Fundamental properties of basic slc-trivial fibrations, II

TL;DR

This paper proves that under certain conditions, the moduli divisor in basic slc-trivial fibrations is linearly trivial and semi-ample when the base is a curve, advancing understanding of their geometric properties.

Contribution

It establishes the linear triviality of the moduli divisor when it is b-numerically trivial and proves semi-ampleness of the moduli part over a curve, extending previous results.

Findings

Moduli $Q$-b-divisor is $Q$-b-linearly trivial if b-numerically trivial.

Moduli part is semi-ample when base is a curve.

Advances understanding of properties of basic slc-trivial fibrations.

Abstract

We prove that if the moduli $\mathbb Q$-b-divisor of a basic slc-trivial fibration is b-numerically trivial then it is $\mathbb Q$-b-linearly trivial. As a consequence, we prove that the moduli part of a basic slc-trivial fibration is semi-ample when the base space is a curve.