SMOOTH

Smoothness of the invariant Hilbert scheme of affine spherical varieties for the existence of wonderful varieties

 Partecipanti

Mappa

 Word cloud

Esplora la "nuvola delle parole (Word Cloud) per avere un'idea di massima del progetto.

 Obiettivo del progetto (Objective)

'Let G be a reductive linear algebraic group over the complex numbers. A G-variety, an algebraic variety with an algebraic action of the group G, is said to be spherical if it is normal and has an open orbit for a maximal connected solvable subgroup of G. We aim to complete the classification of spherical varieties by proving Luna's conjecture on a special class of spherical varieties, called wonderful. To a wonderful variety one can naturally associate an invariant combinatorial object in terms of roots and weights, called spherical system. Luna's conjecture states that there exists a one-to-one correspondence between isomorphism classes of wonderful varieties and spherical systems. Given a spherical system, here we want to provide the corresponding wonderful variety by studying the geometric properties of a certain algebraic scheme, called invariant Hilbert scheme, recently introduced by Alexeev and Brion. The given reductive group G acts linearly on the ring of regular functions of any affine spherical G-variety, the corresponding linear representation is multiplicity-free. The invariant Hilbert scheme of Alexeev and Brion parameterises the affine spherical G-varieties with a fixed multiplicity-free representation in their ring of regular functions. It is endowed with an action of a maximal torus of the group G. Given a spherical system, the strategy is to define a suitable multiplicity-free representation and study the corresponding invariant Hilbert scheme. Via deformation theory arguments we want to prove that under certain conditions the considered invariant Hilbert scheme has an open orbit for the toric action. By a standard procedure, called spherical closure, one can associate to any spherical variety a wonderful variety. Here we want to prove that to an affine spherical variety corresponding to a generic point in the invariant Hilbert scheme it is associated a wonderful variety with the given spherical system.'

Introduzione (Teaser)

Spherical varieties are special complex variations in algebraic geometry. They form a wide class among notable algebraic varieties found in nature.

Descrizione progetto (Article)

The 'Smoothness of the invariant Hilbert scheme of affine spherical varieties for the existence of wonderful varieties' (Smooth) project aimed at classifying a special class of spherical varieties, called wonderful, by proving Luna's conjecture. This theory states that there is a one-to-one correspondence between equally shaped classes of wonderful varieties and spherical systems. A spherical system is a coordinate system for three-dimensional space that is useful for analysing systems with some degree of symmetry about a point, such as within a sphere.

Researchers proposed to provide the corresponding wonderful variety to a given spherical system by studying the geometric properties of the invariant Hilbert scheme, used for classifying problems of certain algebraic varieties. However, on commencement of this EU-funded project, partners discovered that the particular method of research was already in use by others at an advanced stage of development. They therefore decided to continue work on classifying spherical varieties via Luna's conjecture, but with different methods.

Studies extensively analysing the combinatorics (i.e. measurements of structures in an algebraic context) of the so-called spherical systems resulted in a better understanding of their interplay with the geometry of wonderful varieties. This marked the beginning of the development of a more complete theory of wonderful varieties. Smooth researchers managed to solve technical problems that presented during their efforts to generalise Luna's original approach to the classification. Importantly, they also succeeded in formulating the strategy for fully proving Luna's conjecture, which until now has been only partially proven under certain hypotheses.

Smooth project work offered a constructive approach to the classification, providing an algorithm to associate a wonderful variety to a given spherical system.