The Hilbert scheme of hyperelliptic Jacobians and moduli of Picard sheaves

Abstract

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 ${\mathcal{ℳ}}_g\to {\mathcal{𝒜}}_g$ 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.