The Geometry of Syzygies

by Diamond, Fred; Sturmfels, Bernd; Eisenbud, David

• ISBN13: 9780387222325
• ISBN10: 0387222324
• Format: Paperback
• Copyright: 2005-11-30
• Publisher: Springer Verlag

Summary

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.

Author Biography

The author taught at Brandeis University for twenty-seven years, with sabbatical time spent in Paris, Bonn, and Berkeley, and became Director of the Mathematical Sciences Research Institute in Berkeley in the Summer of 1997. At the same time he joined the faculty of UC Berkeley as Professor of Mathematics. In 2003 he became President of the American Mathematical Society. He currently serves on several editorial boards (Annals of Mathematics, Bulletin du Soci+¬t+¬ Math+¬matique de France, Springer-Verlag's book series Algorithms and Computation in Mathematics).

Table of Contents

Preface: Algebra and Geometry	ix
What Are Syzygies?	x
The Geometric Content of Syzygies	xi
What Does Solving Linear Equations Mean?	xii
Experiment and Computation	xiii
What's In This Book?	xiv
Prerequisites	xv
How Did This Book Come About?	xv
Other Books	xvi
Thanks	xvi
Notation	xvi
Free Resolutions and Hilbert Functions	1	(14)
The Generation of Invariants	1	(1)
Enter Hilbert	2	(1)
The Study of Syzygies	3	(2)
The Hilbert Function Becomes Polynomial	4	(1)
Minimal Free Resolutions	5	(5)
Describing Resolutions: Betti Diagrams	7	(1)
Properties of the Graded Betti Numbers	8	(1)
The Information in the Hilbert Function	9	(1)
Exercises	10	(5)
First Examples of Free Resolutions	15	(16)
Monomial Ideals and Simplicial Complexes	15	(5)
Simplicial Complexes	15	(1)
Labeling by Monomials	16	(2)
Syzygies of Monomial Ideals	18	(2)
Bounds on Betti Numbers and Proof of Hilbert's Syzygy Theorem	20	(2)
Geometry from Syzygies: Seven Points in P3	22	(5)
The Hilbert Polynomial and Function	23	(1)
...and Other Information in the Resolution	24	(3)
Exercises	27	(4)
Points in P2	31	(24)
The Ideal of a Finite Set of Points	32	(7)
Examples	39	(3)
Existence of Sets of Points with Given Invariants	42	(5)
Exercises	47	(8)
Castelnuovo--Mumford Regularity	55	(18)
Definition and First Applications	55	(3)
Characterizations of Regularity: Cohomology	58	(7)
The Regularity of a Cohen--Macaulay Module	65	(2)
The Regularity of a Coherent Sheaf	67	(1)
Exercises	68	(5)
The Regularity of Projective Curves	73	(16)
A General Regularity Conjecture	73	(2)
Proof of the Gruson--Lazarsfeld--Peskine Theorem	75	(10)
Exercises	85	(4)
Linear Series and 1-Generic Matrices	89	(30)
Rational Normal Curves	90	(2)
Where'd That Matrix Come From?	91	(1)
1-Generic Matrices	92	(3)
Linear Series	95	(8)
Elliptic Normal Curves	103	(10)
Exercises	113	(6)
Linear Complexes and the Linear Syzygy Theorem	119	(26)
Linear Syzygies	120	(4)
The Bernstein--Gelfand--Gelfand Correspondence	124	(6)
Exterior Minors and Annihilators	130	(5)
Proof of the Linear Syzygy Theorem	135	(1)
More about the Exterior Algebra and BGG	136	(7)
Exercises	143	(2)
Curves of High Degree	145	(32)
The Cohen--Macaulay Property	146	(7)
The Restricted Tautological Bundle	148	(5)
Strands of the Resolution	153	(16)
The Cubic Strand	155	(4)
The Quadratic Strand	159	(10)
Conjectures and Problems	169	(2)
Exercises	171	(6)
Clifford Index and Canonical Embedding	177	(10)
The Cohen--Macaulay Property and the Clifford Index	177	(3)
Green's Conjecture	180	(5)
Exercises	185	(2)
Appendix 1 Introduction to Local Cohomology	187	(14)
Definitions and Tools	187	(8)
Local Cohomology and Sheaf Cohomology	195	(3)
Vanishing and Nonvanishing Theorems	198	(1)
Exercises	199	(2)
Appendix 2 A Jog Through Commutative Algebra	201	(26)
Associated Primes and Primary Decomposition	202	(3)
Dimension and Depth	205	(3)
Projective Dimension and Regular Local Rings	208	(2)
Normalization: Resolution of Singularities for Curves	210	(3)
The Cohen--Macaulay Property	213	(4)
The Koszul Complex	217	(3)
Fitting Ideals and Other Determinantal Ideals	220	(2)
The Eagon--Northcott Complex and Scrolls	222	(5)
References	227	(10)
Index	237