  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.
