g. Let L be a general linear form and let r be the zeroscheme cut out by the corresponding hyperplane. We write H(r, t) for the Hilbert function of r in HL = ]pm-I. Let KL be the submodule of the deficiency module M(C) annihilated by L (see Definition 2.1.1). Let I!» o. Then l
9=
L[d - H(r, i)] - dim KL
(6.11)
;=1
The argument is almost identical to the proof of Proposition 1.4.2; use the fact that now b,.H(C, t + 1) = dim(Kdt + H(r, t + 1). Harris [86] shows that this is equal to L~=o[(i - l)b,.H(r, i)] + 1 - dim K L . Remark 6.4.1 The "decreasing type" description of the possible Hilbert functions for r given in Remark 1.4.6 is central to both problems that we will discuss in this section. We saw in Remark 1.4.6 that any zeroscheme, Z, in ]p2 will have a first difference function that is non-increasing in the range t ~ d2 , (where d2 is the degree of the second generator of the ideal of Z), and that if Z has the Uniform Position Property (U.P.P. - cf. Definition 1.4.5) then it must in fact be strictly decreasing in this range until it reaches zero. 0 Suppose that we are interested in finding an upper bound for the genus of C , in terms of its degree d and the fact that C lies on an irreducible surface of degree k. From (6.11) above, 9 :::; L~=I[d-H(r, i)], with equality if and only if C is arithmetically Cohen-Macaulay.
179
6.4. APPLICATIONS
Now, we would like to determine the largest possible value for the righthand side of the inequality, given the degree d. Clearly the idea is to have the growth of H(f, i) be as slow as possible, subject to the requirement that it have decreasing type. With no condition on the surfaces on which C lies, this happens when the first difference function (I.e. the h-vector) is 0, ift~-l; 1, if t = 0; tlH(f, t) = 2, ifl ~ t ~ r~l 1
1 8,
0,
°
= r~l If t > r21 ~ft
-
where 8 = or 1, depending on whether d is odd or even, respectively. Castelnuovo's bound ([45] or [88] pp. 351-352) follows from this:
<{ g-
(f - 1f ' ( d; 1) (d;3) ,
if d is even; if d is odd
and this bound is achieved if and only if C is arithmetically CohenMacaulay and lies on a quadric surface. Harris' idea was that Castelnuovo's bound above, and the extremal curves obtained, really come about by liaison. The curves are either the complete intersection of a quadric and a surface of degree ~ (if d is even), or else residual to a line in the complete intersection of a quadric and a surface of degree d!l (if d is odd). In either case this gives both the bound and produces the sharp examples. Now we impose the condition that C lies on an irreducible surface of degree k. Harris analyzed the possible Hilbert functions that could arise, as above, once he realized that they had to be of decreasing type. He broke the argument into two cases, depending on whether d > k(k - 1) or d ~ k(k - 1). His conclusion in either case was that the greatest genus of a curve lying on an irreducible surface S of degree k is that of a curve residual to a plane curve. However, the particular complete intersection required for the link depends on whether d > k(k - 1) or d ~ k(k - 1). (Only in the former case is S necessarily one of the surfaces involved in the complete intersection.) Furthermore, not only can some extremal curves be obtained in this way by liaison, but in fact any extremal curve must be residual to a plane curve. Specifically, Harris proves the following. Let 2 7r(d l) = -d , 2l
1 + -d(l 2
4)
+1-
-to ( l 2
to -
1 + -to) l
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CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
where 0 ::; f ::; l - 1, - f = d(mod l). Then the genus 9 of an irreducible curve C lying on an irreducible surface S of degree k satisfies
g::;
{
7r(d, k), 7r(d,ld klJ+l),
if d > k(k - 1); ifd::;k(k-l).
Harris' work, then, described "extremal" kinds of Hilbert functions of decreasing type, showed that they must actually occur as the Hilbert function of the general hyperplane section of an integral, arithmetically Cohen-Macaulay curve, and showed that these curves are extremal with respect to his bound on the genus. But what about the other Hilbert functions of decreasing type? In [87], Harris and Eisenbud ask what may be the Hilbert function of a set of points in r- 1 with V.P.P., and whether every such function actually occurs as the Hilbert function of the general hyperplane section of some integral curve in r. In the case r = 3, we describe the answer given in [133]. As mentioned in Remark 1.4.6, it follows from work of Davis [54] that if a zeroscheme in Jll'2 has Hilbert function which is not of decreasing type, then it cannot have V.P.P. In [133], Maggioni and Ragusa show that any Hilbert function of decreasing type occurs as the Hilbert function of the general hyperplane section of some smooth, arithmetically Cohen-Macaulay curve in]p3. Since such a hyperplane section automatically has V.P.P., it follows immediately that both questions of Harris and Eisenbud have been answered for r = 3. (In the case r :::: 4, neither question has yet been answered, although some progress has been made. See for instance [86], [165], [184], [22].) An important fact used by Maggioni and Ragusa is that if Z is a set of points in Jll'2 with V.P.P., then Z lies in a complete intersection of type (d 1 , d2 ) (cf. (10) of Remark 1.4.6). They also use Corollary 5.2.19, which says (in our current situation) how the Hilbert function of a set of points in Jll'2 behaves under liaison. See for instance Example 5.2.20. Now, the idea of Maggioni and Ragusa goes as follows. Start with a Hilbert function of decreasing type. We want to produce a smooth arithmetically Cohen-Macaulay curve C in ]p3 whose general hyperplane section has the given Hilbert function. We reason backwards. If such a smooth curve C were to exist, its general hyperplane section Z would have V.P.P. Hence d1 and d2 can be read from the Hilbert function, and a complete intersection of type (d 1 , d2 ) would link Z to another set of points Z'. The Hilbert function of Z' would be given by the theorem of [56] above. Let us call this new Hilbert function H' (which formally depends only on the given Hilbert function). The complete intersection linking Z to Z' would lift to one containing C, and the residual curve C' would have
6.4. APPLICATIONS
181
a general hyperplane section with Hilbert function H'. (Z' may not be sufficiently "general," but in any case the Hilbert function does not vary with the hyperplane for arithmetically Cohen-Macaulay curves - see for instance Proposition 3.1.1.) Now, Maggioni and Ragusa produce a stick figure G', using a simple construction, whose general hyperplane section has Hilbert function H'. They then show that G' lies on a smooth surface S of degree d1 (where d1 still refers to the Hilbert function of G, not G'). Finally, they consider the linear system Id2 H - G'I on S and show that the general element G of this linear system is smooth. But G is thus directly linked to G' by a complete intersection of type (d 1 , d2 ), and so we have produced the desired curve. Remark 6.4.2 As these two applications show, it is of interest to have results that tell us when we can use liaison to produce smooth curves (or higher dimensional varieties). A very general result (holding for any dimension) is the theorem of Peskine and Szpiro mentioned below (Theorem 6.4.4). For the case of curves in jp3 there is a result of Nollet [168] which shows how one can produce smooth curves starting from "very" non-reduced curves. Specifically, let Y c jp3 be a smooth curve with homogeneous ideally, and let W be the curve whose homogeneous ideal is the saturation of It, for some integer d > 1. Let n be an integer such that Iw(n) is generated by global sections. If m > n then for a general pair of surfaces F and G of degree m containing W, we have
FnG
= WuG
where W c W, Supp(W) = Y and G is a smooth, irreducible curve. In other words, W itself is linked to the union of a smooth curve C and a curve supported on Y, so it is easy to recover the smooth curve G. Nollet also gives formulas for the degrees and arithmetic genera of G and W. Furthermore, he shows that if m » 0 then (F, G) is actually the lowest degree complete intersection containing G. (See also Example 6.4.5.) 0 Remark 6.4.3 The problem addressed in this subsection is a special case of the more general problem of finding all the possible pairs (d, g) of degree and genus that can occur for smooth connected curves in jp3. See [88] IV.6 for an elementary discussion of this problem. Even more difficult is finding all possible triples (d,g,s) such there exists a smooth connected curve of degree d and genus 9 not lying on a surface of degree less than s. This problem goes back to Halphen [85]. See [ll] for a good discussion of the history of this problem, and an approach using stick figures and smoothabili ty.
