# Hyperelliptic Jacobians and isogenies

## TL;DR

This paper investigates the isogeny relations between hyperelliptic Jacobians and other abelian varieties, proving that very general hyperelliptic Jacobians are not isogenous to non-hyperelliptic ones and exploring the structure of subvarieties dominated by hyperelliptic Jacobians.

## Contribution

It establishes new non-isogeny results for hyperelliptic Jacobians and characterizes subvarieties of moduli spaces dominated by hyperelliptic Jacobians.

## Key findings

- Very general hyperelliptic Jacobians of genus ≥ 4 are not isogenous to non-hyperelliptic Jacobians.
- The intermediate Jacobian of a very general cubic threefold is not isogenous to a Jacobian.
- Subvarieties dominated by hyperelliptic Jacobians have dimension at least 2g, with specific loci characterized.

## Abstract

Motivated by results of Mestre and Voisin, in this note we mainly consider abelian varieties isogenous to hyperelliptic Jacobians In the first part we prove that a very general hyperelliptic Jacobian of genus $g\ge 4$ is not isogenous to a non-hyperelliptic Jacobian. As a consequence we obtain that the Intermediate Jacobian of a very general cubic threefold is not isogenous to a Jacobian. Another corollary tells that the Jacobian of a very general $d$-gonal curve of genus $g \ge 4$ is not isogenous to a different Jacobian. In the second part we consider a closed subvariety $\mathcal Y \subset \mathcal A_g$ of the moduli space of principally polarized varieties of dimension $g\ge 3$. We show that if a very general element of $\mathcal Y$ is dominated by a hyperelliptic Jacobian, then $\dim \mathcal Y\ge 2g$. In particular, if the general element in $\mathcal Y$ is simple, its Kummer variety does not contain rational curves. Finally we show that a closed subvariety $\mathcal Y\subset \mathcal M_g$ of dimension $2g-1$ such that the Jacobian of a very general element of $\mathcal Y$ is dominated by a hyperelliptic Jacobian is contained either in the hyperelliptic or in the trigonal locus.