Cox rings and combinatorics

TL;DR

This paper explores how combinatorial structures within the divisor class group of a variety with finitely generated Cox rings can be used to understand geometric properties like singularities, the ample cone, and Fano criteria, enabling explicit calculations.

Contribution

It introduces a combinatorial approach to analyze geometric properties of varieties with finitely generated Cox rings, providing explicit computational methods.

Findings

Describes singularities using combinatorial data

Calculates the ample cone explicitly

Provides simple criteria for Fano varieties

Abstract

For a variety with a finitely generated total coordinate ring, we describe basic geometric properties in terms of certain combinatorial structures living in its divisor class group. For example, we describe the singularities, we calculate the ample cone, and we give simple Fano criteria. As we show by means of several examples, these results allow explicit computations.