On Picard Bundles over Prym Varieties

Let P be the Prym variety associated with a covering π : Y → X between non-singular irreducible projective curves. If P̃ is a principally polarized Prym-Tyurin variety associated with P , we prove that the induced Abel-Prym morphism ρ̃ : Y → P̃ is birational onto its image for genus gX > 2 and deg π 6= 2. We use such result to prove that the Picard bundle over the Prym variety is simple and moreover is stable when ρ̃ is not birational. As a consequence we obtain that the Picard bundle on the moduli space MX(n, ξ) of stable vector bundles with fixed determinant and rank n is simple for gX ≥ 2 and in the case gX = 2 and n = 2 then we prove that the Picard bundle on MX(2, ξ) is stable.