Singularity categories and singular loci of certain quotient singularities

TL;DR

This paper demonstrates that the singularity category of certain quotient singularities precisely captures the reduced singular locus of their spectrum, linking categorical and geometric aspects of these singularities.

Contribution

It establishes a direct connection between the singularity category and the reduced singular locus for quotient singularities arising from finite abelian groups in characteristic zero.

Findings

Singularity category recovers the reduced singular locus.

Provides a categorical perspective on quotient singularities.

Links algebraic and geometric properties of singularities.

Abstract

Let $V$ be a finite dimensional $k$-vector space, where $k$ is an algebraic closed field of characteristic zero. Let $G \subseteq \mathrm{SL}(V)$ be a finite abelian group, and denote by $S$ the $G$-invariant subring of the polynomial ring $k[V]$. It is shown that the singularity category $D_{sg}(S)$ recovers the reduced singular locus of $\mathrm{Spec}(S)$.