Computing resolutions of quotient singularities

TL;DR

This paper introduces an algorithm to compute Cox rings of resolutions for quotient singularities arising from finite groups acting on complex space, enabling systematic analysis of these resolutions without prior explicit knowledge.

Contribution

The authors develop a geometry-based algorithm and implementation to compute Cox rings of resolutions for quotient singularities, including all cases in dimension three with group order up to 12.

Findings

Successfully computed Cox rings for all G in GL(3) with |G| ≤ 12.

Provided examples of resolutions in dimension four.

Validated the algorithm's effectiveness in singularity resolution analysis.

Abstract

Let $G\subseteq GL(n)$ be a finite group without pseudo-reflections. We present an algorithm to compute and verify a candidate for the Cox ring of a resolution $X\rightarrow \mathbb{C}^n/G$, which is based just on the geometry of the singularity $\mathbb{C}^n/G$, without further knowledge of its resolutions. We explain the use of our implementation of the algorithms in Singular. As an application, we determine the Cox rings of resolutions $X\rightarrow \mathbb{C}^3/G$ for all $G\subseteq GL(3)$ with the aforementioned property and of order $|G|\leq 12$. We also provide examples in dimension 4.