In the classical papers from 1974, Pinkham and Gabriel studied deformations of varieties with a 𝔾m𝐺𝑚-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 44-dimensional algebra ℂ[x,y,z]/(x,y,z)2𝐶[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 ℚ𝑄-Gorenstein component maps to deformations to the 2×22×2 matrix algebra. In fact, we constructed this embedding for all 22-dimensional cyclic quotient singularities.
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