Let G G be a connected semisimple group over C \mathbb C , whose simple components have type A \mathsf A or D \mathsf D . We prove that wonderful G G -varieties are classified by means of combinatorial objects called spherical systems. This is a generalization of a known result of Luna for groups of type A \mathsf A ; thanks to another result of Luna, this implies also the classification of all spherical G G -varieties for the groups G G we are considering. For these G G we also prove the smoothness of the embedding of Demazure.

We complete the classification of wonderful varieties initiated by D. Luna. We review the results that reduce the problem to the family of primitive varieties, and report the references where some of them have already been studied. Finally, we analyze the rest case-by-case

We prove that the invariant Hilbert scheme parameterising the equivariant deformations of the affine multicone over a flag variety is, under certain hypotheses, an affine space. More specifically, we obtain that the isomorphism classes of equivariant deformations of such a multicone are in correspondence with the orbits of a well-determined wonderful variety. (C) 2007 Elsevier Inc. All rights reserved

A spherical system is a combinatorial object, arising in the theory of wonderful varieties, defined in terms of a root system. All spherical systems can be obtained by means of some general combinatorial procedures (such as parabolic induction and wonderful fiber product) from the so-called primitive spherical systems. Here we classify the primitive spherical systems. As an application, we prove that the quotients of a spherical system are in correspondence with the so-called distinguished subsets of colors