# Introduction to Toric Varieties

Princeton University Press, 1993 - 157 pages
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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.

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### Table des matières

 Definitions and examples 3 12 Convex polyhedral cones 8 13 Affine toric varieties 15 14 Fans and toric varieties 20 15 Toric varieties from polytopes 23 Singularities and compactness 28 22 Surfaces quotient singularities 31 23 Oneparameter subgroups limit points 36
 Moment maps and the tangent bundle 78 42 Moment map 81 43 Differentials and the tangent bundle 85 44 Serre duality 87 45 Betti numbers 91 Intersection theory 96 52 Cohomology of nonsingular toric varieties 101 53 RiemannRoch theorem 108

 24 Compactness and properness 39 25 Nonsingular surfaces 42 26 Resolution of singularities 45 Orbits topology and line bundles 51 32 Fundamental groups and Euler characteristics 56 33 Divisors 60 34 Line bundles 63 35 Cohomology of line bundles 73
 54 Mixed volumes 114 55 Bézout theorem 121 56 Stanleys theorem 124 Notes 131 References 149 Index of Notation 151 Index 155 Droits d'auteur

### Fréquemment cités

Page 147 - IM Gelfand, MM Kapranov, and AV Zelevinsky, "Newton polytopes of the classical resultant and discriminant,

### Références à ce livre

 Discriminants, Resultants, and Multidimensional DeterminantsAucun aperçu disponible - 1994
 Using Algebraic GeometryAucun aperçu disponible - 2005
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### À propos de l'auteur (1993)

William Fulton is Professor of Mathematics at the University of Chicago.