Related papers: Wonderful varieties of type D

Let G be a semisimple complex algebraic group, and H a wonderful subgroup of G. We prove several results relating the subgroup H to the properties of a combinatorial invariant S of G/H, called its spherical system. It is also possible to…

In the setting of strict wonderful varieties we answer positively to Luna's conjecture, saying that wonderful varieties are classified by combinatorial objects, the so-called spherical systems. In particular, we prove that strict wonderful…

A geometrical realization of wonderful varieties by means of a suitable class of invariant Hilbert schemes is given. As a consequence, Luna's conjecture asserting that wonderful varieties are classified by spherical systems, triples of…

We show that the proof of Luna's conjecture about the classification of general wonderful G-varieties can be reduced to the analysis of finitely many families of primitive cases. We work out all primitive cases arising with any classical…

Let G be a semisimple connected linear algebraic group over C, and X a wonderful G-variety. We study the possibility of realizing X as a closed subvariety of the projective space of a simple G-module. We describe the wonderful varieties…

A subgroup H of an algebraic group G is said to be strongly solvable if H is contained in a Borel subgroup of G. This paper is devoted to establishing relationships between the following three combinatorial classifications of strongly…

We give an introduction to the theory of wonderful G-varieties, with many examples when G is simple of type F4. We present results and open problems about these varieties: on their classification, on their isotropy groups, on morphisms…

Let G be a complex reductive group. A normal G-variety X is called spherical if a Borel subgroup of G has a dense orbit in X. Of particular interest are spherical varieties which are smooth and affine since they form local models for…

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…

We introduce the notion of (homological) G-smoothness for a complex G-variety X, where G is a connected affine algebraic group. This is based on the notion of smoothness for dg algebras and uses a suitable enhancement of the G-equivariant…

We show that many important varieties and sets of varieties of semigroups may be defined by relatively simple and transparent first-order formulas in the lattice of all semigroup varieties.

The invariant Hilbert schemes considered in \cite{BC1} were proved to be affine spaces. The proof relied on the classification of strict wonderful varieties. We obtain in the present article a classification-free proof of the affinity of…

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 classify the smooth projective symmetric G-varieties with Picard number one (and G semisimple). Moreover we prove a criterion for the smoothness of the simple (normal) symmetric varieties whose closed orbit is complete. In particular we…

We prove some fundamental structural results for spherical varieties in arbitrary characteristic. In particular, we study Luna's two types of localization and use it to analyze spherical roots, colors and their interrelation. At the end, we…

Given the Luna-Vust invariants of a spherical variety, we determine the Luna-Vust invariants of the spectrum of its Cox ring. In particular, we deduce an explicit description of the divisor class group of the Cox ring. It follows that every…

Let G be a complex connected reductive group. I. Losev has shown that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring…

A two-orbit variety is a normal complete complex algebraic variety on which a reductive complex algebraic group acts with exactly two orbits. The aim of this paper is to give the classification of all two-orbit varieties and to prove Luna's…

We prove equivalent numerical conditions for a complete spherical variety to admit a toric structure, and for the smoothness of an arbitrary spherical variety along any given G-orbit. The conditions are in terms of spherical skeletons, a…

In this article, we investigate the geometry of reductive group actions on algebraic varieties. Given a connected reductive group $G$, we elaborate on a geometric and combinatorial approach based on Luna-Vust theory to describe every normal…