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Categorical aspects of the Kollár-Shepherd-Barron correspondence

Categorical aspects of the Kollár-Shepherd-Barron correspondence

In the classical papers from 1974, Pinkham and Gabriel studied deformations of varieties with a ${\Bbb G}_m$-action and deformations of finite-dimensional associative algebras. They found the first examples of reducible versal deformation spaces: deformations of the cone over a rational normal quartic and deformations of the $4$-dimensional algebra ${\Bbb C}[x, y, z]/(x, y, z)^2$. In a joint work with Giancarlo Urzua, we describe a remarkable embedding of the first versal deformation space into the second. Under this embedding, the Artin component maps to deformations to the path algebra of the Kronecker quiver, whereas the ${\Bbb Q}$-Gorenstein component maps to deformations to the $2\times 2$ matrix algebra. In fact, we constructed this embedding for all $2$-dimensional cyclic quotient singularities.

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## Department of Mathematics and Statistics