Introduction to Algebraic Geometry

by Brendan Hassett

• ISBN13: 9780521691413
• ISBN10: 0521691419
• Edition: 1st
• Format: Paperback
• Copyright: 2007-05-21
• Publisher: Cambridge University Press

Summary

Algebraic geometry has a reputation for being difficult and inaccessible, even among mathematicians! This must be overcome. The subject is central to pure mathematics, and applications in fields like physics, computer science, statistics, engineering, and computational biology are increasingly important. This book is based on courses given at Rice University and the Chinese University of Hong Kong, introducing algebraic geometry to a diverse audience consisting of advanced undergraduate and beginning graduate students in mathematics, as well as researchers in related fields. For readers with a grasp of linear algebra and elementary abstract algebra, the book covers the fundamental ideas and techniques of the subject and places these in a wider mathematical context. However, a full understanding of algebraic geometry requires a good knowledge of guiding classical examples, and this book offers numerous exercises fleshing out the theory. It introduces Grobner bases early on and offers algorithms for most every technique described. Both students of mathematics and researchers in related areas benefit from the emphasis on computational methods and concrete examples. Book jacket.

Table of Contents

Preface	p. xi
Guiding problems	p. 1
Implicitization	p. 1
Ideal membership	p. 4
Interpolation	p. 5
Exercises	p. 8
Division algorithm and Grobner bases	p. 11
Monomial orders	p. 11
Grobner bases and the division algorithm	p. 13
Normal forms	p. 16
Existence and chain conditions	p. 19
Buchberger's Criterion	p. 22
Syzygies	p. 26
Exercises	p. 29
Affine varieties	p. 33
Ideals and varieties	p. 33
Closed sets and the Zariski topology	p. 38
Coordinate rings and morphisms	p. 39
Rational maps	p. 43
Resolving rational maps	p. 46
Rational and unirational varieties	p. 50
Exercises	p. 53
Elimination	p. 57
Projections and graphs	p. 57
Images of rational maps	p. 61
Secant varieties, joins, and scrolls	p. 65
Exercises	p. 68
Resultants	p. 73
Common roots of univariate polynomials	p. 73
The resultant as a function of the roots	p. 80
Resultants and elimination theory	p. 82
Remarks on higher-dimensional resultants	p. 84
Exercises	p. 87
Irreducible varieties	p. 89
Existence of the decomposition	p. 90
Irreducibility and domains	p. 91
Dominant morphisms	p. 92
Algorithms for intersections of ideals	p. 94
Domains and field extensions	p. 96
Exercises	p. 98
Nullstellensatz	p. 101
Statement of the Nullstellensatz	p. 102
Classification of maximal ideals	p. 103
Transcendence bases	p. 104
Integral elements	p. 106
Proof of Nullstellensatz I	p. 108
Applications	p. 109
Dimension	p. 111
Exercises	p. 112
Primary decomposition	p. 116
Irreducible ideals	p. 116
Quotient ideals	p. 118
Primary ideals	p. 119
Uniqueness of primary decomposition	p. 122
An application to rational maps	p. 128
Exercises	p. 131
Projective geometry	p. 134
Introduction to projective space	p. 134
Homogenization and dehomogenization	p. 137
Projective varieties	p. 140
Equations for projective varieties	p. 141
Projective Nullstellensatz	p. 144
Morphisms of projective varieties	p. 145
Products	p. 154
Abstract varieties	p. 156
Exercises	p. 162
Projective elimination theory	p. 169
Homogeneous equations revisited	p. 170
Projective elimination ideals	p. 171
Computing the projective elimination ideal	p. 174
Images of projective varieties are closed	p. 175
Further elimination results	p. 176
Exercises	p. 177
Parametrizing linear subspaces	p. 181
Dual projective spaces	p. 181
Tangent spaces and dual varieties	p. 182
Grassmannians: Abstract approach	p. 187
Exterior algebra	p. 191
Grassmannians as projective varieties	p. 197
Equations for the Grassmannian	p. 199
Exercises	p. 202
Hilbert polynomials and the Bezout Theorem	p. 207
Hilbert functions defined	p. 207
Hilbert polynomials and algorithms	p. 211
Intersection multiplicities	p. 215
Bezout Theorem	p. 219
Interpolation problems revisited	p. 225
Classification of projective varieties	p. 229
Exercises	p. 231
Notions from abstract algebra	p. 235
Rings and homomorphisms	p. 235
Constructing new rings from old	p. 236
Modules	p. 238
Prime and maximal ideals	p. 239
Factorization of polynomials	p. 240
Field extensions	p. 242
Exercises	p. 244
Bibliography	p. 246
Index	p. 249
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