= Semiampleness of a square-zero rational curve
{title2=$C^2=0\Longrightarrow\kappa(C)=1$}
A <smooth rational curve> $C$ with $C^2=0$ on a <smooth projective surface> is <semiample> and has <Iitaka dimension> one. Its normal bundle is trivial, so the restriction sequences for $mC$ have quotient $\mathcal O_C$ and zero quotient $H^1$. The finite dimensions of $H^1(X,mC)$ decrease and stabilize. Restriction of sections to $C$ is then surjective; a lift of $1$ and the canonical section generate globally. Exactness also gives $h^0(X,mC)-h^0(X,(m-1)C)=1$ eventually.
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