The Hilbert scheme of hyperelliptic Jacobians and moduli of Picard sheaves

TL;DR

This paper investigates the scheme structure of the Hilbert scheme component containing a hyperelliptic curve in its Jacobian, relating it to Torelli morphism fibers and describing the moduli of Picard sheaves on hyperelliptic Jacobians.

Contribution

It computes the scheme structure of the Hilbert scheme component of hyperelliptic curves in Jacobians and characterizes the moduli space of Picard sheaves on these Jacobians.

Findings

Determined the scheme structure of the Hilbert scheme component for hyperelliptic curves.

Related the structure to the ramification of the Torelli morphism.

Described the scheme structure of the moduli space 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 $\textrm{Hilb}_J$ 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.