Introduction to Toric Varieties

by Fulton, William

• ISBN13: 9780691000497
• ISBN10: 0691000492
• Format: Paperback
• Copyright: 1993-07-12
• Publisher: Princeton Univ Pr

Summary

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.

Table of Contents

Definitions and examples
Introduction	p. 3
Convex polyhedral cones	p. 8
Affine toric varieties	p. 15
Fans and toric varieties	p. 20
Toric varieties from polytopes	p. 23
Singularities and compactness
Local properties of toric varieties	p. 28
Surfaces; quotient singularities	p. 31
One-parameter subgroups; limit points	p. 36
Compactness and properness	p. 39
Nonsingular surfaces	p. 42
Resolution of singularities	p. 45
Orbits, topology, and line bundles
Orbits	p. 51
Fundamental groups and Euler characteristics	p. 56
Divisors	p. 60
Line bundles	p. 63
Cohomology of line bundles	p. 73
Moment maps and the tangent bundle
The manifold with singular corners	p. 78
Moment map	p. 81
Differentials and the tangent bundle	p. 85
Serre duality	p. 87
Betti numbers	p. 91
Intersection theory
Chow groups	p. 96
Cohomology of nonsingular toric varieties	p. 101
Riemann-Roch theorem	p. 108
Mixed volumes	p. 114
Bezout theorem	p. 121
Stanley's theorem	p. 124
Notes	p. 131
References	p. 149
Index of Notation	p. 151
Index	p. 155
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