Ground field $\Bbb{C}$. Algebraic category. Smooth surfaces.
Let $S$ be a minimal elliptic surface $p:S\rightarrow C$ the elliptic fibration (general fiber = elliptic curve). Suppose the $m$-canonical system is non-empty and let $D\in \lvert m K \rvert$.
Why can we say that $D.F=0$, where $F$ is a fiber of $p$ ?
It appears to be very obvious, but not for me. Can't it be positive?
Enlighten me, s'il vous plait !
Recall that by adjunction, $2g-2=F^2+K.F$ where $g$ is the genus of $F.$ But in this case, $g=1,$ and $F^2=0,$ so we end up with $F.K=0.$ This implies in particular that any $D\sim mK$ also satisfies $D.F=0.$