Non-complete rational T-varieties of complexity one

TL;DR

This paper extends the combinatorial Cox ring approach to non-complete rational T-varieties of complexity one, providing a classification of factorially graded affine algebras with specific properties.

Contribution

It introduces a description of factorially graded affine algebras of complexity one with constant invertible elements, expanding the combinatorial framework to non-complete cases.

Findings

Describes all factorially graded affine algebras of complexity one with constant invertible elements.

Extends the Cox ring combinatorial approach to non-complete rational T-varieties.

Provides a classification framework for affine T-varieties of complexity one.

Abstract

We consider rational varieties with a torus action of complexity one and extend the combinatorial approach via the Cox ring developed for the complete case in earlier work to the non-complete, e.g. affine, case. This includes in particular a description of all factorially graded affine algebras of complexity one with only constant homogeneous invertible elements in terms of canonical generators and relations.