Slope equalities for genus 5 surface fibrations

TL;DR

This paper provides a geometric proof of slope equalities for genus 5 surface fibrations, identifying trigonal fibres as those with positive Horikawa numbers and explicitly computing their weights.

Contribution

It offers a new geometric proof for genus 5 fibrations, characterizes trigonal fibres via Horikawa numbers, and constructs explicit examples with specified invariants.

Findings

Trigonal fibres are exactly those with positive Horikawa numbers.

Explicit computation of weights for special fibres.

Construction of explicit regular surfaces with given invariants.

Abstract

K. Konno proved a slope equality for fibred surfaces with fibres of odd genus and general fibre of maximal gonality. More precisely he found a relation between the invariants of the fibration and certain weights of special fibres (called the Horikawa numbers). We give an alternative and more geometric proof in the case of a genus 5 fibration, under generality assumptions. In our setting we are able to prove that the fibre with positive Horikawa numbers are precisely the trigonal ones, we compute their weights explicitly and thus we exhibit explicit examples of regular surfaces with assigned invariants and Horikawa numbers.