Algebraic Geometry

Contents......Page 3
Introduction......Page 7
a Rings and ideals......Page 11
b Rings of fractions......Page 15
c Unique factorization......Page 21
d Integral dependence......Page 24
e Tensor Products......Page 30
Exercises......Page 33
a Definition of an algebraic set......Page 35
b The Hilbert basis theorem......Page 36
c The Zariski topology......Page 37
d The Hilbert Nullstellensatz......Page 38
e The correspondence between algebraic sets and radical ideals......Page 39
g Properties of the Zariski topology......Page 43
h Decomposition of an algebraic set into irreducible algebraic sets......Page 44
i Regular functions; the coordinate ring of an algebraic set......Page 47
k Hypersurfaces; finite and quasi-finite maps......Page 48
l Noether normalization theorem......Page 50
m Dimension......Page 52
Exercises......Page 56
a Sheaves......Page 57
b Ringed spaces......Page 58
c The ringed space structure on an algebraic set......Page 59
d Morphisms of ringed spaces......Page 62
e Affine algebraic varieties......Page 63
f The category of affine algebraic varieties......Page 64
g Explicit description of morphisms of affine varieties......Page 65
h Subvarieties......Page 68
i Properties of the regular map Spm(a)......Page 69
j Affine space without coordinates......Page 70
k Birational equivalence......Page 71
l Noether Normalization Theorem......Page 72
m Dimension......Page 73
Exercises......Page 77
a Tangent spaces to plane curves......Page 79
b Tangent cones to plane curves......Page 81
c The local ring at a point on a curve......Page 83
d Tangent spaces to algebraic subsets of Am......Page 84
e The differential of a regular map......Page 86
f Tangent spaces to affine algebraic varieties......Page 87
g Tangent cones......Page 91
h Nonsingular points; the singular locus......Page 92
i Nonsingularity and regularity......Page 94
j Examples of tangent spaces......Page 95
Exercises......Page 96
a Algebraic prevarieties......Page 97
b Regular maps......Page 98
c Algebraic varieties......Page 99
e Subvarieties......Page 101
f Prevarieties obtained by patching......Page 102
g Products of varieties......Page 103
h The separation axiom revisited......Page 108
i Fibred products......Page 110
j Dimension......Page 111
l Rational maps; birational equivalence......Page 113
m Local study......Page 114
n Étale maps......Page 115
o Étale neighbourhoods......Page 118
p Smooth maps......Page 120
q Algebraic varieties as a functors......Page 121
r Rational and unirational varieties......Page 124
Exercises......Page 125
a Algebraic subsets of Pn......Page 127
b The Zariski topology on Pn......Page 131
c Closed subsets of An and Pn......Page 132
e Pn is an algebraic variety......Page 133
f The homogeneous coordinate ring of a projective variety......Page 135
g Regular functions on a projective variety......Page 136
h Maps from projective varieties......Page 137
i Some classical maps of projective varieties......Page 138
k Projective space without coordinates......Page 143
m Grassmann varieties......Page 144
n Bezout's theorem......Page 148
o Hilbert polynomials (sketch)......Page 149
p Dimensions......Page 150
q Products......Page 152
Exercises......Page 153
a Definition and basic properties......Page 155
b Proper maps......Page 157
c Projective varieties are complete......Page 158
d Elimination theory......Page 159
e The rigidity theorem; abelian varieties......Page 163
f Chow's Lemma......Page 165
g Analytic spaces; Chow's theorem......Page 167
h Nagata's Embedding Theorem......Page 168
Exercises......Page 169
a Normal varieties......Page 171
b Regular functions on normal varieties......Page 174
c Finite and quasi-finite maps......Page 176
d The fibres of finite maps......Page 182
e Zariski's main theorem......Page 184
f Stein factorization......Page 189
h Resolution of singularities......Page 190
Exercises......Page 191
a The constructibility theorem......Page 193
b The fibres of morphisms......Page 196
c Flat maps and their fibres......Page 199
d Lines on surfaces......Page 206
f Birational classification......Page 211
Exercises......Page 212
Solutions to the exercises......Page 213
Index......Page 219