Equivariant Cox ring

TL;DR

This paper introduces the equivariant Cox ring for normal varieties with group actions, explores its properties, and establishes finiteness, presentations, and singularity criteria, especially for varieties of complexity one under reductive group actions.

Contribution

It defines the equivariant Cox ring, relates it to the ordinary Cox ring, and provides new results on its structure, finiteness, and singularity properties for varieties of complexity one.

Findings

The equivariant Cox ring is finitely generated and normal for certain varieties.

The subalgebra of U-invariants has a presentation by generators and relations.

The Cox ring of an almost homogeneous G-variety of complexity one has log terminal singularities under certain conditions.

Abstract

We define the equivariant Cox ring of a normal variety with algebraic group action. We study algebraic and geometric aspects of this object and show how it is related to the ordinary Cox ring. Then, we specialize to the case of normal rational varieties of complexity one under the action of a connected reductive group G. We show that the G-equivariant Cox ring is then a finitely generated integral normal G-algebra. Under a mild additional condition, we give a presentation by generators and relations of its subalgebra of U-invariants, where U is the unipotent part of a Borel subgroup of G. The ordinary Cox ring is also finitely generated and canonically isomorphic to the U-equivariant Cox ring, so that it inherits a canonical structure of U-algebra. Relying on a work of Hausen and Herppich, we prove that the subalgebra of U-invariants of the Cox ring is a finitely generated Cox ring of a variety of complexity one under the action of a torus. This yields in particular that this latter algebra is a complete intersection.Characterizing the log terminality of singularities in a finitely generated Cox ring is an interesting question, particularly since the work of Gongyo, Okawa, Sannai and Takagi characterizing varieties of Fano type via singularities of Cox rings ([13]). We provide a criterion of combinatorial nature for the Cox ring of an almost homogeneous G-variety of complexity one to have log terminal singularities.Iteration of Cox rings has been introduced by Arzhantsev, Braun, Hausen and Wrobel in [1]. For log terminal quasicones with a torus action of complexity one, they proved that the iteration sequence is finite with a finitely generated factorial master Cox ring. We prove that the iteration sequence is finite for equivariant and ordinary Cox rings of normal rational G-varieties of complexity one satisfying a mild additional condition (e.g. complete varieties or almost homogeneous varieties). In the almost homogeneous case, we prove that the equivariant and ordinary master Cox rings are finitely generated and factorial.