On the restriction of the moduli part to a reduced divisor

TL;DR

This paper proves that under certain conditions, the restriction of the moduli part to its augmented base locus is semiample for specific fibrations, advancing understanding of the positivity properties of moduli parts in algebraic geometry.

Contribution

It establishes semi-ampleness of the moduli part restricted to its augmented base locus in cases where the base has dimension 2 or dimension 3 with fibers of dimension at most 3.

Findings

Semi-ampleness of the moduli part restriction in specified cases

Extension of semi-ampleness results to new classes of fibrations

Improved understanding of the positivity of moduli parts in algebraic geometry

Abstract

Let $f \colon (X,\Delta) \to Y$ be a fibration such that $K_X + \Delta$ is torsion along the fibres of $f$. Assume that $Y$ has dimension 2, or that $Y$ has dimension 3 and the fibres have dimension at most 3. Then the restriction of the moduli part to its augmented base locus is semiample.