Categorical aspects of the Koll\'ar--Shepherd-Barron correspondence

TL;DR

This paper explores the relationship between surface singularities, their deformations, and associated algebraic structures, revealing conditions under which these deformations simplify to semi-simple algebras and applying findings to classify derived category embeddings.

Contribution

It establishes a link between deformations of surface singularities and Morita equivalences of associated algebras, extending understanding of their categorical and geometric properties.

Findings

Deformations of algebra $ar{R}$ are Morita--equivalent to path algebras of acyclic quivers.

Semi-simplicity of $ar{R}$ corresponds to $Q$-Gorenstein smoothings.

Application to classification of exceptional collections and derived category embeddings.

Abstract

It is well known that a $2$-dimensional cyclic quotient singularity $\overline{W}$ has the same singularity category as a finite dimensional associative algebra $\overline{R}$ introduced by Kalck and Karmazyn. We study the deformations of the algebra $\overline{R}$ induced by the deformations of the surface $\overline{W}$ to a smooth surface. We show that they are Morita--equivalent to path algebras $\hat{R}$ of acyclic quivers for general smoothings within each irreducible component of the versal deformation space of $\overline{W}$ (as described by Koll\'ar and Shepherd-Barron). Furthermore, $\hat{R}$ is semi-simple if and only if the smoothing is $\mathbb{Q}$-Gorenstein (one direction is due to Kawamata). We provide many applications. For example, we describe strong exceptional collections of length $10$ on all Dolgachev surfaces and classify admissible embeddings of derived categories of quivers into derived categories of rational surfaces.