Luna-Vust invariants of Cox rings and skeletons of spherical varieties

Abstract

Given the Luna-Vust invariants of a spherical variety, we determine the Luna-Vust invariants of the spectrum of its Cox ring. In particular, we deduce an explicit description of the divisor class group of the Cox ring. It follows that every spherical variety admits iteration of Cox rings. Moreover, we obtain a combinatorial proof of the fact that the Cox ring is determined by the spherical skeleton, which is a known result following from the description of the Cox ring due to Brion. Finally, we show that a conjectural combinatorial smoothness criterion can be reduced to the case of a factorial affine spherical variety with a fixed point.