Giuliano Gagliardi

The Cox ring of a spherical embedding

Abstract Let G be a connected reductive group and G / H a spherical homogeneous space. We show that the ideal of relations between a natural set of generators of the Cox ring of a G-embedding of G / H can be obtained by homogenizing certain equations which depend only on the homogeneous space. Using this result, we describe some examples of spherical homogeneous spaces such that the Cox ring of any of their G-embeddings is defined by one equation.

A combinatorial smoothness criterion for spherical varieties

We suggest a combinatorial criterion for the smoothness of an arbitrary spherical variety using the classification of multiplicity-free spaces, generalizing an earlier result of Camus for spherical varieties of type A.

Gorenstein spherical Fano varieties

We obtain a combinatorial description of Gorenstein spherical Fano varieties in terms of certain polytopes, generalizing the combinatorial description of Gorenstein toric Fano varieties by reflexive polytopes and its extension to Gorenstein horospherical Fano varieties due to Pasquier. Using this description, we show that the rank of the Picard group of an arbitrary

The generalized Mukai conjecture for symmetric varieties

d

On the algebraic stringy Euler number

d-dimensional

A_k

\mathbb {Q}