Let $(X,0)$ be a normal surface singularity. Suppose that it does not admit a smoothing.


Is it possible that there exists an isolated surface singularity $(Y,0)$ reduced near $0$ which is not irreducible and $(X,0)$ is one of its irreducible components and such that $(Y,0)$ does admit a smoothing? Is there some obstruction for this to happen?


Are these kind of questions treated somewhere?



Smoothings of isolated non-irreducible surface singularities
I will adhere to Mumford's notation as much as possible.
Let $y$ denote the closed point of $Y$, $\mathcal O=\Gamma(Y,\mathcal O\_Y)$, $\widetilde{\mathcal O}=\Gamma(Y-\{y\},\mathcal O\_Y)$ and $Z=\operatorname{Spec}\widetilde{\mathcal O}$, which is the normalization of $Y$ in this situation. Then (Rim) $\widetilde{\mathcal O}$ is local if $Y$
is smoothable, so $Z$ is connected. Then $X=Z$, and then $X=Y$.


