Theta-duality on Prym varieties and a Torelli theorem

Let π : C ~ → C π : C → C be an unramified double covering of irreducible smooth curves and let P P be the attached Prym variety. We prove the scheme-theoretic theta-dual equalities in the Prym variety T ( C ~ ) = V 2 T( C)=V^2 and T ( V 2 ) = C ~ T(V^2)= C , where V 2 V^2 is the Brill-Noether locus of P P associated to π π considered by Welters. As an application we prove a Torelli theorem analogous to the fact that the symmetric product D ( g ) D⁽ᵍ⁾ of a curve D D of genus g g determines the curve.