Introduction to Toric Varieties - Rilegato

Fulton, William

Rilegato

ISBN 10: 0691033323 ISBN 13: 9780691033327

Casa editrice: Princeton Univ Pr, 1993

Riassunto

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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Contenuti

Ch. 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

Dettagli dell?edizione

• Editore: Princeton Univ Pr
• Data di pubblicazione: 1993
• Lingua: Inglese
• ISBN 10: 0691033323
• ISBN 13: 9780691033327
• Rilegatura: Copertina rigida
• Numero di pagine: 180
• Contatto del produttore: non disponibile
• Persona responsabile: non disponibile