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 ${\text{Hilb}}_J$ containing the Abel-Jacobi curve as a point. We relate the result to the ramification (and to the fibres) of the Torelli morphism ${\mathcal{M}}_g\to {\mathcal{A}}_g$ 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.

Improved the exposition according to the referees' suggestions. To appear in Algebra and Number Theory

References in corpus (2)

• The DT/PT correspondence for smooth curves
• Local contributions to Donaldson-Thomas invariants