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Let $f:X \rightarrow C$ be a flat morphism from a complex variety $X$ to a smooth curve $C$. If any fiber $X_{t}=f^{-1}(t)$ is a reduced normal projective variety, is the total space $X$ a normal variety?

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Yes.

See Lemma 4.1.18 in Qing Liu's book "Algebraic geometry and arithmetic curves".

For your convenience:

Let $R$ be a dvr with field of fractions $K$ and residue field $k$. Let $X$ be an $R$-scheme such that $O_X(U)$ is flat over $R$ for all affine open subschemes $U$ of $X$. Suppose that $X_K$ is normal and that $X_k$ is reduced. Then $X$ is normal.

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