Hyperelliptic Jacobians and isogenies

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≥4is 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≥4is not isogenous to a different Jacobian.
In the second part we consider a closed subvariety Y⊂Agof the moduli space of principally polarized varieties of dimension g≥3. We show that if a very general element of Yis dominated by the Jacobian of a curve Cand dimY≥2g, then Cis not hyperelliptic. In particular, if the general element in Yis simple, its Kummer variety does not contain rational curves. Finally we show that a closed subvariety Y⊂Mgof dimension 2g−1such that the Jacobian of a very general element of Yis dominated by a hyperelliptic Jacobian is contained either in the hyperelliptic or in the trigonal locus.