On Fujita's semi-ampleness in the rank one case

TL;DR

This paper proves Fujita's semi-ampleness conjecture for rank one cases, confirming its validity in this specific scenario despite counterexamples in higher ranks.

Contribution

It establishes the truth of Fujita's semi-ampleness conjecture for rank one direct summands, clarifying its limitations in higher ranks.

Findings

Fujita's conjecture holds in rank one cases.

Counterexamples exist in higher rank cases.

The result clarifies the scope of the conjecture.

Abstract

We prove that a conjecture of Fujita on the semi-ampleness is true in the case of rank one direct summand, though it is wrong in higher rank case by Catanese and Dettweiler.