Related papers: Wonderful Varieties: A geometrical realization
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…
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…
Let G be a complex connected semisimple group, whose simple components have type A or D. We prove that wonderful G-varieties are classified by means of combinatorial objects called spherical systems. This is a generalization of a known…
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…
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…
We study equivariant real structures on spherical varieties. We call such a structure canonical if it is equivariant with respect to the involution defining the split real form of the acting reductive group G. We prove the existence and…
We give a survey of the incredibly beautiful amount of geometry involved with the problem of realizing a projective variety as hyperplane section of another variety.
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 suggest to look at formal sentences describing complex algebraic varieties together with their universal covers as topological invariants. We prove that for abelian varieties and Shimura varieties this is indeed a complete invariant,…
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,…
A variety is a category of ordered (finitary) algebras presented by inequations between terms. We characterize categories enriched over the category of posets which are equivalent to a variety. This is quite analogous to Lawvere's classical…
We prove that the set of non-degenerate second order maximally superintegrable systems in the complex Euclidean plane carries a natural structure of a projective variety, equipped with a linear isometry group action. This is done by…
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 define varieties of algebras for an arbitrary endofunctor on a cocomplete category using pairs of natural transformations. This approach is proved to be equivalent to the one of equational classes defined by equation arrows. Free…
We invent the notion of a {\it dimension of a variety} $V$ as the cardinality of all its proper {\it derived} subvarieties (of the same type). The dimensions of varieties of lattices, varieties of regular bands and other general algebraic…
We introduce a new family of invariants of real algebraic sets defined in terms of the topology of their complexifications and compute some of these invariants for spheres. This allows us to completely classify topological isomorphism…
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…
We study the cone of effective divisors and the total coordinate ring of wonderful varieties, with applications to their automorphism group. We show that the total coordinate ring of any spherical variety is obtained from that of the…
We define the wonderful compactification of an arrangement of subvarieties. Given a complex nonsingular algebraic variety $Y$ and certain collection $\mathcal{G}$ of subvarieties of $Y$, the wonderful compactification $Y_\mathcal{G}$ can be…