Algebraic varieties

G. Kempf0521426138, 9780521426138

In this book, Professor Kempf gives an introduction to the theory of algebraic varieties from a sheaf theoretic standpoint. By taking this view he is able to give a clean and lucid account of the subject, which will be easily accessible to all newcomers to algebraic varieties.

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Table of contents :
Contents……Page 3
Introduction……Page 6
1.1 Spaces with functions……Page 8
1.2 Varieties……Page 9
1.3 The existence of affine varieties……Page 11
1.4 The nullstellensatz……Page 12
1.5 The rest of the proof of existence of affine varieties / subvarieties……Page 15
1.6 A^n and P^n……Page 17
1.7 Determinantal varieties……Page 18
2.1 The lemma……Page 20
2.2 The Hilbert basis theorem……Page 22
2.3 Irreducible components……Page 23
2.4 Affine and finite morphisms……Page 25
2.5 Dimension……Page 27
2.6 Hypersurfaces and the principal ideal theorem……Page 28
3.1 Products……Page 32
3.2 Products of projective varieties……Page 34
3.3 Graphs of morphisms and separatedness……Page 35
3.4 Algebraic groups……Page 37
3.5 Cones and projective varieties……Page 38
3.6 A little more dimension theory……Page 39
3.7 Complete varieties……Page 40
3.8 Chow’s lemma……Page 41
3.9 The group law on an elliptic curve……Page 42
3.10 Blown up A^n at the origin……Page 43
4.1 The definition of presheaves and sheaves……Page 45
4.2 The construction of sheaves……Page 49
4.3 Abelian sheaves and flabby sheaves……Page 53
4.4 Direct limits of sheaves……Page 57
5.1 Sheaves of rings and modules……Page 61
5.2 Quasi-coherent sheaves on affine varieties……Page 63
5.3 Coherent sheaves……Page 65
5.4 Quasi-coherent sheaves on projective varieties……Page 68
5.5 Invertible sheaves……Page 69
5.6 Operations on sheaves that change spaces……Page 72
5.7 Morphisms to projective space and affine morphisms……Page 75
6.1 The Zariski cotangent space and smoothness……Page 77
6.2 Tangent cones……Page 79
6.3 The sheaf of differentials……Page 82
6.4 Morphisms……Page 87
6.5 The construction of affine morphisms and normalization……Page 89
6.6 Bertini’s theorem……Page 90
7.1 Introduction to curves……Page 92
7.2 Valuation criterions……Page 94
7.3 The construction of all smooth curves……Page 95
7.4 Coherent sheaves on smooth curves……Page 97
7.5 Morphisms between smooth complete curves……Page 99
7.6 Special morphisms between curves……Page 101
7.7 Principal parts and the Cousin problem……Page 103
8.1 The definition of cohomology……Page 105
8.2 Cohomology of affines……Page 107
8.3 Higher direct images……Page 109
8.4 Beginning the study of the cohomology of curves……Page 111
8.5 The Riemann-Roch theorem……Page 113
8.6 First applications of the Riemann-Roch theorem……Page 115
8.7 Residues and the trace homomorphism……Page 117
9.1 The cohomology of A^n – {0} and P^n……Page 120
9.2 Cech cohomology and the Kunneth formula……Page 121
9.3 Cohomology of projective varieties……Page 123
9.4 The direct images of flat sheaves……Page 125
9.5 Families of cohomology groups……Page 127
10.1 Embedding in projective space……Page 131
10.2 Cohomological characterization of affine varieties……Page 132
10.3 Computing the genus of a plane curve and Bezout’s theorem……Page 133
10.4 Elliptic curves……Page 135
10.5 Locally free coherent sheaves on P^1……Page 136
10.6 Regularity in codimension one……Page 137
10.7 One dimensional algebraic groups……Page 138
10.8 Correspondences……Page 139
10.9 The Riemann-Roch theorem for surfaces……Page 146
A.l Localization……Page 148
A.2 Direct limits……Page 150
A.3 Eigenvectors……Page 151
Bibliography……Page 153
Glossary of notation……Page 156
Index……Page 162