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

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.