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It is well known that a $2$-dimensional cyclic quotient singularity$\overline{W}$ has the same singularity category as a finite dimensionalassociative algebra $\overline{R}$ introduced by Kalck and Karmazyn. We studythe deformations of the algebra $\overline{R}$ induced by the deformations ofthe surface $\overline{W}$ to a smooth surface. We show that they areMorita--equivalent to path algebras $\hat{R}$ of acyclic quivers for generalsmoothings 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 manyapplications. For example, we describe strong exceptional collections of length$10$ on all Dolgachev surfaces and classify admissible embeddings of derivedcategories of quivers into derived categories of rational surfaces.

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