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 HilbJ containing the Abel–Jacobi embedding as a point. We relate the result to the ramification (and to the fibres) of the Torelli morphism Mg → Ag 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.