Séminaire Algèbre Géométrie Algébrique Topologie Algébrique :
Le 01 avril 2021 à 11:30 -
Présentée par Vezier Antoine - Univ. Grenoble Alpes
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.