# Crepant resolutions of 3-dimensional quotient singularities via Cox rings

## TL;DR

This paper investigates Cox rings of crepant resolutions of 3D quotient singularities, providing insights into their structure, the number of resolutions, and relations, especially for groups with elements of age 2.

## Contribution

It offers explicit analysis of Cox rings for crepant resolutions of $C^3/G$, revealing geometric structures and relationships between different resolutions.

## Key findings

- Identified Cox rings for specific quotient singularities
- Determined the number of crepant resolutions in examples
- Analyzed relations between different resolutions

## Abstract

We study Cox rings of crepant resolutions of quotient singularities $\mathbb{C}^3/G$ where $G$ is a finite subgroup of $SL(3,\mathbb{C})$. We use them to obtain information on the geometric structure of these resolutions, number of different resolutions and relations between them. In particular, we treat explicitly several examples where $G$ contains elements of age 2.