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 $\mathbb{P}^1\_{\mathbb{C}}$.



Flat family of normal schemes over a normal base
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. 


