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.



Are Du Val singularities smoothable?
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](http://dx.doi.org/10.1007/s002220000101), *Invent. Math*. **143**, No. 1, 29-76 (2001). [ZBL1060.14520](https://zbmath.org/?q=an:1060.14520), 


where they are called *rational double points*.