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CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
The traditional approach to this problem has been to divide the possible degrees into different ranges and in each range give a maximum and minimum genus, and see if curves exist for all genera in between. A value for which there is no curve is called a Halphen gap. Some parts of the problem are still open. We would like to mention the paper [119], which uses liaison to construct "good" (i.e. unobstructed of maximal corank) curves near the maximal genus in range C and the "upper part" of range B, and thus showing that there are no Halphen gaps there. This is similar to the approach of Harris. D
6.4.2
Smooth Surfaces in
r4 and Threefolds in
lP'5
A (special case of a) conjecture of Hartshorne [89] states that any smooth co dimension two subvariety of]pn, for n ~ 6, must be a complete intersection. Smooth curves in ]p'3 are rather well known (for instance, it is known what pairs (d, g) of degrees and genera can occur). Hence it is of interest to study surfaces in JP4 and threefolds in lP'5, to see what smooth varieties exist. An excellent reference for this subject is the paper [57]. The main problem in this area is to classify surfaces in JP4 or threefolds in F, including existence and uniqueness results. According to [21] (page 325), "Adjunction Theory is a major tool to get maximal lists of all possible cases. Liaison is the major tool used to get existence results." The classification is generally done according to the degree. The idea, then, is to start with known varieties, and link to get new (smooth) varieties. The main fact which is used to produce these varieties is a theorem of Peskine and Szpiro ([174] Proposition 4.1) which gives a necessary condition for the residual variety to be smooth. (More precisely, it says that the singular locus of the residual is "small.") We give a special case of this theorem, quoted from [57] (Theorem 2.1), which deals with the situation in which we are interested:
c ]pn, n ~ 5, be a local complete intersection of codimension two. Let m be a twist such that Ix(m) is globally generated. Thenforeverypaird l ,d2 ~ m there existformsFi E HO(Ix(d i )), i = 1,2, such that the corresponding hypersurfaces VI and V2 intersect properly and link X to a variety X'. Furthermore, X' is a local complete intersection with no component in common with X, and X' is nonsingular outside a set of positive codimension in Sing X.
Theorem 6.4.4 Let X
Example 6.4.5 Note in particular that if X is smooth, or if X is a local complete intersection with a zero-dimensional singular locus, then X' is
6.4. APPLICATIONS
183
smooth and has no component in common with X. These conclusions cannot be expected to hold in general without hypotheses of this sort, even for curves in ]p'3. For example, consider the ideal (Xl, X2)3. By Example 6.1.5, this is a saturated ideal defining a scheme X of degree 6 supported on a line, A. Then we claim that any complete intersection containing X will link X to a residual X' which is non-reduced and has a common component with X, namely the line A. Indeed, any element of Ix vanishes to order at least three on the line A, so any complete intersection will vanish to order at least 9 on A, and thus X' has a component of degree at least 3 supported on A. D Example 6.4.6 As an example of Theorem 6.4.4 in action, we mention the paper of Miro-Roig [157]. The object of [157] is to complete the list of the degrees for which there exist smooth threefolds in F which are not arithmetically Cohen-Macaulay. (The Hilbert-Burch matrices, hence the degrees, of smooth arithmetically Cohen-Macaulay codimension two subvarieties of F are known - for instance cf. [49] and [36].) It was shown by Banica [14] that for any odd integer d 2 7 and any even integer d = 2k > 8 with k = 58 + 1,58 + 2,58 + 3 or 58 + 4 there exist smooth threefolds in F of degree d which are not arithmetically Cohen-Macaulay. (In smaller degrees they are all arithmetically Cohen-Macaulay.) Hence it remained to consider multiples of 10. It was shown by Beltrametti, Schneider and Sommese [19] that any smooth threefold of degree 10 is arithmetically Cohen-Macaulay. Miro-Roig's idea was to use liaison (and Theorem 6.4.4 in particular) to produce smooth, non-arithmetically Cohen-Macaulay examples for degrees d = lOn, n > 1. Her starting point is the smooth non-arithmetically Cohen-Macaulay threefold Y C F of degree 12 having a locally free resolution
0-+ 0 EB 0(1)3 -+ 0(3) -+ Iy(6) -+ 0 where 0 is the cotangent bundle of F (cf. [14]). By studying the cohomology of 0, and using Theorem 1.1.5, one obtains that Iy(6) is globally generated. Hence Theorem 6.4.4 gives that a general complete intersection of type (6,7) yields a residual X which is smooth. Note that X has degree 30, is not arithmetically Cohen-Macaulay (since the property of being arithmetically Cohen-Macaulay is preserved under liaison), and one can check that Ix(8) is globally generated. From X, Miro-Roig produces her smooth threefolds of degree lOn, n 2 5, by linking with general hypersurfaces of degree 10 and n+3. For degrees 20 and 40 she uses the same idea, but starts with different Y. We omit the details. D
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CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
The classification of smooth surfaces in ]p4 and threefolds in jp'5 is not yet complete. A very nice table summarizing the state of affairs to date is given in §7 of [57]. It covers surfaces in ]p4 up to degree 15 and threefolds in F up to degree 18. That paper, together with the book [21], give most of the necessary background , as well as a very extensive set of references, from which the reader can learn more about this fascinating subject , using many different approaches (in addition to liaison and adjunction theory). Another useful source is the paper [177].
