On (2,2)-decomposable genus 4 Jacobians

TL;DR

This paper characterizes when genus 4 Jacobians are (2,2)-decomposable, linking them to Prym varieties and Igusa quartic geometry, and explores their isogenies to genus 2 Jacobians, revealing both expected and higher-dimensional families.

Contribution

It provides a geometric and algebraic description of (2,2)-decomposable genus 4 Jacobians, connecting Prym varieties, Igusa quartic, and isogenies, with new insights into their moduli.

Findings

Characterization of (2,2)-decomposable genus 4 Jacobians via Prym varieties.

Description of the locus in terms of Igusa quartic geometry.

Identification of a high-dimensional family of hyperelliptic genus 4 Jacobians.

Abstract

We consider the question of when a Jacobian of a curve of genus $2g$ admits a $(2,2)$-isogeny to two polarized dimension $g$ abelian varieties. We find that one of them must be a Jacobian itself and, if the associated curve is hyperelliptic, so is the other. For $g=2$ this allows us to describe $(2,2)$-decomposable genus $4$ Jacobians in terms of Prym varieties. We describe the locus of such genus $4$ curves in terms of the geometry of the Igusa quartic threefold. We also explain how our characterization relates to Prym varieties of unramified double covers of plane quartic curves, and we describe this Prym map in terms of $6$ and $7$ points in $\mathbb{P}^3$. We also investigate which genus $4$ Jacobians admit a $2$-isogeny to the square of a genus $2$ Jacobian and give a full description of the hyperelliptic ones. While most of the families we find are of the expected dimension $1$, we also find a family of unexpectedly high dimension~$2$.