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I am interested in when a Du Val surface singularity is smoothable. By Du Val singularity, I mean (the germ of a) isolated double point surface singularity admitting a resolution by blowups of isolated double points over the original. By smoothable, I mean when does there exist a flat family containing the singularity as one of the fibers, and a smooth germ as another fiber.

Any solution/reference would be appreciated.

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Du Val singularities are indeed smoothable and, in fact, more is true: they are of class $T$, namely, they are quotient singularities admitting a $1$-parameter $\mathbb{Q}$-Gorenstein smoothing.

See Definition 1.1 and Proposition 1.2 in the paper

M. Manetti: On the moduli space of diffeomorphic algebraic surfaces, Invent. Math. 143, No. 1, 29-76 (2001). ZBL1060.14520,

where they are called rational double points.

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Du Val singularities are hypersurface singularities, hence they can be smoothed --- just replace the defining equation $F(x,y,z) = 0$ by the equation $F(x,y,z) = \epsilon$.

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