Gorensteinness and iteration of Cox rings for Fano type varieties

TL;DR

This paper proves that finitely generated Cox rings are Gorenstein and characterizes Fano type varieties via their Cox rings, showing that Cox ring iteration terminates with a factorial master Cox ring, and extends these results to certain bundles.

Contribution

It establishes the Gorenstein property of Cox rings and characterizes Fano type varieties through their Cox rings, also demonstrating finite Cox ring iteration for specific varieties.

Findings

Finitely generated Cox rings are Gorenstein.

Fano type varieties have Gorenstein canonical quasicone Cox rings.

Cox ring iteration is finite for varieties of Fano type with Kawamata log terminal quasicones.

Abstract

We 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 quasicones, iteration of Cox rings is finite with factorial master Cox ring. Moreover, we prove a relative version of Cox ring iteration for almost principal solvable $G$-bundles.