In this question, all varieties are supposed to be over an algebraically closed field $k$.
Hypothesis: X is a smooth projective surface and $f:X\longrightarrow \mathbb P^1$ is a morphism with we following properties (maybe some conditions are redundant but for completeness I write the complete list):
- $f$ is flat, proper and has a section.
- There is an open dense subset $U \subseteq\mathbb P^1 $ such that the fiber $X_u$ is a smooth projective curve (i.e. integral, separated scheme of finite type) for every $u\in U$.
- All fibers are irreducible (and hence connected).
- The singular fibers can have only one node as singularities (multiple nodes are not allowed)
I'd like to show (if true) that all fibers are reduced. Pactically it remains to show that the singular fibers are reduced.