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.
Categories: math.AG