The Cox ring of a complexity-one horospherical variety

Kevin Langlois,Ronan Terpereau

doi:10.1007/s00013-016-0979-y

Kevin Langlois, Ronan Terpereau

Open Access

https://doi.org/10.1007/s00013-016-0979-y

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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.

Highlights

All algebraic varieties and algebraic groups considered in this article are defined over an algebraically closed field k of characteristic zero
The homogeneous space G/H may be realized as the total space of a principal Tbundle over the flag variety G/P, where P = NG(H) is the parabolic subgroup normalizing H and T is the algebraic torus P/H
We consider a specific class of G-varieties: the complexity-one horospherical G-varieties, that is, the normal G-varieties whose general orbit is horospherical and of codimension one

Summary

Introduction

All algebraic varieties and algebraic groups considered in this article are defined over an algebraically closed field k of characteristic zero.Let G be a connected -connected reductive algebraic group (i.e., a direct product of a torus and a connected -connected semisimple group), and let H ⊆ G be a closed subgroup. Let us note that any projective Q-factorial normal variety X, with finitely generated class group Cl(X), is completely determined (up to isomorphism) by the data of its Cox ring, as a Cl(X)-graded algebra, and an ample class (see [1, §1.6.3]). A description of the Cox ring for algebraic varieties with torus action is given in [12].

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