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

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.