Let $(X,0)\subset(\mathbb{C}^N,0)$ be the (formal) germ of a singular space (isolated singularity). Let $\mathbb{P}T_{(X,0)}\subset\mathbb{P}^{N-1}$ be its projectivized tangent cone (considered as a subscheme). Quite often $\mathbb{P}T_{(X,0)}$ is itself singular. Suppose $\mathbb{P}T_{(X,0)}$ is smoothable, i.e. there exists a flat embedded family $\{Y_\epsilon\}_\epsilon$ of subvarieties of $\mathbb{P}^{N-1}$ whose central fibre coincides with $\mathbb{P}T_{(X,0)}$, while the generic fibre is smooth.
Then $Cone(\mathbb{P}T_{(X,0)})$ deforms naturally to $Cone(Y_\epsilon)$. Does $(X,0)$ deform to some germ whose tangent cone is $Cone(Y_\epsilon)$? This certainly happens for hypersurfaces, more generally for locally complete intersections. Does it hold for Cohen-Macaulay singularities? At least for Gorenstein types? Or, e.g. in the case when $\mathbb{P}T_{(X,0)}\subset\mathbb{P}^{N-1}$ is projectively normal?
The same can be stated in com.alg. language: given a local (non-regular) ring $(R,m)$ and the associated graded ring $gr_m R$. Are the deformations of $gr_m R$ that preserve the multiplicity induced from those of $R$?
(Probably this is some standard question in com.alg/deformation theory...)
