Gorensteinness and iteration of Cox rings for Fano type varieties

Abstract

AbstractWe show that finitely generated Cox rings are Gorenstein. This leads to a refined characterization of varieties of Fano type: they are exactly those projective varieties with Gorenstein canonical quasicone Cox ring. We then show that for varieties of Fano type and Kawamata log terminal quasiconesX, iteration of Cox rings is finite with factorial master Cox ring. In particular, even if the class group has torsion, we can express suchXas quotients of a factorial canonical quasicone by a solvable reductive group.