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Let $f \colon X \to Y$ be a flat morphism of schemes over $\mathbb{C}$. Suppose that $Y$ is normal and that the fibers over the closed points of $Y$ are all normal.

  1. Can I say something about the fibers over non-closed points?

  2. Is it true that $X$ is normal?

In the local setting, a positive answer to 2 can be found in Matsumura "Commutative ring theory", Theorem 23.9 and its corollary. However, the normality of all the fibers is required.

I am interested, in particular, in the case where the fibers over the closed point are all isomorphic to $\mathbb{P}^1_{\mathbb{C}}$.

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Let's focus on the case where all the closed fibers are $\mathbf{P}^1$. I'm not sure if it's automatic, so I will also assume the generic fibre to be $\mathbf{P}^1$. Then, by flatness, all the fibres are (possible singular) curves over $\mathbf{C}$. The arithmetic genus is constant in the fibres. Also, the morphism is smooth over the set of closed points. Therefore it is smooth everywhere. In particular, all the fibres are $\mathbf{P}^1$. –  Ari Nov 30 '11 at 10:52
    
Thank you Ariyan. I don't understand why you say that all the fibers in this case are schemes over $\mathbb{C}$. –  origal Nov 30 '11 at 11:29
    
Hi origal. I don't understand it either. Please ignore that. It should read, the fibre over $y$ in $Y$ is a (possibly singular) curve over the residue field $k(y)$. (For some reason, this morning, I thought this would always be $\mathbf{C}$.) The conclusion then reads that the fibre over $y$ is $\mathbf{P}^1_{k(y)}$. In view of Laurent Moret-Bailly's answer below, note that we do not use that $Y$ is normal. –  Ari Nov 30 '11 at 13:30
    
I think there is one more problem: what is constant in a flat family is the arithmetic genus, then, in this case $h^0(X_y,\mathcal{O}_{X_y})-h^1(X_y,\mathcal{O}_{X_y})=1$ (with $X_y$ fiber over $y$). But is it true that $h^0(X_y,\mathcal{O}_{X_y})=1$ if the variety $X_y$ is not defined over $\mathbb{C}$? I think we need $k(y)$ to be algebraically closed. –  origal Nov 30 '11 at 15:17
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1 Answer

up vote 8 down vote accepted

The fibers are normal. This follows (without the normality assumption on $Y$) from EGA IV (12.1.6) which says that the set those $x\in X$ where the fiber is normal is open, hence if its complement were nonempty, it would contain a closed point.

Hence, by your remark (or by EGA IV (6.8.3)), $X$ is normal.

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Perfect, thank you! –  origal Nov 30 '11 at 11:36
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