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. |