  We prove by means of the study of the infinitesimal variation of Hodge
structure and a generalization of the classical Babbage-Enriques-Petri theorem
that the Jacobian variety of a generic element of a $k$ codimensional
subvariety of $\mathcal M_g$ is not isogenous to a distinct Jacobian if
$g>3k+4$. We extend this result to $k=1, g\ge 5$ by using degeneration methods.
