Factorially graded rings of complexity one

TL;DR

This paper characterizes factorially graded rings of complexity one, providing explicit descriptions of their generators, relations, and Cox rings for certain rational varieties with torus actions.

Contribution

It offers a comprehensive description of finitely generated normal algebras with complexity one gradings satisfying UFD conditions, including explicit Cox ring classifications.

Findings

Describes generators and relations of these algebras

Classifies Cox rings of certain rational varieties

Provides explicit algebraic structures for complexity one cases

Abstract

We consider finitely generated normal algebras over an algebraically closed field of characteristic zero that come with a complexity one grading by a finitely generated abelian group such that the conditions of a UFD are satisfied for homogeneous elements. Our main results describe these algebras in terms of generators and relations. We apply this to write down explicitly the possible Cox rings of normal complete rational varieties with a complexity one torus action.