6.4.3
Possible Degrees and Genera in a Co dimension Two Even Liaison Class
In this section and in Section 6.4.4, we describe some consequences of the Lazarsfeld-Rao Property (Theorem 6.3.1). In this section we look at some simple consequences, and in Section 6.4.4 we consider a more surprising consequence. Let C be an even liaison class of co dimension two subschemes of r. The Lazarsfeld-Rao Property says, in effect, that up to deformation, all of C is described once you know all possible basic double links that can be performed. (See the discussion beginning on page 165. We will use the notation from that discussion without comment .) And essentially, all you need to know to do this is the Hilbert function of the minimal elements. We will give some idea of the extent to which this can be done. As a first observation, notice that all minimal elements of C (i.e. all elements of CO) have the same Hilbert function (see the proof of Proposition 6.3.5) . In particular, they all have the same degree and arithmetic genus, and in any given degree the size of the corresponding components of the saturated ideals is the same (so they lie on the same number of hypersurfaces of given degree). Let us assume we know the Hilbert function of any minimal element VI of C. Assume that we have performed a basic double link, using F2 E Iv! of degree d2 and a general element Fl E Sd!, arriving at a scheme Z E Cd! . Then from (4) on page 166, we can compute the cohomology of the vector bundle g, and hence the Hilbert function of Z . In particular we know the degree of Z (namely deg Z = deg VI +d 1d2 ) and the arithmetic genus of Z. But we also know all the possible basic double links that can be performed on Z . So with only the initial information of the Hilbert function of VI, we can (in principle) describe all the possible sequences of basic double links that can be performed starting with VI. Hence we can describe all the possible degrees and arithmetic genera of elements of the even liaison
185
6.4. APPLICATIONS
class£. In practice, it is not so easy to carry out the above process from scratch. However, there are several useful facts from [34] §5 which make the job much easier. Before stating these facts, we introduce the following notation for basic double links in co dimension two (see page 165): Let VI be a co dimension two su bscheme of IF and F2 E I VI a form of degree d2. Choose FI ESdI such that FI and F2 form a regular sequence. Let Z be the scheme obtained by performing the corresponding Basic Double Linkage. Then we write
(by Proposition 6.3.3, up to deformation it is enough to know the degrees dl and d2 ). The first fact, which can be proved fairly quickly using Proposition 6.3.3, is the following: Lemma 6.4.7 Let VI be a codimension two subscheme of IF. Consider the basic double links VI : (a, b + c) -+ Z and VI : (a, b) -+ YI
:
(a, c) -+ Z'.
Then Z and Z' are in the same shift of the same even liaison class, and there is an irreducible fiat family of codimension two subschemes of IF to which both belong.
In particular, the basic double link VI: (a,b) -+ Z
is equivalent (up to deformation) to the sequence of basic double links VI : (a, 1) -+ ZI : (a, 1) -+ ... -+ Zb-I : (a, 1) -+ Zb.
Hence any sequence of basic double links is equivalent (up to deformation) to one (longer, in general) where the polynomials playing the role of FI all have degree 1. The next fact can also be proved using similar methods. (The proof can be found in the proof of Lemma 5.2 of [34].)
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CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
Lemma 6.4.8 Assume that a> b and that there exists a sequence of basic double links VI : (a, 1) --t Y : (b, 1) --t Z. (It is not always the case that such a sequence exists, even if the first basic double link exists.) Then the sequence
VI : (a - 1,1) --t Y' : (b + 1,1) --t Z' also exists, Z and Z' are in the same shift of the same even liaison class, and there is an irreducible fiat family of codimension two subschemes of ]pn to which both belong.
Using these facts, the following very useful theorem is proved in [34] . It severely restricts the possible sequences of basic double links that one
has to consider in the above program of describing the entire even liaison class. It will also playa crucial role in Section 6.4.4. Theorem 6.4.9 ([34] Corollary 5.3) Let £ be an even liaison class of codimension two subschemes of]pn. Let X E £0 and let X' E £h . Let X = Xo : (b l , h) --t Xl : (b 2, h) --t .. . --t Xp -
1 :
(bp, fp) --t Xp
be a sequence of basic double links, where Xp E £h has the same Hilbert function as X'. Let s = o:(lx) (see Definition 2.2.1). Then there exists another sequence of basic double links
X
= Yo : (s,b)
--t Y1
:
(92,1) --t ... --t Yr -
1 :
(9r.1) --t Yr
where
(1) b 2: 0, s < 92 < 93 < ... < 9r and b + r - 1 = h; (2) deg X' = deg X
+ sb + g2 + ... + gr,'
(3) Xp and Y,. have the same Hilbert function (as that of X') and are in £h (hence we have the desired deformation). Moreover, the sequence (b; g2, ... ,9r) is uniquely determined by X'.
