= Restriction ampleness implies semiampleness for an effective divisor
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 $X$.
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