The Cox ring of a spherical embedding

TL;DR

This paper investigates the structure of Cox rings for spherical embeddings, showing how their relations can be derived from homogeneous space equations, and provides examples where the Cox ring is defined by a single equation.

Contribution

It introduces a method to obtain Cox ring relations via homogenization of equations depending only on the homogeneous space, advancing understanding of spherical embeddings.

Findings

Relations between Cox ring generators can be obtained by homogenizing space-dependent equations.

Some spherical homogeneous spaces have Cox rings defined by a single equation.

The approach simplifies the description of Cox rings for certain embeddings.

Abstract

Let G be a connected reductive group and G/H a spherical homogeneous space. We show that the ideal of relations between a natural set of generators of the Cox ring of a G-embedding of G/H can be obtained by homogenizing certain equations which depend only on the homogeneous space. Using this result, we describe some examples of spherical homogeneous spaces such that the Cox ring of any of their G-embeddings is defined by one equation.