On factoriality of Cox rings

TL;DR

This paper provides a new proof that Cox rings are factorial when the divisor class group is finitely generated and free, and discusses how torsion in the class group affects factoriality.

Contribution

It introduces a proof of factoriality of Cox rings under certain conditions and analyzes the impact of torsion in the divisor class group.

Findings

Cox rings are factorial if the divisor class group is finitely generated and free.

Torsion in the divisor class group can cause loss of factoriality.

The proof utilizes the concept of graded factoriality.

Abstract

Generalized Cox's construction associates with an algebraic variety a remarkable invariant -- its total coordinate ring, or Cox ring. In this note we give a new proof of factoriality of the Cox ring when the divisor class group of the variety is finitely generated and free. The proof is based on a notion of graded factoriality. We show that if the divisor class group has torsion, then the Cox ring is again factorially graded, but factoriality may be lost.