## HYPERELLIPTIC JACOBIANS AND ISOGENIES

Juan Carlos Naranjo
Dept Matemàtiques i Informàtica
Universitat de Barcelona
Abstract:
We will report on a joint work with Gian Pietro Pirola (see arXiv:1705.10154). We mainly will consider abelian varieties isogenous to hyperelliptic Jacobians. In the first part of the talk we will prove that a very general hyperelliptic Jacobian of genus
$g ≥ 4$
is not isogenous to a non-hyperelliptic Jacobian. As a consequence we will obtain that the Intermediate Jacobian of a very general cubic threefold is not isogenous to a Jacobian. Another corollary is that the Jacobian of a very general
$d$
-gonal curve of genus
$g ≥ 4$
is not isogenous to a different Jacobian.
In the second part we will consider a closed subvariety
$Y\subset Ag$
of the moduli space of principally polarized varieties of dimension
$g ≥ 3$
. We will show that if a very general element of
$Y$
is dominated by the Jacobian of a curve
$C$
and
${\rm dim} Y\geq 2g$
, then
$C$
is not hyperelliptic. In particular, if the general element in
$Y$
is simple, its Kummer variety does not contain rational curves. Finally, if time permits, we will show that a closed subvariety
$Y\subset Mg$
of dimension
$2g-1$
such that the Jacobian of a very general element of
$Y$
is dominated by a hyperelliptic Jacobian is contained either in the hyperelliptic or in the trigonal locus.