We have already seen the degree of the scheme obtained by a basic double link. It is also of interest to know the genus. Since basic double links are a special case of Liaison Addition, it is possible to use Theorem 3.2.3 (d) to compute the arithmetic genus of the curve obtained from a given curve by performing a basic double link. There are other ways to do this, for instance using the fact that a basic double link really is a sequence of two links (cf. page 138), and then applying Corollary 5.2.14 twice. For simplicity we compute only the case of curves in Jr3. One obtains
6.4. APPLICATIONS
187
Lemma 6.4.10 Let C : (a, b) -+ Y, where C and Yare curves in jp3. Let g( C) and g(Y) denote the arithmetic genera of C and Y respectively. Then 1 g(Y) = b· deg C + 2ab(a + b - 4) + g(C). Example 6.4.11 We illustrate the ideas above by giving a complete list of all possible degrees and arithmetic genera of Buchsbaum (non-aCM) curves in jp3 of degree ::::: 10. Recall the notation from Section 3.3: if C is a Buchsbaum curve, we assume that the module has components of dimension n1 > 0, n2 ~ 0···, n r-1 ~ 0, nr > 0 respectively, and we let N = dimkM(C) = n1 + ... + n r . From Remark 3.3.1 (c), the following are the only three possibilities for the dimensions of the components of
M(C): (a) r = 1, n1 = 1 (in this case, the minimal curve C 1 has degree 2 and arithmetic genus -1); (b) r = 1, n1 = 2 (in this case, the minimal curve C 2 has degree 8 and arithmetic genus 5); (c) r = 2, n1 = 1, n2 = 1 (in this case, the minimal curve C3 has degree 10 and arithmetic genus 10). The genera in the last two cases can be computed using Theorem 3.2.3 (d). Now, to know which Buchsbaum curves of degree::::: 10 exist, it is enough to know what sequences of basic double links can be performed on the above three curves and still obtain a resulting curve of degree::::: 10, keeping in mind Theorem 6.4.9. Clearly no such basic double link can be done to C 2 or C3 since their degrees are already too large. (Recall that their homogeneous ideals start in degree 4, by Corollary 2.2.6, so the result of the smallest basic double links on C 2 and C3 would have degrees 12 and 14, respectively.) To compute the arithmetic genera we use Lemma 6.4.10. To give our list of possible Buchsbaum curves of degree::::: 10, we modify slightly the notation of Theorem 6.4.9 for basic double links, denoting a typical sequence by X : (b; g2, ... ,gr) (b ~ 0, s < g2 < ... < gr)' For example,
C1 : (3; 0) {::::::::} C1 : (2,3) -+ Z C1 :(0;4) {::::::::} C1 :(4,1)-+Z C 1 : (3; 4, 5) {::::::::} C 1 : (2,3) -+ Y1 : (4,1) -+ Y2 : (5,1) -+ Z.
188
CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
So, for instance, C 1 : (0; 4) is the union of a plane curve of degree 4 and two skew lines, each meeting the plane curve once, while C 1 : (3; 0) is the union of a complete intersection of type (2,3) and two skew lines, each meeting the complete intersection in 3 points. Then Figure 6.1 gives all the possible pairs (d, g) that can occur, and it gives all the possible sequences of basic double links (in view of Theorem 6.4.9) . Notice that this does not tell us which of the corresponding families contain integral curves. (See [2], [138], [166], [171] and [172], [169].) 0
degree
genus
2 3 4
-1
5 6 7
8
0 1 3
C1 C1 C1 C1
4 6 5 6
C 1 :(1;3)
10
8 9
10
C1 (does not exist)
8 9
sequence of BDL's
15 10 11 13 15 21
: : : :
(1; 0) (0; 3) (2; 0) (0; 4)
C1 : (0; 5) C2 C1 :(1;4)
C 1 : (0; 2,4) C1 : (3; 0) C 1 : (0; 6) C 1 : (0; 3, 4) C1 : (2; 3) C1 : (1; 5) C1 : (0; 7) C3 C1 : (2; 4) C 1 : (0; 3, 5) C1 :(1;6) C1 : (4; 0) C 1 : (0; 8)
Figure 6.1: Buchsbaum (non-aCM) curves in jp>3 of degree ~ 10 As a final illustration of how the whole even liaison class C can be described from knowledge of the minimal elements, we consider curves in jp>3 and the problem of finding maximal rank curves in C. (See page 37
6.4. APPLICATIONS
189
for the definition. The main idea behind the notion of "maximal rank" is that the deficiency module must end before the ideal can begin.) Of course since all the elements of £0 have the same Hilbert function and the same deficiency module, £0 contains some maximal rank curves if and only if every element of £0 has maximal rank. To see whether there exist curves in £ of maximal rank, it is enough to consider only the minimal curves and any curves obtained from them by sequences of basic double links (because of the Lazarsfeld-Rao Property). Now, using the facts about basic double links starting on page 165, it is not hard to check the following: if we perform the basic double link X : (a, b) -t Z, then in passing from X to Z the deficiency module moves to the right exactly b places, while the beginning of the ideal moves to the right at most b places (and exactly b places if a is sufficiently large, namely a = deg F2 2 (deg Fd + a(Ix) = b + a(Ix )). Hence it follows that £ contains maximal rank curves in every shift (of infinitely many degrees in each shift) if and only if the minimal elements of £ have maximal rank. (This was first observed in [31].)
6.4.4
Stick Figures
It is a classical problem whether every smooth curve in p3 specializes to a stick figure. This problem was solved recently by Hartshorne [91], who gave examples of smooth curves in ]p'3 that do not have this property. The problem has a long and distinguished history, including work by many of the important names in Algebraic Geometry. At the suggestion of H.G. Zeuthen it was even made a prize problem, with a gold medal, by the Royal Danish Academy of Arts and Sciences, and it is often referred to as Zeuthen's problem. See [91] for an account of the history of this problem, and of course for the solution. Recall that a stick figure is a union of lines such that at most two of the lines meet in any point. See [93] for background on stick figures. It was shown by Gaeta [71] that every arithmetically Cohen-Macaulay curve specializes to a stick figure. (See also [37] for an analogous statement in higher dimension, still co dimension two.) In this section we describe the approach taken in [35] to prove that every Buchsbaum curve in p3 specializes to a stick figure. The idea is to show how the results on basic double links from Section 6.4.3 can be used, in conjunction with the results on Buchsbaum curves in Chapter 3 and the Lazarsfeld-Rao Property, to extend an idea of Bolondi [29]. Bolondi's idea, briefly, is the following. Let £ be an even liaison class
190
CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
of curves in jp3 and assume that CO contains a stick figure Co. Assume furthermore that in the initial degree of the ideal of this curve (i.e. 0: (Ico) ) there is a polynomial consisting of a product of linear forms; that is, a union of planes. Hence in any degree d for which (Ico)d is not zero, there is a union of d planes containing Co . Then given any curve C in C, we can choose our sequence of basic double links (guaranteed by the Lazarsfeld-Rao Property) so that each basic double link Ci : (a, b) -+ CHI uses polynomials Fl and F2 (see page 165) which are unions of planes (by abuse of notation), and Fl can be chosen generally enough so that the intersection of Fl and F2 is a reduced union of lines. Hence C specializes to a reduced union of lines. The problem is that this union of lines may be forced to have more than two of the lines meeting in a point. A simple example is to let Co be the degree 9 curve consisting of four general lines on a plane L 1 , four general lines on a plane L 2 , and the line of intersection of Ll and L 2 • Then LIL2 is the only surface of degree 2 containing Co , so the basic double link Co : (2, 1) -+ C 1 results in a curve C 1 with a triple point. The first step in resolving this problem, used already by Bolondi, is to assume that the minimal element is hyperplanar, i.e. that the reduced union of planes of minimal degree on which it lies (by assumption) has the further properties that any three planes meet in a point and no component of C lies on the intersection of any two of the planes. (See also page 73.) We saw in Section 3.3 that for any Buchsbaum even liaison class C we can construct a hyperplanar stick figure in Co. (This is needed for Theorem 6.4.13.) It is slightly more subtle to see why this is not enough. Consider the following example.
