The Geometry of Syzygies A second course in Commutative Algebra and Algebraic Geometry David Eisenbud University of California, Berkeley with the collaboration of Freddy Bonnin, Clement´ Caubel and Hel´ ene` Maugendre For a current version of this manuscript-in-progress, see www.msri.org/people/staff/de/ready.pdf Copyright David Eisenbud, 2002 ii Contents 0 Preface: Algebra and Geometry xi 0A What are syzygies? . xii 0B The Geometric Content of Syzygies . xiii 0C What does it mean to solve linear equations? . xiv 0D Experiment and Computation . xvi 0E What’s In This Book? . xvii 0F Prerequisites . xix 0G How did this book come about? . xix 0H Other Books . 1 0I Thanks . 1 0J Notation . 1 1 Free resolutions and Hilbert functions 3 1A Hilbert’s contributions . 3 1A.1 The generation of invariants . 3 1A.2 The study of syzygies . 5 1A.3 The Hilbert function becomes polynomial . 7 iii iv CONTENTS 1B Minimal free resolutions . 8 1B.1 Describing resolutions: Betti diagrams . 11 1B.2 Properties of the graded Betti numbers . 12 1B.3 The information in the Hilbert function . 13 1C Exercises . 14 2 First Examples of Free Resolutions 19 2A Monomial ideals and simplicial complexes . 19 2A.1 Syzygies of monomial ideals . 23 2A.2 Examples . 25 2A.3 Bounds on Betti numbers and proof of Hilbert’s Syzygy Theorem . 26 2B Geometry from syzygies: seven points in P3 .......... 29 2B.1 The Hilbert polynomial and function. 29 2B.2 . and other information in the resolution . 31 2C Exercises . 34 3 Points in P2 39 3A The ideal of a finite set of points . 40 3A.1 The Hilbert-Burch Theorem . 41 3A.2 Invariants of the resolution . 46 3B Examples . 49 3B.1 Points on a conic . 49 3B.2 Four non-colinear points . 51 3C Existence of sets with given invariants . 53 CONTENTS v 3C.1 The existence of monomial ideals with given numerical invariants . 54 3C.2 Points from a monomial ideal . 54 3D Exercises . 58 4 Castelnuovo-Mumford Regularity 67 4A Definition and First Applications . 67 4A.1 The Interpolation Problem . 68 4A.2 When does the Hilbert function become a polynomial? 69 4B Characterizations of regularity . 71 4B.1 Cohomology . 71 4B.2 Solution of the Interpolation Problem . 78 4B.3 The regularity of a Cohen-Macaulay module . 79 4B.4 The regularity of a coherent sheaf . 82 4C Exercises . 84 5 The regularity of projective curves 89 5A The Gruson-Lazarsfeld-Peskine Theorem . 89 5A.1 A general regularity conjecture . 90 5B Proof of Theorem 5.1 . 92 5B.1 Fitting ideals . 92 5B.2 Linear presentations . 94 5B.3 Regularity and the Eagon-Northcott complex . 97 5B.4 Filtering the restricted tautological bundle . 99 5B.5 General line bundles . 103 vi CONTENTS 5C Exercises . 104 6 Linear Series and One-generic Matrices 107 6A Rational normal curves . 108 6A.1 Where’d that matrix come from? . 109 6B 1-Generic Matrices . 111 6C Linear Series . 114 6C.1 Ampleness . 117 6C.2 Matrices from pairs of linear series. 119 6C.3 Linear subcomplexes and mapping cones . 123 6D Elliptic normal curves . 125 6E Exercises . 137 7 Linear Complexes and the Linear Syzygy Theorem 145 7A Linear Syzygies . 146 7A.1 The linear strand of a complex . 146 7A.2 Green’s Linear Syzygy Theorem . 148 7B The Bernstein-Gel’fand-Gel’fand correspondence . 151 7B.1 Graded Modules and Linear Free Complexes . 151 7B.2 What it means to be the linear strand of a resolution . 153 7B.3 Identifying the linear strand . 157 7C Exterior minors and annihilators . 158 7C.1 Definitions . 159 7C.2 Description by multilinear algebra . 160 7C.3 How to handle exterior minors . 162 CONTENTS vii 7D Proof of the Linear Syzygy Theorem . 164 7E More about the Exterior Algebra and BGG . 166 7E.1 Gorenstein property and Tate Resolutions . 166 7E.2 Where BGG Leads . 169 7F Exercises . 174 8 Curves of High Degree 177 8.1 The Cohen-Macaulay Property . 178 8.2 The restricted tautological bundle . 181 8A Strands of the Resolution . 