I'm reading Beauville's book "Complex Algebraic Surfaces" and I have a question about Step 3 of the proof of Proposition VI.2:
Suppose we have a minimal surface $S$ with $K_S^2<0$ and Albanese map $p:S\to B$ with connected fibers to a smooth curve $B$. Suppose that we also have an irreducible curve $C$ on $S$ such that $C\cdot K_S<-1$ and $|C+K_S|=\emptyset$. Moreover, $C$ is the section of $p$. Then by Riemann-Roch we conclude that $h^0(C)\ge 2$.
Then Beauville says that: $C$ moves in its linear equivalence class. Let $F$ be a generic fiber of $p$; then the point $C\cap F$ moves linearly on $F$ and so $F$ must be rational.
I do not understand what does he mean by "$C\cap F$ moves linearly on $F$" and how does it give us that "$F$ must be rational"?
Thank you in advance.
If you vary $C$ in a pencil, it cuts out a pencil of degree $1$ divisors on $F$, and this pencil induces a degree $1$ map $F \to \mathbb P^1$. Such maps are necessarily isomorphisms.