What is $h^0(\mathcal O_F)$ where $F$ is a fiber of a normal surface over a smooth curve? Lately I am studying the bend-and-break, and I follow the proof in the following note written by Olivier Debarre:
http://www.math.ens.fr/~debarre/M2.pdf
There is a detail that I just cannot go through in the top of page 92, the situation here is: there is a normal surface $S$ and a smooth curve $C$ with a flat morphism $\pi: S \to C$, and we have a fiber $F$ of $\pi$ in $S$ over $C$ which is not integral and has no embedded point. I already know that the genus of the fiber $F$ which is a curve in $S$ is $0$. But I just can't understand why this implies that $h^0(F, \mathcal O_F) = 1$. 
Hope somebody can help me get through this, thanks!
 A: I believe that you misinterpreted the top of page 92 of Debarre's notes.
At that stage Debarre works with a flat morphism $\pi: S\to \overline{T}$, where $S$ is a normal surface and the general fiber of $\pi$ is a smooth rational curve.
He then assumes that every fiber of $\pi$ is integral. Then flatness etc. implies that every fiber of $\pi$ is a smooth rational curve (middle of page 91) and hence $S$ is a ruled surface over $T$. This implies that the two sections $T_0$ and $T_\infty$ have both negative self-intersection and one easily deduces a contradiction from this  (first paragraph of page 92).
This implies that $\pi$ has at least one fiber $F$ which is not integral, hence this fiber $F$ is either a multiple fiber of has several irreducible components, or a combination of this. 
However, $\pi$ is proper and flat, and the generic fiber is connected. This implies that every fiber of $\pi$ is connected. 
(See e.g., Number of irreducible and connected components constant in flat families )
In particular, $F$ is a connected, but it may be nonreduced, and hence $h^0(\mathcal{O}_F)$ can be larger than one.
Using that $\pi$ is flat you get that every fiber is a connected curve with arithmetic genus $0$. In particular, every irreducible component of $F_{red}$ is a rational curve.
Debarre then consider $S''$, a resolution of singularities of $S$. In this way he introduces further rational curves, and obtains that every fiber of $S''\to T$ consists of rational curves only. 
