Smoothing cones over K3 surfaces
  We prove that the affine cone over a general primitively polarised K3 surface of genus g is smoothable if and only if g\le 10 or g=12. We also give several examples of singularities with special behaviour, such as surfaces whose affine cone is smoothable even though the projective cone is not. 
Stephen Coughlan, Taro Sano
math.AG