Example 6.4.12 In this example we will use in a heavy way the fact that a basic double link is the union of a given curve and a certain complete intersection (see page 165). Let Co be a set of two skew lines and consider the sequence of basic double links Co: (20,1) -+ C 1 : (15,1) -+ C 2 : (4, 1) -+ C3 . C 1 is the union of Co and 20 lines on a plane L 1 • C 2 is the union of C 1 and 15 lines on a plane L 2. However, because 15 < 20, the surface F2 of degree 15 (containing Cd used in this latter basic double link contains Ll as a component. Hence the line of intersection of Ll and L2 is a component of C2 · Next, any surface of degree 4 containing C2 contains both Ll and L2 as components. Hence C3 contains a triple point (coming from the two facts written in italics). 0
6.4. APPLICATIONS
191
The idea used in [35] is that this type of obstruction can be avoided by using Theorem 6.4.9, which allows us to assume that the sequence of basic double links is done in strictly increasing degree. One checks that this is the only thing that can go wrong, and so we have
Theorem 6.4.13 ([35]) Let £ be an even liaison class of curves in lP'3 and assume that there is a hyperplanar stick figure Co E £0. Then every curve in £ specializes to a stick figure. As a corollary of Theorem 6.4.13 we have the promised result for Buchsbaum curves (thanks to the construction in Section 3.3):
Corollary 6.4.14 ([35]) Every Buchsbaum curve in lP'3 specializes to a stick figure. In [35] there is a more general theorem, generalizing the notion of a stick figure to codimension two subschemes of projective space. This result is applied in a similar way to certain Buchsbaum surfaces in F , although the result is not as strong as the one above for curves.
Remark 6.4.15 Thanks to Remark 3.3.1, we can use the same techniques to prove a result in the opposite extreme: given any line A C JP3, every Buchsbaum curve in lP'3 specializes to a curve supported on A (even staying in the same liaison class throughout the deformation). Liebling [129] has proved the following more general result. Let M be a graded k[w, x, y, z]-module of finite length such that (x , y)M = o. Let £, be the even liaison class of curves associated to M. Then every element of £, specializes to a curve supported on the line x = y = o. As with the Lazarsfeld-Rao deformations, the deformations in this theorem are through curves all in £, and with fixed Hilbert function. It is not known which curves have this property of specializing to a curve supported on a line. 0
6.4.5
Low Rank Vector Bundles and Schemes Defined by a Small Number of Equations
In this section we give a very brief description of how liaison theory can be applied to the study of vector bundles of low rank on F, and to the study of schemes defined by a small number of equations. We refer to Peterson's
192
CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
thesis [175], Chapter 3, for a much fuller discussion of this topic; almost all of this section is based on that discussion, and is just intended to give a flavor of what can be done. How many equations "should" it take to define a scheme? That is, given a scheme V, what is the ideal with the smallest number of generators whose saturation is Iv? Consider for instance curves in p3 (locally Cohen-Macaulay and equidimensional, as always). If C is a complete intersection then of course its saturated homogeneous ideal is generated by two elements (and in particular C is defined scheme-theoretically by two elements). It is also possible that the saturated ideal of C may be generated by three elements: for instance, a twisted cubic is such a curve. These are called almost complete intersections. On the other hand, a result of Peskine and Szpiro says that as long as C C p3 is at least a local complete intersection then C is defined schemetheoretically by four equations (see below). An intuitive argument for this is as follows. Choose two general polynomials FI and F2 E Ie of sufficiently large degree, defining a complete intersection X which links C to some curve C' having no component in common with C. At "most" (but not all) points of C, FI and F2 are enough to cut out C locally. Choose a third general polynomial F3 E Ie. This is enough to cut out C at the remaining points. However, (FI' F2 , F3 ) vanishes not only on C but also at the points of intersection of F3 with C'; hence we need a fourth polynomial to eliminate those points. The interesting question is to describe those curves in p3 which are cut out scheme-theoretically by three polynomials. Such a curve is called a quasi-complete intersection. Among these, of course, are the almost complete intersections. In general, any co dimension d subscheme of lP" whose saturated homogeneous ideal is generated by d + 1 elements is called an almost complete intersection. If V is defined scheme-theoretically by d + 1 equations (i.e. if there is an ideal with d + 1 generators whose saturation is Iv) then V is said to be a quasi-complete intersection. The observation of Peskine and Szpiro referred to above actually says that if V is a local complete intersection in lP" then V can in any case be defined scheme-theoretically by n + 1 equations ([174] pp. 301-302). A rank r vector bundle £ on lP" is said to be of low rank if r < n. Liaison provides a connection between low rank vector bundles and codimension two schemes which can be defined by a (not necessarily saturated) ideal with a small number of generators, namely n or fewer. In one direction, given a scheme V which is scheme-theoretically de-
193
6.4. APPLICATIONS
fined by m equations, then we have an exact sequence m
o -+ £ -+ EB OJl>n( -ai) -+ Iv -+ O.
