Variétés horosphériques de Fano

Abstract

Résumé. -Une variété horosphérique est une variété algébrique normale dans laquelle un groupe algébrique réductif opère avec une orbite ouverte fibrée en tores sur une variété de drapeaux. En particulier, les variétés toriques et les variétés de drapeaux sont horosphériques. Dans cet article, on classifie les variétés horosphériques de Fano en termes de certains polytopes rationnels qui généralisent les polytopes ré-flexifs considérés par V. Batyrev. Puis on obtient une majoration du degré des variétés horosphériques lisses de Fano, analogue à celle donnée par O. Debarre dans le cas torique. On étend un résultat récent de C. Casagrande: les variétés horosphériques Q-factorielles de Fano ont leur nombre de Picard majoré par deux fois la dimension.Abstract (Fano horospherical varieties). -A horospherical variety is a normal algebraic variety where a reductive algebraic group acts with an open orbit which is a torus bundle over a flag variety. For example, toric varieties and flag varieties are horospherical. In this paper, we classify Fano horospherical varieties in terms of certain rational polytopes that generalize the reflexive polytopes considered by V. Batyrev. Then, we obtain an upper bound on the degree of smooth Fano horospherical varieties, analogus to that given by O. Debarre in the toric case. We extend a recent result of C. Casagrande: the Picard number of any Fano Q-factorial horospherical variety is bounded by twice the dimension.

The Pseudo-Index of Horospherical Fano Varieties

“…In fact, the curves C µ and C α,v are exactly the irreducible B-stable curves in X and they generate the cone of effective 1-cycles. Now, using the definition of the polytope Q given in [8,Ch.3] and the intersection formulas given in [3,Ch.3.2], one can prove the following lemma.…”

“…. , a un u n ) is a basis of N R but not of N (see [8,Proposition 4.4]). This means that there exist a non-zero element u 0 = n i=1 λ i a u i u i of N such that ∀i ∈ {1, .…”

“…We will use arguments similar to the ones used by C. Casagrande, and we will use also a few results on horospherical varieties from [8]. So, let us summarize the theory of Fano horospherical varieties (See [8] for more details).…”

An approach of the minimal model program for horospherical varieties via moment polytopes

“…If D is Cartier and only ample, then (n − 1)D is very ample ([Pas06,Theorem 0.3]). In particular, the assumptions Cartier and very ample, instead of Q-Cartier and ample, are not really restrictive.…”

“…First, sinceQ is lattice polytope of maximal dimension, the Pas06,Lemma 5.1], the closure X ′ of this G-orbit is normal, and then a G/H-embedding. Now, if χ 0 is a vertex of Q, the intersection of the closure of the Gorbit with the affine space ∩P(⊕ χ∈(pv 0 +M )∩pQ V (χ)) v * χ 0 =0 is the B − -stable affine variety whose structure ring is the B − -module generated by the rational…”