By Michel Brion, Laurent Bonavero
ISBN-10: 2856291228
ISBN-13: 9782856291221
Résumé :
Géométrie des variétés toriques
Ce quantity rassemble des textes issus de l'école d'été « Géométrie des variétés toriques » (Grenoble, 19 juin-7 juillet 2000). Ils reprennent, sous une forme plus détaillée, des cours et des exposés de séminaire des deuxième et troisième semaines de l'école, los angeles première semaine ayant été consacrée à des cours introductifs. On trouvera dans l'article de D. Cox un landscape des travaux récents en géométrie torique et de leurs purposes, qui met en viewpoint les autres textes du présent volume.
Mots clefs : Variétés toriques
Abstract:
This quantity gathers texts originated in the summertime tuition ``Geometry of Toric Varieties'' (Grenoble, June 19-July 7, 2000). those are multiplied types of lectures brought throughout the moment and 3rd weeks of the college, the 1st week having been dedicated to introductory lectures. The paper by means of D. Cox is an summary of modern paintings in toric types and its purposes, placing into standpoint the opposite contributions of the current volume.
Key phrases: Toric varieties
Class. math. : 14M25
Table of Contents
* D. A. Cox -- replace on toric geometry
* W. Bruns and J. Gubeladze -- Semigroup algebras and discrete geometry
* A. Craw and M. Reid -- tips on how to calculate A-Hilb C3
* D. I. Dais -- Resolving three-d toric singularities
* D. I. Dais -- Crepant resolutions of Gorenstein toric singularities and top sure theorem
* J. Hausen -- generating strong quotients by means of embedding into toric varieties
* Y. Ito -- unique McKay correspondence
* Y. Tschinkel -- Lectures on peak zeta features of toric varieties
* J. A. Wiśniewski -- Toric Mori thought and Fano manifolds
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Extra resources for Geometry of toric varieties
Sample text
125 1. Introduction These notes, composed for the Summer School on Toric Geometry at Grenoble, June/July 2000, contain a major part of the joint work of the authors. In Section 3 we study a problem that clearly belongs to the area of discrete geometry or, more precisely, to the combinatorics of finitely generated rational cones and their Hilbert bases. Our motivation in taking up this problem was the attempt 2000 Mathematics Subject Classification.
In 1997, Shokurov [221] conjectured that if we have a projective log variety (X, D), D = di Di , with KX + D numerically trivial and at worst log canonical singularities, then di rank N S(X) + dim(X). Furthermore, equality should hold if and only if X is a toric variety and the Di are the torus-invariant divisors on X. Shokurov proved this for surfaces and then Prohkorov [208] proved a special case in dimension 3. 18. Other Results of Interest. — Here is a selection of some of the many interesting papers dealing with toric varieties: – In [18], Altmann computes the torsion submodule of Ω1Y , where Y is any affine toric variety.
Now let K be a field. Then we can form the semigroup algebra K[S]. Since S is finitely generated as a semigroup, K[S] is finitely generated as a Kalgebra. When an embedding S → Zn is given, it induces an embedding K[S] → K[Zn ], and upon the choice of a basis in Zn , the algebra K[Zn ] can be identified with the Laurent polynomial ring K[X1±1 , . . , Xn±1 ]. Under this identification, K[S] has the monomial basis X a , a ∈ S ⊂ Zn (where we use the notation X a = X1a1 · · · Xnan ). If we identify S with the semigroup K-basis of K[S], then there is a conflict of notation: addition in the semigroup turns into multiplication in the ring.
Geometry of toric varieties by Michel Brion, Laurent Bonavero
by Michael
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