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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 !

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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 $D.F=0.$

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    $\begingroup$ oh... blessed Adjuction! (or genus formula in this case). Thank you Andrew $\endgroup$ – Heitor Fontana May 17 '13 at 21:15
  • $\begingroup$ By the way, in this example can we then conclude $D^2=0$ ? Also, can we say $D=\sum m_i F_i$ with $m_i>0$ where the $F_i$ are fibers of $p$ ? $\endgroup$ – Heitor Fontana May 18 '13 at 9:19
  • $\begingroup$ Dear @Heitor, the first question seems to be true, via Beauville Prop IX.3, for example, but the same proposition only claims the second for Kodaira dimension one, i.e., for "honest" elliptic surfaces and not bi-elliptic surfaces. $\endgroup$ – Andrew May 21 '13 at 16:03

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