(6.12)
i=l
The surjection comes from the fact that V is defined scheme-theoretically by m equations. Since we do not assume that these m polynomials define a saturated ideal, we do not have that H; (£) = 0 (compare with the sequence (6.2) and the discussion immediately following it). However, since V is locally Cohen-Macaulay and equidimensional, £ is at least locally free. (We do not distinguish between locally free sheaves and vector bundles. See [88] Exercise 11.5.18.) If m :S n, then r = rk £ = m - 1 < n. In the other direction, given a vector bundle £ of rank m - 1, for large d we have an exact sequence
o -+ ( r - 2 -+ £ (d)
-+ Iv' (6) -+ 0
where 6 can be computed from a Chern class calculation, and V' has codimension two. Then link V' by a complete intersection (Fl' F2 ), where deg Fi = ai, i = 1,2, and let V be the residual. This gives, after twisting by -6 and applying Proposition 5.2.10, the exact sequence 0-+ £v (6-d- al -a2) -+ 0(6 -al -a2)m-2 EB o( -a2)EBO( -ad -+ Iv -+ 0 (where we write 0 for OJl>n). Hence V is defined scheme-theoretically by m equations. It is natural to ask if there is any connection between a set of minimal generators for the saturated ideal of a scheme V (or the number I/(Iv) of such generators) and a set of polynomials which defines V schemetheoretically. In this regard we have the following useful theorem of Portelli and Spangher:
Theorem 6.4.16 ([178]) Let I be the homogeneous ideal of a scheme V in IF. Let s = I/(Iv). Let t denote the minimum number of elements required to generate V scheme-theoretically. Then there exists a minimal system gl, ... , gs of homogeneous generators of I such that gl,···, gt define V scheme-theoretically. Peterson uses this result to prove the following theorem:
Theorem 6.4.17 ([175]) Let V be an equidimensional, locally CohenMacaulay codimension two subscheme of]pn with homogeneous ideal Iv and ideal sheaf Iv. If V is a quasi-complete intersection then I/(Iv) (See page 133 for notation).
1/
(cMn-2) v (V))
= 3.
194
CHAPTER 6. LIAISON THEORY IN CODIMENSION TWO
We now show how Peterson uses this theorem to reprove a result from [39] about quasi-complete intersection Buchsbaum curves in JP3; this result brings together several of the topics which we have seen in these chapters.
Corollary 6.4.18 ([39]) Let C be a Buchsbaum, non-aCM curve in JP3. Let N = dimkM (C) . If C is a quasi-complete intersection then v (Ie ) = 4 and N
= 1.
Proof: Notice that since M (C) and MV (C) have trivial module structure,
we have v( M V (C)) = N . Recall also the result of Bresinsky, Schenzel and Vogel [39] that v(Ie) ~ 3N + 1 (see page 53 for the proof). Hence Theorem 6.4.17 gives 3 = v(Ie) - N ~ 2N + 1. Since in any case N > 0, we get the desired result. 0 Finally, it is observed in [39] and in [175] that the connection (described above) between vector bundles of low rank and schemes defined scheme-theoretically by a small number of equations easily gives the following corollary. Let £ be a rank two vector bundle on 1P'3 with N = LnEZ h 1(1P'3 , £ (n)). If N > 1 then no section of £ can have a Buchsbaum curve as zero scheme.
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Index (Mi)(V), 9, 10 e(C), 175, 176 Go, 140, 141 G l , 91, 109, 131, 132, 142
H(c, n), 111
£,106 6.j,27 6.H,27 m,l diam M, 8 Pf(f),94
H(Y, t), 7
Pfn - l (f),94
H2(F), 6 H!(F), 1, 6 i(N), 57
/-li,3 11(1),51 II(M),3 IId(M) , 3 ¢i : 51 ---+ Hom (Mi' Mi+d, 2 ¢i,F : Mi ---+ Mi+d, 2
GT) 91
K A ,49 K F ,48 K v ,10
M(d),3 M(V), 10,47
Vl '" V2 , 104
M
£0, 138 £h, 138, 165, 167, 169
3 M VS , 3 Ta(C), 15, 174 To(C) , 15, 174 5,1 5 d ,1 Sl(X), 175 t l (X),175 Vk ,
Wi,T) 2
[A: J], 47 [A:M J], 47
a(1),51 a (Iv ),29 a(V), 29 a(X), 175 a(N),57 J, 7
M,6
X
F V ,3
aCM,lO adjunction formula, 34 algebraic linkage, 104, 111 almost complete intersection, 192 alternating, 94 arithmetic genus bound for, 178 of aCM curve, 28 of basic double link, 187 of linked curves, 122 of minimal element, 184 possible in an even liaison class, 184 arithmetically Cohen-Macaulay, 9, 10, 110, 159, 178
212 codimension two, 51 Cohen-Macaulay type, 11 deficiency modules are zero, 11 generators of ideal, 30 hypersurface section, 19, 20, 23 invariant under linkage, 134 licci, 151 projective dimension, 10 scheme is connected, 13 threefolds, 110 arithmetically Gorenstein, 11, 12, 55, 59, 77, 164, 168 codimension two, 81 curve, 56 symmetric h-vector, 79 Artinian reduction, 27, 28, 126 Auslander-Buchsbaum Theorem, 10 Basic Double G-Linkage, 140, 141 Basic Double Linkage, 14, 71, 137, 140, 141, 165, 184, 185 biliaison, 106 elementary, 140, 142 bilinked, 106 Buchsbaum k-Buchsbaum, 8, 42, 54, 65 2-Buchsbaum, 9, 54 curve, 37, 40, 52, 53, 62, 72, 111, 143, 167, 177, 187, 194 generalize aCM, 42, 52, 62, 63, 67 hyperplane section, 13, 19, 62 index, 8, 42 locally CM and equidimensional, 63 maximal rank curves, 64
INDEX minimal curve in jp3, 66 preserved under linkage, 135 quasi-Buchsbaum, 41, 63 scheme is connected, 13, 64 subscheme of pn, 41 type, 52, 53, 66 Buchsbaum-Rim complex, 84 Buchsbaum-Rim sheaf, 84 canonical module, 10, 12, 33, 91 Castelnuovo's bound, 179 Castelnuovo-Mumford regularity, 7, 30 Cayley-Bacharach property, 80 CI-liaison, 103, 106 CI-liaison class, 106 CI-link, 106 CoCoA, 9, 110 Cohen-Macaulay ring, 10 Cohen-Macaulay type, 12, 22, 56, 59 complete intersection, 35, 55, 81, 108 arithmetic genus of (curve), 36 degree of, 36 is aCM, 35 is arithmetically Gorenstein, 35 resolution of, 35 complete intersection liaison, 101, 106 connectedness, 13 aCM scheme, 13 Buchsbaum curve, 13, 64 decreasing type, 32, 97, 178, 180 deficiency module, 9 Buchsbaum curve, 63, 64 invariant of liaison, 134 measures failure to be aCM, 10
