Notes on hyperelliptic fibrations of genus 3, I

TL;DR

This paper investigates the structure of hyperelliptic fibrations of genus 3, providing a structure theorem for certain cases, conditions for existence, and examples including minimal regular surfaces with specific invariants.

Contribution

It offers a structure theorem for hyperelliptic genus 3 fibrations with 2-connected fibers and explores existence, uniqueness, and examples, extending prior work.

Findings

Established a structure theorem for hyperelliptic fibrations of genus 3 with 2-connected fibers.

Provided conditions for the existence of such fibrations over the projective line.

Presented examples including minimal regular surfaces with specific invariants.

Abstract

We shall study the structure of hyperelliptic fibrations of genus 3, from the view point given by Catanese and Pignatelli in arXiv:math/0503294. In this part I, we shall give a structure theorem for such fibrations for the case of f : S \to B with all fibers 2-connected. We shall also give, for the case of B projective line, sufficient conditions for the existence of such fibrations from the view point of our structure theorem, prove the uniqueness of the deformation type and the simply connectedness of S for some cases, and give some examples including those with minimal regular S with geometric genus 4 and the first Chern number 8. The last example turns out to be a member of the family M_0 given in Bauer--Pignatelli arXiv:math/0603094.