Locally non-trivial fibred surfaces with maximal unitary rank

TL;DR

This paper investigates fibred surfaces with maximal unitary rank, establishing slope inequalities and constraints, especially for genus 6, to deepen understanding of their geometric properties and limitations.

Contribution

It provides new slope inequalities for extremal fibred surfaces with maximal unitary rank and applies these to constrain the genus 6 case.

Findings

Proves a strong slope inequality for extremal fibred surfaces.

Shows the index of such surfaces is always positive.

Imposes restrictions on the classes of the relative canonical divisor.

Abstract

Let $f\colon S\to B$ a locally non-trivial fibred surface with fibres of genus $g$. Let $u_f$ be its unitary rank, i.e. the rank of the flat unitary part in the second Fujita decomposition. We study in detail the case when $u_f$ is maximal, i.e. $u_f=g-1$. In this case necessarily $g\leq 6$, but examples in genus $5$ and $6$ are not known, and conjecturally do not exist. We prove a strong slope inequality for these extremal cases. We then use this inequality, together with results on trigonal curves, to give new constraints on the case $g=6$, $u_f=5$. In particular, we prove that the index of the surface is always strictly positive and give strong limitations on the possible classes of the relative canonical divisor