Introduction to algebraic geometry

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 9
Preface......Page 13
1.1 Implicitization......Page 15
1.2 Ideal membership......Page 18
1.3 Interpolation......Page 19
1.4 Exercises......Page 22
2.1 Monomial orders......Page 25
2.2 Gröbner bases and the division algorithm......Page 27
2.3 Normal forms......Page 30
2.4 Existence and chain conditions......Page 33
2.5 Buchberger’s Criterion......Page 36
2.6 Syzygies......Page 40
2.7 Exercises......Page 43
3.1 Ideals and varieties......Page 47
3.2 Closed sets and the Zariski topology......Page 52
3.3 Coordinate rings and morphisms......Page 53
3.4 Rational maps......Page 57
3.5 Resolving rational maps......Page 60
3.6 Rational and unirational varieties......Page 65
3.7 Exercises......Page 67
4.1 Projections and graphs......Page 71
4.2 Images of rational maps......Page 75
4.3 Secant varieties, joins, and scrolls......Page 79
4.4 Exercises......Page 82
5.1 Common roots of univariate polynomials......Page 87
5.2 The resultant as a function of the roots......Page 94
5.3 Resultants and elimination theory......Page 96
5.4 Remarks on higher-dimensional resultants......Page 98
5.5 Exercises......Page 101
6 Irreducible varieties......Page 103
6.1 Existence of the decomposition......Page 104
6.2 Irreducibility and domains......Page 105
6.3 Dominant morphisms......Page 106
6.4 Algorithms for intersections of ideals......Page 108
6.5 Domains and field extensions......Page 110
6.6 Exercises......Page 112
7 Nullstellensatz......Page 115
7.1 Statement of the Nullstellensatz......Page 116
7.2 Classification of maximal ideals......Page 117
7.3 Transcendence bases......Page 118
7.4 Integral elements......Page 120
7.5 Proof of Nullstellensatz I......Page 122
7.6 Applications......Page 123
7.7 Dimension......Page 125
7.8 Exercises......Page 126
8.1 Irreducible ideals......Page 130
8.2 Quotient ideals......Page 132
8.3 Primary ideals......Page 133
8.4 Uniqueness of primary decomposition......Page 136
8.5 An application to rational maps......Page 142
8.6 Exercises......Page 145
9.1 Introduction to projective space......Page 148
9.2 Homogenization and dehomogenization......Page 151
9.3 Projective varieties......Page 154
9.4 Equations for projective varieties......Page 155
9.5 Projective Nullstellensatz......Page 158
9.6 Morphisms of projective varieties......Page 159
9.7 Products......Page 168
9.8 Abstract varieties......Page 170
9.9 Exercises......Page 176
10 Projective elimination theory......Page 183
10.1 Homogeneous equations revisited......Page 184
10.2 Projective elimination ideals......Page 185
10.3 Computing the projective elimination ideal......Page 188
10.4 Images of projective varieties are closed......Page 189
10.5 Further elimination results......Page 190
10.6 Exercises......Page 191
11.1 Dual projective spaces......Page 195
11.2 Tangent spaces and dual varieties......Page 196
11.3 Grassmannians: Abstract approach......Page 201
11.4 Exterior algebra......Page 205
11.5 Grassmannians as projective varieties......Page 211
11.6 Equations for the Grassmannian......Page 213
11.7 Exercises......Page 216
12.1 Hilbert functions defined......Page 221
12.2 Hilbert polynomials and algorithms......Page 225
12.3 Intersection multiplicities......Page 229
12.4 Bezout Theorem......Page 233
12.5 Interpolation problems revisited......Page 239
12.6 Classification of projective varieties......Page 243
12.7 Exercises......Page 245
A.1 Rings and homomorphisms......Page 249
A.2 Constructing new rings from old......Page 250
A.3 Modules......Page 252
A.4 Prime and maximal ideals......Page 253
A.5 Factorization of polynomials......Page 254
A.6 Field extensions......Page 256
A.7 Exercises......Page 258
Bibliography......Page 260
Index......Page 263