Notes on hyperelliptic fibrations of genus 3, II

TL;DR

This paper develops a structure theorem for smooth deformations of hyperelliptic genus 3 fibrations to non-hyperelliptic ones, and explores the implications for the moduli space of certain minimal surfaces with specific invariants.

Contribution

It provides a new structure theorem for deformations from hyperelliptic to non-hyperelliptic genus 3 fibrations and identifies new moduli space strata for related minimal surfaces.

Findings

Identified conditions for hyperelliptic fibrations to deform into non-hyperelliptic fibrations.

Discovered two new moduli space strata of dimensions 32 and 30.

Located boundary relations between new strata and previously known strata.

Abstract

This is the part II of the series under the same title. In part I, using the approach developed by Catanese--Pignatelli arXiv:math/0503294, we gave a structure theorem for hyperelliptic genus 3 fibrations all of whose fibers are 2-connected (arXiv:1209.6278 [math.AG]). In this part II, we shall give a structure theorem for smooth deformation families of these to non-hyperelliptic genus 3 fibrations. As an application, we shall give a set of sufficient conditions for our genus 3 hyperelliptic fibration above to allow deformation to non-hyperelliptic fibrations, and use this to study certain minimal regular surfaces with first Chern number 8 and geometric genus 4: we shall find in the moduli space two strata M_0^{sharp} and M_0^{flat}$ (each of dimensions 32 and 30, respectively), and show that Bauer--Pingatelli's stratum M_0 (arXiv:math/0603094) and its 26-dimensional substratum are at the boundary of these new strata M_0^{sharp} and M_0^{flat}, respectively.