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.