  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.
