4 - Selected topics

Summary

We discuss various topics around Cox rings. The first section is devoted to birational maps, for example, blow-ups. We figure out the class of modifications that preserve finite generation and we show how to compute the Cox ring of the modified variety in terms of the Cox ring of the initial one. In Section 4.2, we first introduce a class of quotient presentations dominated by the characteristic space and comprising, for example, the classical affine cones. Then we provide a lifting result for connected subgroups of the automorphism group. In the case of a finitely generated Cox ring, this gives an approach to the whole automorphism group, showing in particular that it is affine algebraic. The topic of Section 4.3 is finite generation of the Cox ring. We provide general criteria and characterizations; for example, the finiteness characterization of Hu and Keel in terms of polyhedral subdivisions of the moving cone is proven and we discuss finite generation of the Cox ring for Fano varieties. In Section 4.4 we consider varieties coming with a torus action. We express their Cox ring in terms of data of the action. In the case of a rational variety with an action of complexity 1, we see that the Cox ring is given by trinomial relations as in Section 3.4. In particular, the constructions given there produce indeed all rational normal complete A₂-varieties with a torus action of complexity 1. Section 4.5 is about almost homogeneous varieties, that is, equivariant open embeddings of homogeneous spaces. We first describe the Cox ring of a homogeneous space G/H. Embeddings X of homogeneous spaces G/H with finitely generated Cox ring and small boundary X \ G/H allow an immediate description via bunched rings and provide a rich example class. Finally, we survey in this section results on the case of wonderful and more general spherical varieties.