The Geometry of Syzygies: A Second Course in Algebraic Geometry and Commutative Algebra

Description

Algebraic Geometry often seems very abstract, but in fact it is full of concrete examples and problems. This side of the subject can be approached through the equations of a variety, and the syzygies of these equations are a necessary part of the study. This book is the first textbook-level account of basic examples and techniques in this area. It illustrates the use of syzygies in many concrete geometric considerations, from interpolation to the study of canonical curves. The text has served as a basis for graduate courses by the author at Berkeley, Brandeis, and in Paris. It is also suitable for self-study by a reader who knows a little commutative algebra and algebraic geometry already. As an aid to the reader, the appendices provide summaries of local cohomology and commutative algebra, tying together examples and major results from a wide range of topics. Geometry of syzygies Contents Preface: Algebra and Geometry What Are Syzygies? The Geometric Content of Syzygies What Does Solving Linear Equations Mean? Experiment and Computation What’s In This Book? Prerequisites How Did This Book Come About? Other Books Thanks Notation 1 Free Resolutions and Hilbert Functions The Generation of Invariants Enter Hilbert 1A The Study of Syzygies The Hilbert Function Becomes Polynomial 1B Minimal Free Resolutions Describing Resolutions: Betti Diagrams Properties of the Graded Betti Numbers The Information in the Hilbert Function 1C Exercises 2 First Examples of Free Resolutions 2A Monomial Ideals and Simplicial Complexes Simplicial Complexes Labeling by Monomials Syzygies of Monomial Ideals 2B Bounds on Betti Numbers and Proof of Hilbert’s Syzygy Theorem 2C Geometry from Syzygies: Seven Points in ℙ³ The Hilbert Polynomial and Function ... and Other Information in the Resolution 2D Exercises 3 Points in ℙ² 3A The Ideal of a Finite Set of Points 3B Examples 3C Existence of Sets of Points with Given Invariants 3D Exercises 4 Castelnuovo–Mumford Regularity 4A Definition and First Applications 4B Characterizations of Regularity: Cohomology 4C The Regularity of a Cohen–Macaulay Module 4D The Regularity of a Coherent Sheaf 4E Exercises 5 The Regularity of Projective Curves 5A A General Regularity Conjecture 5B Proof of the Gruson–Lazarsfeld–Peskine Theorem 5C Exercises 6 Linear Series and 1-Generic Matrices 6A Rational Normal Curves 6A.1 Where’d That Matrix Come From? 6B 1-Generic Matrices 6C Linear Series 6D Elliptic Normal Curves 6E Exercises 7 Linear Complexes and the Linear Syzygy Theorem 7A Linear Syzygies 7B The Bernstein–Gelfand–Gelfand Correspondence 7C Exterior Minors and Annihilators 7D Proof of the Linear Syzygy Theorem 7E More about the Exterior Algebra and BGG 7F Exercises 8 Curves of High Degree 8A The Cohen–Macaulay Property 8A.1 The Restricted Tautological Bundle 8B Strands of the Resolution 8B.1 The Cubic Strand 8B.2 The Quadratic Strand 8C Conjectures and Problems 8D Exercises 9 Clifford Index and Canonical Embedding 9A The Cohen–Macaulay Property and the Clifford Index 9B Green’s Conjecture 9C Exercises Appendix 1 Introduction to Local Cohomology A1A Definitions and Tools Local Cohomology and Ext Local Cohomology and Čech cohomology Change of Rings Local Duality A1B Local Cohomology and Sheaf Cohomology A1C Vanishing and Nonvanishing Theorems A1D Exercises Appendix 2 A Jog Through Commutative Algebra A2A Associated Primes and Primary Decomposition A2B Dimension and Depth A2C Projective Dimension and Regular Local Rings A2D Normalization: Resolution of Singularities for Curves A2E The Cohen–Macaulay Property A2F The Koszul Complex A2G Fitting Ideals and Other Determinantal Ideals A2H The Eagon–Northcott Complex and Scrolls References Index