INDEX
213
measures failure to be locally CM and equidimensional, 12 shift of, 14, 54 leftmost, 14, 25, 66, 138 degeneracy locus, 2, 143 degree of a complete intersection, 36 of minimal element, 184 possible in an even liaison class, 184 determinantal ideal, 155 good, 156 standard, 156 diameter, 8, 43, 54, 64 directly CI-linked, 104 directly G-linked, 104 dissocie, 1 domination, 174 double line, 16, 42, 56, 66, 108, 129 double-six configuration, 147, 149 dualizing sheaf, 116 Dubreil's Theorem, 51
G-liaison class, 106 G-link, 106 generalized divisors, 107 geometric linkage, 103, 111 geometrically CI-linked, 103 geometrically G-linked, 103 glicci, 109, 131, 155, 157 good linear configuration, 75, 99 Gorenstein in codimension 1, 91 Gorenstein liaison, 101, 106 graded Betti numbers, 4, 21, 36, 98 graded module degeneracy loci, 2 diameter of, 8 dual of, 3 minimal free resolution of, 3, 4 minimal generators of, 3 multiplication by a homogeneous polynomial, 2 of finite length, 5 shift of, 3 structure, 2, 3
embedded components, 71 equal cohomology, 175 even liaison, 106, 165 even liaison class, 15, 106 structure, 137 extremal curve, 16
h-vector, 28, 29, 31, 33, 79, 80, 82, 179 regularity, 30 Halphen gap, 182 Hartshorne-Rao module, 9 Hartshorne-Schenzel Theorem, 132 Hilbert function, 7, 25, 62, 110, 147, 169, 178, 184 decreasing type, 32, 178, 180 hyperplane section, 177 hypersurface section, 20 under liaison, 126 Hilbert polynomial, 7, 36, 121123 Hilbert scheme, 16
first difference function, 27 first infinitesimal neighborhood, 121 flag, 111 free sheaf, 1 G-biliaison elementary, 142 G-bilinked, 131, 132 G-liaison, 103, 106
214 Hilbert series, 28 Hilbert Syzygy Theorem, 4 Hilbert's Nullstellensatz, 27 Hilbert-Burch matrix, 11, 50, 96, 99, 153, 154, 183 theorem, 95, 153 homogeneous coordinate ring, 10 hyperplane section, 17, 34, 143, 145, 178 general, 25, 130 of an integral curve, 25 , 178 hypersurface section, 17, 48, 56 preserves aCM, 19 preserves graded Betti numbers, 21 preserves linkage, 124 ideal minimal generator, 18 saturated, 5, 7, 16, 158, 193 ideal quotient, 17, 47, 87, 93, 104, 113 index of speciality, 167, 175, 176 initial degree, 51, 57, 63 integral curve genus of, 1l0, 178 hyperplane section of, 25, 178 inverse systems, 83 Koszul resolution, 35 Lazarsfeld-Rao Property, 41 , 42, 66, 72, 75, 138, 149, 165167, 184 leftmost shift, 14, 26 liaison, 104, 106 class, 106 nontrivial equivalence relation, 134
INDEX
Liaison Addition, 40, 42, 66, 68, 69, 163 licci, 109, 151, 154, 156, 159, 162 lifting a link, 128-130, 180 a polynomial, 63 the aCM property, 55 linear general position, 81 linearly equivalent curves, 181 linearly equivalent divisors, 132, 140, 142 are CI-bilinked, 109 are G-bilinked, 131 linkage, 104 algebraic, 104 class, 106 direct, 103, 104 even, 106 geometric, 103 LR-Property, 139, 152 Macaulay, 9, 24, 44, 46, 110 mapping cone, 4, 119 maximal minors, 153 maximal rank curve, 37, 59, 149, 188 Buchsbaum curves, 64 minimal element, 14, 25, 26, 41 , 72, 138, 139, 150, 165, 166, 171,174,184,188 Buchsbaum curves, 66, 187 Hilbert function, 184 minimal free resolution, 4, 8, 21, 57 infinite, 24 pure, 156 minimal generating set, 3 minimal shift, 138 obstructedness, 111
INDEX Pfaffian, 94 Porteous' formula, 2, 149 projective dimension, 10 pure resolution, 156 quadric surface, 146, 164 quasi-Buchsbaum, 41, 63 quasi-complete intersection, 192 rational quartic curve, 37 rational sextic curve, 148 rational surface, 131 reflexive, 21, 84 regularity m-regular, 7 Castelnuovo-Mumford, 7, 8 connection to minimal free resolutions, 8 residual, 93, 99, 103, 104 residual intersections, 108 resolution, free, 4 Riemann-Roch theorem, 147, 149 saturated, 6 saturation, 6, 7, 16 secant variety, 130 sectional genus, 29 self-linked, 104, 105 sheaf ideal, 5 structure, 5 sheafification, 5 examples of, 5 perserves exactness, 5 shift, 3 skew lines, 36, 68, 145, 146 skew symmetric, 94 socle, 79 socle degree, 79, 126 Socle Lemma, 57, 146 stably equivalent, 136, 159, 162
215 stick figure, 40, 42, 72, 74, 97-99, 111 , 152, 181 , 189 hyperplanar, 73, 74, 190 subcanonical divisor, 91, 118, 131 subextremal curve, 16 surfaces in jp4 , 163, 177 syzygy module, 4 threefolds in IF, 110, 177 top dimensional component, 85 top dimensional part, 85, 87, 115, 116 Uniform Position Property, 25,31 , 178, 180 Uniform Resolution Property, 25 unmixed, 113 unobstructedness, 111, 182 vector bundle, 170 low rank, 192 zeroscheme, 10, 11, 67 Zeuthen's problem, 42, 152, 189
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A. Weinstein Department of Mathematics University of California Berkeley, CA 94720 USA
Progress in Mathematics is a series of books intended for professional mathematicians and scientists, encompassing all areas of pure mathematics. This distinguished series, which began in 1979, includes authored monographs and edited collections of papers on important research developments as well as expositions of particular subject areas. We encourage preparation of manuscripts in some form ofTEX for delivery in camera-ready copy which leads to rapid publication, or in electronic form for interfacing with laser printers or typesetters. Proposals should be sent directly to the editors or to: Birkhauser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U. S. A. 100 101 102 103 104
