The Cox ring of a complexity-one horospherical variety

TL;DR

This paper provides a detailed presentation of Cox rings for complete rational complexity-one horospherical varieties, generalizing the concept of homogeneous coordinate rings to a broader class of algebraic varieties.

Contribution

It introduces explicit generators and relations for Cox rings of these varieties, advancing understanding of their algebraic structure.

Findings

Explicit generators for Cox rings are identified.

Relations among generators are established.

The structure of Cox rings for these varieties is clarified.

Abstract

Cox rings are intrinsic objects naturally generalizing homogeneous coordinate rings of projective spaces. A complexity-one horospherical variety is a normal variety equipped with a reductive group action whose general orbit is horospherical and of codimension one. In this note, we provide a presentation by generators and relations for the Cox rings of complete rational complexity-one horospherical varieties.