Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

The Cox ring of a complexity one almost homogeneous variety.

The Cox ring of an algebraic variety (satisfying some natural conditions) is a very rich invariant. It was introduced by Cox in 1995 for the study of toric varieties, and then generalized to normal varieties by Arzhantsev, Berchtold and Hausen. Later, Hu and Keel discovered that the normal varieties with a finitely generated Cox ring define a class of varieties whose birational geometry is particularly well understood. They called them the Mori Dream Spaces (MDS) by virtue of their good behaviour with respect to the minimal model program of Mori. A first problem is to find natural conditions for a normal variety to be an MDS. A second one is to describe the Cox ring of a given MDS: find a presentation by generators and relations, give the nature of its singularities, etc...
Among algebraic varieties equipped with an action of an (affine) algebraic group, a particularly well understood class consists of normal varieties of complexity at most one: a connected reductive group is acting in such a way that the minimal codimension of an orbit of a Borel subgroup is at most one. The normal varieties of complexity zero are the spherical varieties (e.g. a toric variety is spherical). In 2007, Brion proved that spherical varieties are MDS, and gave a description of their Cox ring by generators and relations. A normal variety of complexity one is an MDS if and only if it is a rational variety (e.g. a normal rational surface with an effective $\mathbb{G}_m$-action or a normal SL$_2$-threefold with a dense orbit). This provides a natural class of MDS with group action for which the second problem has only been solved in very particular cases.
In this talk, we will detail the construction of the Cox ring of a normal variety, and explain its importance in algebraic geometry. Then, we will describe the Cox ring of a complexity one almost homogeneous variety (i.e. it is normal with a dense orbit), together with the methods developed to obtain this description.