Introduction to Toric Varieties

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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

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Expressions et termes fréquents

Fréquemment cités

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

À propos de l'auteur (1993)

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

Informations bibliographiques