Singularity categories and singular loci of certain quotient singularities
Xiaojun Chen, Jieheng Zeng
TL;DR
The paper shows that for quotient singularities S = k[V]^G with G a finite abelian subgroup of SL(V), the reduced singular locus $\sqrt{\mathrm{Sing}(\Spec(S))}$ is completely determined by the singularity category $D_{sg}(S)$. The authors develop a framework linking $D_{sg}(S)$ to noncommutative resolutions via contraction algebras $\Lambda_{\mathrm{con}}$ and their centers, constructing an inverse system of centers whose limit recovers $\varphi(S)/\sqrt{\varphi(S)\cap(\Lambda e\Lambda)}$, the coordinate ring of the reduced singular locus. The main result asserts that a triangle equivalence $D_{sg}(S_{1}) \cong D_{sg}(S_{2})$ implies $\sqrt{\mathrm{Sing}(\Spec(S_{1}))} \cong \sqrt{\mathrm{Sing}(\Spec(S_{2}))}$, providing a reconstruction theorem for the reduced singular locus from singularity categories. The approach combines McKay quivers, noncommutative resolutions, contraction algebras, and an explicit inverse-system analysis, and is illustrated with concrete examples highlighting the method’s effectiveness in abelian quotient settings.
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 $\mathrm{Spec}(S)$.