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TAYLOR. Pseudodifferential Operators and Nonlinear PDE BARKER/SALLY (eds). Harmonic Analysis on Reductive Groups DAVID (ed). Seminaire de Theorie des Nombres, Paris 1989-90 ANGER !PORTENIER. Radon Integrals ADAMS !BARBASCHNOGAN. The Langlands Classification and Irreducible Characters for Real Reductive Groups DRAO/WALLACH(eds). New Developments in Lie Theory and Their Applications BUSER. Geometry and Spectra of Compact Riemann Surfaces BRYLINSKl. Loop Spaces, Characteristic Classes and Geometric Quantization DAVID (ed). Seminaire de Theorie des Nombres, Paris 1990-91 EYSSETIE/GALLIGO (eds). Computational Algebraic Geometry LUSZTIG. Introduction to Quantum Groups SCHWARZ. Morse Homology DONG/LEPOWSKY. Generalized Vertex Algebras and Relative Vertex Operators MOEGLIN/WALDSPURGER. Decomposition spectrale et series d'Eisenstein
114 BERENSTEIN/GAYNIDRAS/Y GER. Residue Currents and Bezout Identities 115 BABELON/CARTIER/KOSMANNSCHWARZBACH (eds). Integrable Systems, The Verdier Memorial Conference: Actes du Colloque International de Luminy 116 DAVID (ed). Seminaire de Theorie des Nombres, Paris 1991-92 117 AUDIN/LaFONTAINE (eds). Holomorphic Curves in Symplectic Geometry 118 V AISMAN. Lectures on the Geometry of Poisson Manifolds 119 JOSEPH! MEURAT!MIGNON!PRUM/ RENTSCHLER (eds). First European Congress of Mathematics, July, 1992, Vol. I 120 JOSEPH/ MEURAT!MIGNON!PRUM/ RENTSCHL.ER (eds). First European Congress of Mathematics, July, 1992, Vol. II 121 JOSEPH/ MEURAT!MIGNON!PRUM/ RENTSCHLER (eds). First European Congress of Mathematics, July, 1992, Vol. III (Round Tables) 122 GUILLEMIN. Moment Maps and Combinatorial Invariants of T"-spaces
123 BRYLINSKl!BRYLINSKl/GUILLEMIN/KAC (eds). Lie Theory and Geometry: In Honor of Bertram Kostant 124 AEBISCHER/BORER/KALIN/LEUENBERGER! REIMANN (eds). Symplectic Geometry 125 LUBOTZKY. Discrete Groups, Expanding Graphs and Invariant Measures 126 RIESEL. Prime Numbers and Computer Methods for Factorization 127 HORMANDER. Notions of Convexity 128 SCHMIDT. Dynamical Systems of Algebraic Origin 129 DUKGRAAF!FABERNAN DER GEER (eds). The Moduli Space of Curves 130 DUISTERMAAT. Fourier Integral Operators 131 GINDIKIN/LEPOWSKY/WILSON (eds). Functional Analysis on the Eve of the 21st Century. In Honor of the Eightieth Birthday of I. M. Gelfand, Vol. 1 132 GINDIKlN/LEPOWSKY/WILSON (eds.) Functional Analysis on the Eve of the 21st Century. In Honor of the Eightieth Birthday of I. M. Gelfand, Vol. 2 133 HOFER/TAUBES/WEINSTEIN/ZEHNDER(eds). The floer Memorial Volume 134 CAMPILLO LOPEZ!NARVAEZ MACARRO (eds) Algebraic Geometry and Singularities 135 AMREIN!BOUTET DE MONvEL/GEORGESCU. Co-Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians 136 BROTO/CASACUBERTA!MISLIN (eds). Algebraic Topology: New Trends in Localization and Periodicity 137 VIGNERAS. Representations l-modulaires d'un groupe reductif p-adique avec I p 138 BERNDT!DIAMOND/HILDEBRAND (eds). Analytic Number Theory, Vol. 1 In Honor of Heini Halberstam 139 BERNDT!DIAMOND/HILDEBRAND (eds). Analytic Number Theory, Vol. 2 In Honor of Heini Halberstam 140 KNAPP. Lie Groups Beyond an Introduction 141 CABANES (eds). Finite Reductive Groups: Related Structures and Representations 142 MONK. Cardinal Invariants on Boolean Algebras 143 GONZALEZ-VEGA/RECIO (eds). Algorithms in Algebraic Geometry and Applications
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144 BELLAICHE/RISLER (eds). Sub-Riemannian Geometry 145 ALBERT!BROUZET!DUFOUR (eds). Integrable Systems and Foliations Feuilletages et Systemes Integrables 146 JARDINE. Generalized Etale Cohomology 147 DIBIASE. Fatou TypeTheorems. Maximal Functions and Approach Regions 148 HUANG. Two-Dimensional Conformal Geometry and Vertex Operator Algebras 149 SOURIAU. Structure of Dynamical Systems. A Symplectic View of Physics 150 SHIOTA. Geometry of Subanalytic and Semialgebraic Sets 151 HUMMEL. Gromov's Compactness Theorem For Pseudo-holomorphic Curves 152 GROMOV. Metric Structures for Riemannian and Non-Riemannian Spaces 153 BUESCU. Exotic Attractors: From Liapunov Stability to Riddled Basins 154 BOTTCHER/KARLOVICH. Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators 155 DRAGOMIR/ORNEA. Locally Conformal Kahler Geometry 156 GUIvARCH/JI/TAYLOR. Compactifications of Symmetric Spaces 157 MURTY/MURTY. Non-vanishing of Lfunctions and Applications 158 nRAoNOGAN/WOLF (eds). Geometry and Representation Theory of Real and p-adic Groups 159 THANGAVELU. Harmonic Analysis on the Heisenberg Group 160 KASHIwARA!MATSUO/SAITO/SATAKE (eds). Topological Field Theory, Primitive Forms and Related Topics 161 SAGAN/STANLEY (eds). Mathematical Essays in honor of Gian-Carlo Rota 162 ARNOLD/GREUEL/STEENBRINK (eds). Singularities. The Brieskorn Anniversary Volume 163 BERNDT/SCHMIDT. Elements of the Representation Theory of the Jacobi Group 164 ROUSSARIE. Birfurcations of Planar Vector Fields and Hilbert's Sixteenth Problem 165 MIGLIORE. Introduction to Liaison Theory and Deficiency Modules 166 ELIAS/GIRAL!MIRO-RoIG/ZARZUELA. Six Lectures on Commutative Algebra
