c © 2001 Universitat de Barcelona On Picard bundles over Prym varieties

Let π:Y → X be a covering between non-singular irreducible projective curves. The Jacobian J(Y) has two natural subvarieties, namely, the Prym variety P and the variety π∗(J(X)). We prove that the restriction of the Picard bundle to the subvariety π∗(J(X)) is stable. Moreover, if P ̃ is a principally polarized Prym-Tyurin variety associated with P, we prove that the induced Abel-Prym morphism ρ̃:Y → P ̃ is birational to its image for genus gX> 2 and deg π = 2. We use this result to prove that the Picard bundle over the Prym variety is simple and moreover is stable when ρ ̃ is not birational onto its image