The Artin Component and Simultaneous Resolution via Reconstruction Algebras of Type $A$

TL;DR

This paper introduces a method to construct deformation spaces and achieve simultaneous resolutions of certain surface singularities using noncommutative algebraic techniques, extending classical results to broader settings.

Contribution

It recovers the Artin component of deformation spaces via reconstruction algebra quivers and deforms relations to obtain simultaneous resolutions, generalizing previous work.

Findings

Reconstruction algebra quivers encode the Artin component.

Deformed relations and GIT variation produce simultaneous resolutions.

Extends classical singularity resolution techniques to new algebraic settings.

Abstract

This paper uses noncommutative resolutions of non-Gorenstein singularities to construct classical deformation spaces, by recovering the Artin component of the deformation space of a cyclic surface singularity using only the quiver of the corresponding reconstruction algebra. The relations of the reconstruction algebra are then deformed, and the deformed relations together with variation of the GIT quotient achieve the simultaneous resolution. This extends work of Brieskorn, Kronheimer, Grothendieck, Cassens-Slodowy and Crawley-Boevey-Holland into the setting of singularities $\mathbb{C}^2/H$ with $H\leq\mathrm{GL}(2,\mathbb{C})$, and furthermore gives a prediction for what is true more generally.