Semiample divisor

A Cartier divisor is semiample if some positive multiple is a basepoint-free divisor, equivalently its divisor line bundle has a globally generated positive power. Such a power defines a Kodaira map $f$ with $\mathcal{O}(mD)=f^{\ast }\mathcal{O}(1)$. Semiampleness implies nefness but need not imply ampleness: a fibre divisor of a morphism to a curve is a basic example.

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.

A semiample divisor on an integral projective variety is ample if it has positive degree on every integral projective curve. Its Kodaira map cannot contract a positive-dimensional fibre, since such a fibre contains a curve and the pulled-back hyperplane bundle has degree zero there. Thus the morphism is proper and a quasi-finite morphism, hence a finite morphism. The finite pullback of an ample line bundle is ample.

If $E$ is an effective Cartier divisor on a projective scheme and ${\mathcal{O}}_E(E)$ is ample, then $E$ is semiample. The divisor restriction exact sequence and Serre vanishing make $H^1(X,(m-1)E)\to H^1(X,mE)$ surjective for large $m$. Their finite dimensions stabilize, so restriction on global sections is eventually surjective. Lift generators on $E$; off $E$, the canonical section of $mE$ generates. Together these generate ${\mathcal{O}}_X(mE)$, including on nonreduced $X$.