Flexible varieties and Cox rings

The Cox ring of an algebraic variety X is a useful invariant which
may be applied, in particular, to the study of the automorphism group Aut(X).
In this way D.Cox (1995) gave a description of the automorphism group of a complete
simplicial toric variety. In a joint paper with S.Gaifullin we find all toric varieties
with a transitive action of a semisimple algebraic group G.

The second part of the talk is based on a joint work with H.Flenner, S.Kaliman, F.Kutzschebauch and M.Zaidenberg.
Let us say that an affine variety X is flexible if the tangent space at any smooth point on X is generated
by sections of locally nilpotent vector fields. Suppose that dim X>1. We prove that flexibility implies that the action of the group
of special automorphisms on the smooth locus of X is infinitely transitive .
Clearly, every affine homogeneous space of a semisimple group G is flexible. Assume now that G acts on an
affine variety X with an open orbit. We show that X is flexible provided it is smooth. Moreover,
using a description of the Cox ring of a normal affine SL(2)-embedding due to V.Batyrev and F.Haddad (2008),
we prove flexibility for arbitrary X and G=SL(2).

Exposé dans les cadres du rencontre
Journées de la géométrie affine à  Grenoble,
Grenoble, 9-10 mai 2011