I am reading Beauville's chapter IX on Elliptic surfaces.

Let $S$ be a minimal elliptic surface with $\kappa=1$ and $p:S\rightarrow C$ be the *elliptic fibration*.

We know $K^2=0$. Suppose the $m$-canonical system is non-empty and let $D\in \lvert m K \rvert$.

Then $D.F=0$, where $F$ is a fiber of $p$, by the genus formula. So the components of $D$ are contained in the fibers. I don't get the following:

Since $K^2=0$ we have $D=\sum r_i F_i \quad $ with $r_i\in\Bbb{Q}$ and $F_i$ fibers.

(he says this comes from Proposition VII.4, but this is probably a misprint, since that proposition has nothing to do with this situation. I thought it could follow from VIII.3 or VIII.4, but it doesn't seem any obvious to me -actually it seems that we'd get $D=r F$, which he does not use).

Sigh... I don't see why this holds. Any hint gentlemen? Thank you.