Completion of local ring

Suppose we have a variety $X$ over an algebraically closed char 0 field $K$, and a point $p\in X$. If the completion of the local ring of $X$ at $p$ is isomorphic to $K[[x_1,\ldots,x_n]]/(f)$ for some polynomial $f$, what does this mean for the geometry of $X$ at $p$? Does there exist a nbhd $U$ of $p$ isomorphic to the affine variety defined by $f$?

The completed local ring depends on information which is a lot more "local" than what the Zariski topology sees. For instance, for any smooth point $x$ in a variety $X$ with $\dim_x X = n$, the completed local ring will just be $K[[x_1,\cdots,x_n]]$. So the only chance of nontrivial $f$ would be if the point were singular. Even here, the completed local ring doesn't see much global information.

Consider the following two examples: $V(xy)$ and $V(y^2=x^2(x^2-1)(x-2))$ both as subschemes of $\Bbb A^2_K$. It can be computed that the completed local ring at the origin of both of these is $K[[x,y]]/(xy)$, but there's no isomorphism between any open sets of the two varieties - up to shrinking the open set in $V(xy)$ so that it is only contained in one of the irreducible components, this would correspond to a birational map between curves of different genus, which is clearly nonsense.