On the Jacobian locus in the Prym locus and geodesics

Abstract

In the paper we consider the Jacobian locus Jg and the Prym locus Pg+1, in the moduli space Ag of principally polarized abelian varieties of dimension g, for g ≥ 7, and we study the extrinsic geometry of Jg ⊂ Pg+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 Ag, starting at a Jacobian variety [JC] ∈ Ag of a curve [C] ∈ Mg and with direction ζ ∈ T [J C] Jg. We prove that for a general JC, any geodesic of this kind is not contained in Jg and even in Pg+1, if ζ has rank k < Cliff C − 3, where Cliff C denotes the Clifford index of C.