The class group of a minimal model of a quotient singularity

TL;DR

This paper characterizes the class group of minimal models of quotient singularities arising from finite group actions on vector spaces, showing it is governed by the junior elements of the group.

Contribution

It provides a complete description of the class group of minimal models of linear quotient singularities using the junior elements of the acting group.

Findings

Class group is determined by junior elements in G

Minimal models are $Q$-factorial terminalizations

Explicit description of class groups for quotient singularities

Abstract

Let $V$ be a finite-dimensional vector space over the complex numbers and let $G\leq \operatorname{SL}(V)$ be a finite group. We describe the class group of a minimal model (that is, $\mathbb Q$-factorial terminalization) of the linear quotient $V/G$. We prove that such a class group is completely controlled by the junior elements contained in $G$.