Slope equality of non-hyperelliptic Eisenbud--Harris special fibrations of genus $4$

TL;DR

This paper introduces the Horikawa index and local signature for fibered surfaces with non-hyperelliptic genus 4 curves, providing new tools for understanding their geometric properties.

Contribution

It develops the Horikawa index and local signature specifically for non-hyperelliptic genus 4 fibrations with a unique trigonal structure, advancing the study of their slope equality.

Findings

Defined the Horikawa index for these fibrations

Established the local signature for non-hyperelliptic genus 4 surfaces

Provided insights into slope equality for special fibrations

Abstract

The Horikawa index and the local signature are introduced for relatively minimal fibered surfaces whose general fiber is a non-hyperelliptic curve of genus $4$ with unique trigonal structure.