Introduction to Toric Varieties

ISBN-10: 0691000492
ISBN-13: 9780691000497
Edition: 1993
Authors: William Fulton
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Description: 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  More...

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Book details

List price: $75.00
Copyright year: 1993
Publisher: Princeton University Press
Publication date: 8/1/1993
Binding: Paperback
Pages: 180
Size: 6.50" wide x 9.50" long x 0.50" tall
Weight: 0.594
Language: English

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.

Definitions and examples
Introduction
Convex polyhedral cones
Affine toric varieties
Fans and toric varieties
Toric varieties from polytopes
Singularities and compactness
Local properties of toric varieties
Surfaces; quotient singularities
One-parameter subgroups; limit points
Compactness and properness
Nonsingular surfaces
Resolution of singularities
Orbits, topology, and line bundles
Orbits
Fundamental groups and Euler characteristics
Divisors
Line bundles
Cohomology of line bundles
Moment maps and the tangent bundle
The manifold with singular corners
Moment map
Differentials and the tangent bundle
Serre duality
Betti numbers
Intersection theory
Chow groups
Cohomology of nonsingular toric varieties
Riemann-Roch theorem
Mixed volumes
Bezout theorem
Stanley's theorem
Notes
References
Index of Notation
Index

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