Factorially graded rings and Cox rings

TL;DR

This paper investigates the algebraic structure of Cox rings of normal varieties, focusing on their factorial grading property, and provides criteria for their algebraic detection and construction.

Contribution

It introduces a criterion that reduces the factorial grading property of Cox rings to factoriality, enabling algebraic detection and construction.

Findings

Established a criterion linking factorial grading to factoriality.

Provided methods to detect Cox rings algebraically.

Facilitated construction of Cox rings through algebraic criteria.

Abstract

Cox rings of normal varieties are factorially graded, i.e. homogeneous elements allow a unique decomposition into homogeneous factors. We study this property from an algebraic point of view and give a criterion which in a sense reduces it to factoriality. This will allow us to detect and construct Cox rings in a purely algebraic manner.