On the Jacobian locus in the Prym locus and geodesics

Let Jg be the Jacobian locus and let Pg+1 be the Prym locus, in the moduli space Ag of principally polarized abelian varieties of dimension g, for g ≥ 7. We study the extrinsic geometry of Jg Pg+1, under the inclusion provided by the theory of generalized Prym varieties introduced by Beauville. More precisely, we address the problem if certain geodesic curves, defined with respect to the Siegel metric of Ag, starting at a Jacobian variety [JC] ∈ Ag of a curve [C] ∈ Mg and with direction ζ ∈ T[JC]Jg, are locally contained in Pg+1. The result is that for a general JC, the answer is negative, if the rank k of ζ is such that k < Cliff C-3, where Cliff C denotes the Clifford index of C.