A characterization of semiampleness and contractions of relative curves

Abstract

I give a cohomological characterization of semiample line bundles. The result is a generalization of both the Fujita-Zariski Theorem on semiampleness and the Grothendieck-Serre Criterion for ampleness. As an application of the Fujita-Zariski Theorem I characterize contractible curves in 1-dimensional families.