  In the paper we consider the Jacobian locus $\overline{J_g}$ and the Prym
locus $\overline{P_{g+1}}$, in the moduli space $A_g$ of principally polarized
abelian varieties of dimension $g$, for $g\geq 7$, and we study the extrinsic
geometry of $\overline{J_g}\subset \overline{P_{g+1}}$, under the inclusion
provided by the theory of generalized Prym varieties as introduced by
Beauville. More precisely, we study certain geodesic curves with respect to the
Siegel metric of $A_g$, starting at a Jacobian variety $[JC]\in A_g$ of a curve
$[C]\in M_g$ and with direction $\zeta\in T_{[JC]}J_g$. We prove that for a
general $JC$, any geodesic of this kind is not contained in $\overline{J_g}$
and even in $\overline{P_{g+1}}$, if $\zeta$ has rank $k<\Cliff C-3$, where
$\Cliff C$ denotes the Clifford index of $C$.
