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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 $\textrm{Hilb}_J$ containing the Abel-Jacobi curve as a point. We
relate the result to the ramification (and to the fibres) of the Torelli
morphism $\mathcal M_g\rightarrow \mathcal A_g$ 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.