Vol. 14, No. 6, 2020

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The Hilbert scheme of hyperelliptic Jacobians and moduli of Picard sheaves

Andrea T. Ricolfi

Vol. 14 (2020), No. 6, 1381–1397
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 ${\mathsc{ℳ}}_{g}\to {\mathsc{𝒜}}_{g}$ 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.

Keywords
Jacobian, Torelli morphism, Hilbert schemes, Picard sheaves, Fourier–Mukai transform
Mathematical Subject Classification 2010
Primary: 14C05
Secondary: 14H40, 14K10
Milestones
Received: 14 September 2018
Revised: 31 December 2019
Accepted: 10 February 2020
Published: 30 July 2020
Authors
 Andrea T. Ricolfi Max Planck Institut für Mathematik Bonn Germany Scuola Internazionale Superiore di Studi Avanzati Trieste Italy