Hello all, I am attempting to understand the proof of Lemma III.8 of Beauville's Complex Algebraic Surfaces:

Let $S$ be a minimal surface, $C$ a smooth curve, $p:S\rightarrow C$ a morphism with generic fibre isomorphic to $\mathbb{P}^1$. Then $S$ is geometrically ruled by $p$.

The proof begins by selecting an arbitrary fibre $F$ of $p$, and observing that $F^2=0$ and $F.K=-2$, where $K$ is the canonical divisor of $S$.

My question is how do we know that $F.K=-2$? The tools that come to mind for deducing this all involve knowing the genus of $F$, which we are trying to calculate.

Thank you for any responses.