Algebraic varieties / George R. Kempf.

Format:

Book

Author/Creator:

Kempf, George.

Series:

London Mathematical Society lecture note series ; 172.

London Mathematical Society lecture note series

Language:

English

Subjects (All):

Algebraic varieties.

Physical Description:

x, 163 pages ; 23 cm.

Place of Publication:

Cambridge, UK : Cambridge University Press, [1993]

Summary:

In this book Professor Kempf gives an introduction to the theory of algebraic functions on 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, graduate students or experts from other fields alike. Anyone who goes on to study schemes will find that this book is an ideal preparatory text.

Contents:

1 Algebraic varieties: definition and existence 1

1.1 Spaces with functions 1

1.2 Varieties 2

1.3 The existence of affine varieties 4

1.4 The nullstellensatz 5

1.5 The rest of the proof of existence of affine varieties / subvarieties 8

1.6 A[superscript n] and P[superscript n] 10

1.7 Determinantal varieties 11

2 The preparation lemma and some consequences 13

2.1 The lemma 13

2.2 The Hilbert basis theorem 15

2.3 Irreducible components 16

2.4 Affine and finite morphisms 18

2.5 Dimension 20

2.6 Hypersurfaces and the principal ideal theorem 21

3 Products; separated and complete varieties 25

3.1 Products 25

3.2 Products of projective varieties 27

3.3 Graphs of morphisms and separatedness 28

3.4 Algebraic groups 30

3.5 Cones and projective varieties 31

3.6 A little more dimension theory 32

3.7 Complete varieties 33

3.8 Chow's lemma 34

3.9 The group law on an elliptic curve 35

3.10 Blown up A[superscript n] at the origin 36

4 Sheaves 38

4.1 The definition of presheaves and sheaves 38

4.2 The construction of sheaves 42

4.3 Abelian sheaves and flabby sheaves 46

4.4 Direct limits of sheaves 50

5 Sheaves in algebraic geometry 54

5.1 Sheaves of rings and modules 54

5.2 Quasi-coherent sheaves on affine varieties 56

5.3 Coherent sheaves 58

5.4 Quasi-coherent sheaves on projective varieties 61

5.5 Invertible sheaves 62

5.6 Operations on sheaves that change spaces 65

5.7 Morphisms to projective space and affine morphisms 68

6 Smooth varieties and morphisms 70

6.1 The Zariski cotangent space and smoothness 70

6.2 Tangent cones 72

6.3 The sheaf of differentials 75

6.4 Morphisms 80

6.5 The construction of affine morphisms and normalization 82

6.6 Bertini's theorem 83

7 Curves 85

7.1 Introduction to curves 85

7.2 Valuation criterions 87

7.3 The construction of all smooth curves 88

7.4 Coherent sheaves on smooth curves 90

7.5 Morphisms between smooth complete curves 92

7.6 Special morphisms between curves 94

7.7 Principal parts and the Cousin problem 96

8 Cohomology and the Riemann-Roch theorem 98

8.1 The definition of cohomology 98

8.2 Cohomology of affines 100

8.3 Higher direct images 102

8.4 Beginning the study of the cohomology of curves 104

8.5 The Riemann-Roch theorem 106

8.6 First applications of the Riemann-Roch theorem 108

8.7 Residues and the trace homomorphism 110

9 General cohomology 113

9.1 The cohomology of A[superscript n] - 0 and P[superscript n] 113

9.2 Cech cohomology and the Kunneth formula 114

9.3 Cohomology of projective varieties 116

9.4 The direct images of flat sheaves 118

9.5 Families of cohomology groups 120

10 Applications 124

10.1 Embedding in projective space 124

10.2 Cohomological characterization of affine varieties 125

10.3 Computing the genus of a plane curve and Bezout's theorem 126

10.4 Elliptic curves 128

10.5 Locally free coherent sheaves on P[superscript 1] 129

10.6 Regularity in codimension one 130

10.7 One dimensional algebraic groups 131

10.8 Correspondences 132

10.9 The Reimann-Roch theorem for surfaces 139

Appendix 139

A.1 Localization 141

A.2 Direct limits 143

A.3 Eigenvectors 144.

Notes:

Includes bibliographical references (pages [146]-148) and index.

ISBN:

0521426138

OCLC:

30975169