Hyperelliptic Jacobians and isogenies

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.

New version. Accepted in Adavances in Mathematics