2022/04/27 by Tevelev, Jenia, Urzúa, Giancarlo
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2204.13225
It is well known that a 2-dimensional cyclic quotient singularity W has the same singularity category as a finite dimensional associative algebra R introduced by Kalck and Karmazyn. We study the deformations of the algebra R induced by the deformations of the surface W to a smooth surface. We show that they are Morita--equivalent to path algebras R of acyclic quivers for general smoothings within each irreducible component of the versal deformation space of W (as described by Kollár and Shepherd-Barron). Furthermore, R is semi-simple if and only if the smoothing is 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.