Cox rings of minimal resolutions of surface quotient singularities

TL;DR

This paper studies Cox rings of minimal resolutions of surface quotient singularities, providing explicit descriptions and equations for these rings and the resolutions within toric varieties.

Contribution

It offers two novel descriptions of Cox rings for these resolutions, including explicit equations and generator sets, enhancing understanding of their algebraic structure.

Findings

Cox rings are described as hypersurfaces in affine space.

Explicit generators for Cox rings are identified.

Minimal resolutions are embedded as divisors in toric varieties.

Abstract

We investigate Cox rings of minimal resolutions of surface quotient singularities and provide two descriptions of these rings. The first one is the equation for the spectrum of a Cox ring, which is a hypersurface in an affine space. The second is the set of generators of the Cox ring viewed as a subring of the coordinate ring of a product of a torus and another surface quotient singularity. In addition, we obtain an explicit description of the minimal resolution as a divisor in a toric variety.