187 8A.1 The Cubic Strand . 190 8A.2 The Quadratic Strand . 194 8B Conjectures and Problems . 208 8C Exercises . 211 9 Clifford Index and Canonical Embedding 217 9A The Clifford Index . 218 9B Green’s Conjecture . 221 9C Exercises . 226 10 Appendix A: Introduction to Local Cohomology 229 10A Definitions and Tools . 230 10B Local cohomology and sheaf cohomology . 237 10C Vanishing and nonvanishing theorems . 240 10D Exercises . 244 viii CONTENTS 11 Appendix B: A Jog Through Commutative Algebra 247 11A Associated Primes and primary decomposition. 249 11A.1 Motivation and Definitions . 249 11A.2 Results . 250 11A.3 Examples . 252 11B Dimension and Depth . 253 11B.1 Motivation and Definitions . 253 11B.2 Results . 254 11B.3 Examples . 256 11C Projective dimension . 257 11C.1 Motivation and Definitions . 257 11C.2 Results . 257 11C.3 Examples . 258 11D Normalization . 259 11D.1 Motivation and Definitions . 259 11D.2 Results . 261 11D.3 Examples . 262 11E The Cohen-Macaulay property . 264 11E.1 Motivation and Definitions . 264 11E.2 Results . 267 11E.3 Examples . 268 11F The Koszul complex . 270 11F.1 Motivation and Definitions . 270 11F.2 Results . 272 CONTENTS ix 11F.3 Examples . 273 11G Fitting ideals, determinantal ideals . 274 11G.1 Motivation and Definitions . 274 11G.2 Results . 274 11G.3 Examples . 275 11H The Eagon-Northcott complex and scrolls . 277 11H.1 Motivation and Definitions . 277 11H.2 Results . 279 11H.3 Examples . 281 x CONTENTS Chapter 0 Preface: Algebra and Geometry Syzygy, ancient Greek συζυγια: yoke, pair, copulation, conjunction—OED This book describes some aspects of the relation between the geometry of projective algebraic varieties and the algebra of their equations. It is intended as a (rather algebraic) second course in algebraic geometry and commutative algebra, such as I have taught at Brandeis University, the Intitut Poincar´e in Paris, and Berkeley. Implicit in the very name Algebraic Geometry is the relation between geome- try and equations. The qualitative study of systems of polynomial equations is also the fundamental subject of Commutative Algebra. But when we ac- tually study algebraic varieties or rings, we often know a great deal before finding out anything about their equations. Conversely, given a system of equations, it can be extremely difficult to analyze the geometry of the corre- sponding variety or their other qualitative properties. Nevertheless, there is a growing body of results relating fundamental properties in Algebraic Ge- ometry and Commutative Algebra to the structure of equations. The theory of syzygies offers a microscope for enlarging our view of equations. This book is concerned with the qualitative geometric theory of syzygies: it describes some aspects of the geometry of a projective variety that cor- respond to the numbers and degrees of its syzygies or to its having some structural property such as being determinantal, or more generally having a free resolution with some particularly simple structure. xi xii CHAPTER 0. PREFACE: ALGEBRA AND GEOMETRY 0A What are syzygies? In algebraic geometry over a field K we study the geometry of varieties through properties of the polynomial ring S = K[x0, . , xr] and its ideals. It turns out that to study ideals effectively we we also need to study more general graded modules over S. The simplest way to describe a module is by generators and relations. We may think of a set M ⊂ M of generators for an S-module M as a map from a free S-module F = SM onto M sending the basis element of F corresponding to a generator m ∈ M to the element m ∈ M. When M is graded, we keep the grading in view by insisting that the chosen generators be homogeneous. Let M1 be the kernel of the map F → M; it is called the module of syzygies of M (corresponding to the given choice of generators), and a syzygy of M is an element of M1–that.