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.



The intersection multiplicity of the canonical divisor of a surface with a fibre of a map to a curve
You use that S is a minimal surface. Thus relatively minimal. Thus, it has no minus one curves. Also, the arithmetic genus is constant in the fibers. The latter is a fairly general fact about fibered surfaces.


