= Solution
Put $L=\mathcal O_X(D)$ and let $s_D$ be the canonical <global section> cutting out the <effective Cartier divisor> $D$. The <divisor restriction exact sequence> gives
$$
0\longrightarrow L^{m-1}\xrightarrow{s_D}L^m\longrightarrow L^m|_D\longrightarrow0.
$$
The hypothesis says $L|_D$ is an <ample line bundle>. By <Serre vanishing>, $H^1(D,L^m|_D)=0$ for all sufficiently large $m$, so the <long exact sequence in sheaf cohomology> makes
$$
H^1(X,L^{m-1})\longrightarrow H^1(X,L^m)
$$
surjective for all such $m$. These are finite-dimensional <vector spaces>. Their dimensions form a nonincreasing sequence of nonnegative <integers>, so the maps are isomorphisms from some point on. Exactness then implies that the restriction
$$
H^0(X,L^m)\longrightarrow H^0(D,L^m|_D)
$$
is surjective for all sufficiently large $m$.
Choose such an $m$ for which $L^m|_D$ is also a <globally generated line bundle>. Lift a generating collection of its <global sections> to $X$. At each point of $D$, one lift has nonzero image in the one-dimensional residue-field fibre, hence generates the stalk of $L^m$ by <Nakayama lemma>. Outside $D$, $s_D^m$ is nowhere zero and generates $L^m$. Together these sections generate it everywhere. Therefore
$$
\boxed{\mathcal O_X(mD)\text{ is globally generated for }m\gg0;\quad D\text{ is semiample}.}
$$
This proof works on an arbitrary <projective scheme> because the defining section of an <effective Cartier divisor> is a <non-zero-divisor>. If $D$ is empty, $L\cong\mathcal O_X$ and the conclusion is immediate. <Semiampleness> is the conclusion: the pullback of a line avoiding the centre of a <blowup of a smooth algebraic surface> of $\mathbb P^2$ satisfies the hypothesis on its support but has zero intersection with the exceptional curve, and therefore is not <ample>.
Back to article page
