Computing resolutions of quotient singularities

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 $X\to {\mathbb{C}}^n/G$, 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 $X\to {\mathbb{C}}^3/G$ for all $G\subseteq GL(3)$ with the aforementioned property and of order $|G|\le 12$. We also provide examples in dimension 4.

arXiv commentv3: 20 pages; section 2 rewritten