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Let $\mathcal M$ be a (reduced) quasi-projective scheme over a DVR (of mixed caracteristics), $R$. Suppose that the generic fiber $\mathcal M_{\eta_R}$ is (nonempty) smooth and irreducible of dimension $n>0$ and that the special fiber $\mathcal M_{x}$ is (nonempty) smooth of dimension $n$ also. Can one find an irreducible component $\mathcal M'\subset \mathcal M$ such that $\mathcal M'_{\eta_R}$ contains an open subscheme of $\mathcal M_{\eta_R}$ and $\mathcal M'_x$ contains also an open subscheme of $\mathcal M_x$?

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Not necessarily.

E.g. let $\mathcal M_{\eta}$ be a curve over $K$ (the fraction field of $R$), let $\mathcal M_x$ be a curve over $k$ (the residue field of $R$), and let $\mathcal M$ be the disjoint union of $\mathcal M_{\eta}$ and $\mathcal M_x$.

You'll need an extra flatness or properness assumption to get what you're asking for.

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