Title: 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.