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Annals of Mathematics Studies: Introduction to Toric Varieties by William Fulton (1993, Trade Paperback)
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About this product
Product Identifiers
PublisherPrinceton University Press
ISBN-100691000492
ISBN-139780691000497
eBay Product ID (ePID)943591
Product Key Features
Number of Pages180 Pages
LanguageEnglish
Publication NameIntroduction to Toric Varieties
SubjectCombinatorics, Geometry / Algebraic
Publication Year1993
TypeTextbook
AuthorWilliam Fulton
Subject AreaMathematics
SeriesAnnals of Mathematics Studies
FormatTrade Paperback
Dimensions
Item Height0.7 in
Item Weight8 Oz
Item Length9.2 in
Item Width6.1 in
Additional Product Features
Intended AudienceCollege Audience
LCCN93-011045
Series Volume Number131
IllustratedYes
Table Of ContentCh. 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
SynopsisToric 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., Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects. This text aims to develop the foundations of the study of toric varieties, and describe these relations and applications. It includes Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope., 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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