Introduction to Toric Varieties

Produktbeschreibung

Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic geometry notions such as singularities, birational maps, cycles, homology, intersection theory, and Riemann-Roch translate into simple facts about polytopes, toric varieties provide a marvelous source of examples in algebraic geometry. In the other direction, general facts from algebraic geometry have implications for such polytopes, such as to the problem of the number of lattice points they contain. In spite of the fact that toric varieties are very special in the spectrum of all algebraic varieties, they provide a remarkably useful testing ground for general theories. The aim of this mini-course is to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications. The text concludes with Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope. Although some general theorems are quoted without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry. 

Inhaltsverzeichnis

Ch. 1

Definitions and examples

1.1

Introduction

3

1.2

Convex polyhedral cones

8

1.3

Affine toric varieties

15

1.4

Fans and toric varieties

20

1.5

Toric varieties from polytopes

23

Ch. 2

Singularities and compactness

2.1

Local properties of toric varieties

28

2.2

Surfaces; quotient singularities

31

2.3

One-parameter subgroups; limit points

36

2.4

Compactness and properness

39

2.5

Nonsingular surfaces

42

2.6

Resolution of singularities

45

Ch. 3

Orbits, topology, and line bundles

3.1

Orbits

51

3.2

Fundamental groups and Euler characteristics

56

3.3

Divisors

60

3.4

Line bundles

63

3.5

Cohomology of line bundles

73

Ch. 4

Moment maps and the tangent bundle

4.1

The manifold with singular corners

78

4.2

Moment map

81

4.3

Differentials and the tangent bundle

85

4.4

Serre duality

87

4.5

Betti numbers

91

Ch. 5

Intersection theory

5.1

Chow groups

96

5.2

Cohomology of nonsingular toric varieties

101

5.3

Riemann-Roch theorem

108

5.4

Mixed volumes

114

5.5

Bezout theorem

121

5.6

Stanley's theorem

124

Notes

131

References

149

Index of Notation

151

Index

155 

Autoreninfo

William Fulton