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Let $X\rightarrow Spec(R)$ an irreducible projective scheme over a dvr. Suppose the generic fiber $X_{\eta}$ is smooth (over the field $Frac(R)$) and irreducible. Is it true that $X$ is integral (i.e. reduced)?

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  • $\begingroup$ For smthg related to your question, see Prop 4.3.8 p. 137 in Liu book (this shows eg that if you assume that $X$ is flat over $R$ and $X_\eta$ is integral, then so is $X$). $\endgroup$ – Damian Rössler Aug 3 '15 at 13:08
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Let $R = k[t]$ ($k$ a field) and let $X$ be the scheme which is $\mathbb{P}^{2}_{k[t]}$ with an embedded point lying over $0.$ Then $X$ is irreducible, but not reduced. However, the generic fiber $X_{\eta}$ is $\mathbb{P}^{2}_{k(t)},$ which is smooth over $k(t)$ and irreducible.

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