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.