The Hilbert scheme of hyperelliptic Jacobians and moduli of Picard sheaves

Let C be a hyperelliptic curve embedded in its Jacobian J via an Abel-Jacobi map. We compute the scheme structure of the Hilbert scheme component of Hilb(J) containing the Abel-Jacobi embedding as a point. We relate the result to the ramification (and to the fibres) of the Torelli morphism M-g -> A(g) along the hyperelliptic locus. As an application, we determine the scheme structure of the moduli space of Picard sheaves (introduced by Mukai) on a hyperelliptic Jacobian.