Monodromy of Kodaira Fibrations of Genus $3$

TL;DR

This paper studies the monodromy groups of genus 3 Kodaira fibrations, classifying which groups can occur when these fibrations are realized as complete intersection curves in specific subvarieties of the moduli space.

Contribution

It provides a classification of possible monodromy groups for genus 3 Kodaira fibrations arising from certain geometric constructions.

Findings

Classified the connected monodromy groups for genus 3 Kodaira fibrations.

Identified which monodromy groups can arise from fibrations with Jacobians having extra endomorphisms.

Connected monodromy groups are constrained by the geometry of the moduli space and endomorphism conditions.

Abstract

A Kodaira fibration is a non-isotrivial fibration $f\colon S\rightarrow B$ from a smooth algebraic surface $S$ to a smooth algebraic curve $B$ such that all fibers are smooth algebraic curves of genus $g$. Such fibrations arise as complete curves inside the moduli space $\mathcal{M}_g$ of genus $g$ algebraic curves. We investigate here the possible connected monodromy groups of a Kodaira fibration in the case $g=3$ and classify which such groups can arise from a Kodaira fibration obtained as a general complete intersection curve inside a subvariety of $\mathcal{M}_3$ parametrizing curves whose Jacobians have extra endomorphisms.