Cox rings and combinatorics II

TL;DR

This paper explores the structure of varieties with finitely generated Cox rings, extending combinatorial methods to cases with torsion and analyzing how Cox rings change under modifications like blow ups, enabling explicit constructions from minimal cases.

Contribution

It generalizes combinatorial approaches to Cox rings to include torsion and characterizes Cox ring transformations under specific modifications, facilitating explicit constructions.

Findings

Extended combinatorial methods to torsion cases.

Described Cox ring changes under certain modifications.

Provided a finite construction process from minimal models.

Abstract

We study varieties with a finitely generated Cox ring. In a first part, we generalize a combinatorial approach developed in earlier work for varieties with a torsion free divisor class group to the case of torsion. Then we turn to modifications, e.g., blow ups, and the question how the Cox ring changes under such maps. We answer this question for a certain class of modifications induced from modifications of ambient toric varieties. Moreover, we show that every variety with finitely generated Cox ring can be explicitly constructed in a finite series of toric ambient modifications from a combinatorially minimal one.