Suppose $R$ is a localization of a normal closed point of a variety of dimension $n$ over an algebraically closed field $k$ with maximal ideal $\mathfrak{m}$. Suppose also that the associated graded $\operatorname{Gr}_{\mathfrak{m}} R$ is isomorphic to the degree $d$ Veronese subring $S_d$ of $k[X_1,\ldots,X_n]$. Is this enough to conclude that the completion of $R$ is isomorphic to the completed local ring of $S_d$ at the origin? Are there counterexamples?


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