P-Divisors of Cox Rings

TL;DR

This paper explores the structure of Cox rings of Mori Dream Spaces by describing them as polyhedral divisors, linking algebraic and geometric properties through torus actions and stability conditions.

Contribution

It introduces a novel description of Cox rings as polyhedral divisors, connecting their equivariant structure with stability and multiplicity concepts.

Findings

Cox rings can be represented as polyhedral divisors.

The shape of polyhedral coefficients encodes stability and multiplicity information.

This approach bridges algebraic torus actions with geometric stability analysis.

Abstract

The Cox ring of a so-called Mori Dream Space (MDS) is finitely generated and it is graded over the divisor class group. Hence the spectrum of the Cox ring comes with an action of an algebraic torus whose GIT quotient is the variety in question. We present the associated description of this Cox ring as a polyhedral divisor. Via the shape of its polyhedral coefficients, it connects the equivariant structure of the Cox ring with the world of stable loci and stable multiplicities of linear systems.