On the slope of hyperelliptic fibrations with positive relative irregularity

TL;DR

This paper establishes a sharp lower bound on the slope of hyperelliptic fibrations with positive relative irregularity, confirming conjectures related to the slope's behavior and the ampleness of the direct image sheaf.

Contribution

It provides a precise lower bound on the slope for hyperelliptic fibrations, proving conjectures by Barja, Stoppino, and Xiao regarding slope bounds and sheaf ampleness.

Findings

Proved a sharp lower bound on the slope  for hyperelliptic fibrations.

Confirmed that the slope  is an increasing function of the relative irregularity q_f.

Established the ampleness of the direct image of the relative canonical sheaf when <4.

Abstract

Let $f:\, S \to B$ be a locally non-trivial relatively minimal fibration of hyperelliptic curves of genus $g\geq 2$ with relative irregularity $q_f$. We show a sharp lower bound on the slope $\lambda_f$ of $f$. As a consequence, we prove a conjecture of Barja and Stoppino on the lower bound of $\lambda_f$ as an increasing function of $q_f$ in this case, and we also prove a conjecture of Xiao on the ampleness of the direct image of the relative canonical sheaf if $\lambda_f<4$.