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Suppose I have a curve $C$ over a field $k$ and that $p$ is a singular point of $C$. Let $f : X \to C$ be the desingularization of $C$ at $p$. Then for each $s \in f^{-1}(p)$ we have a map of local rings $\mathcal O_{C,p} \to \mathcal O_{X,p}$ and also of their completions: $\widehat{\mathcal O_{C,p}} \to \widehat{\mathcal O_{X,s}}$. My question is to which extend the complete local ring $\widehat{\mathcal O_{C,p}}$ determines the desingularization. I.e. is it possible to determine the rings $\widehat{\mathcal O_{X,s}}$ and the maps $\widehat{\mathcal O_{C,p}} \to \widehat{\mathcal O_{X,s}}$ just knowing $\widehat{\mathcal O_{C,p}}$? In particular can one determine the number of points and the fields of definition of the points in $f^{-1}(p)$ just knowing $\widehat{\mathcal O_{C,p}}$

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$X$ is the normalization of $C$; the ring $\prod_{f(s)=p}\widehat{\mathcal{O}_{X,s}}$ is just the normalization of $\widehat{\mathcal{O}_{C,p}}$ (that is, its integral closure in the total ring of fractions). It is completely determined by $\widehat{\mathcal{O}_{C,p}}$.

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