= Vanishing-section ampleness criterion
A <Cartier divisor> is <ample> if for every positive-dimensional integral closed <subvariety> some positive multiple restricts to a bundle with a nonzero section having a nonempty zero locus. Inductively the divisor is <ample> on all lower-dimensional subschemes. On an integral component the chosen section cuts out a nonempty <effective Cartier divisor> $E$ whose restriction bundle $\mathcal O_E(E)$ is <ample>. The <restriction ampleness implies semiampleness for an effective divisor> lemma makes the original divisor <semiample>. On a curve the vanishing section forces positive degree, so the <semiample and curve-positive ampleness criterion> proves <ampleness>.
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