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)?

## 1 Answer

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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.