On smoothness of minimal models of quotient singularities by finite subgroups of $SL_n(\mathbb{C})$

TL;DR

This paper establishes conditions under which quotient singularities of the form ^n/G have crepant resolutions, provides a method to compute Cox rings of minimal models, and classifies symplectically imprimitive quotient singularities with symplectic resolutions.

Contribution

It generalizes Verbitsky's result on crepant resolutions, introduces an explicit Cox ring computation procedure, and completes classification of certain symplectic quotient singularities.

Findings

A quotient singularity ^n/G has a crepant resolution iff G is generated by junior elements.

An explicit procedure to compute Cox rings of minimal models of ^n/G.

Classification of symplectically imprimitive quotient singularities with projective symplectic resolutions.

Abstract

We prove that a quotient singularity $\mathbb{C}^n/G $ by a finite subgroup $G\subset SL_n(\mathbb{C})$ has a crepant resolution only if $G $ is generated by junior elements. This is a generalization of the result of Verbitsky [V]. We also give a procedure to compute the Cox ring of a minimal model of a given $\mathbb{C}^n/G$ explicitly from information of $G$. As an application, we investigate the smoothness of minimal models of some quotient singularities. Together with work of Bellamy and Schedler, this completes the classification of symplectically imprimitive quotient singularities which admit projective symplectic resolutions.