A characterization of semiampleness and contractions of relative curves

TL;DR

This paper provides a cohomological characterization of semiample line bundles, generalizing key criteria for semiampleness and ampleness, and applies these results to characterize contractible curves in families.

Contribution

It introduces a unified cohomological criterion for semiample line bundles, extending classical theorems and applying them to curve contractions in families.

Findings

Cohomological criterion for semiample line bundles

Generalization of Fujita-Zariski and Grothendieck-Serre theorems

Characterization of contractible curves in families

Abstract

A line bundle with a base-point-free multiple is called semiample. I give a cohomological characterization of semiample line bundles. The result is a common generalization of the Fujita-Zariski criterion for semiampleness and the Grothendieck-Serre characterization of ampleness. Applying the Fujita-Zariski Theorem, I characterize contractible curves in 1-dimensional families.