The Hilbert scheme of hyperelliptic Jacobians and moduli of Picard sheaves

Ricolfi, Andrea T.

doi:10.2140/ant.2020.14.1381

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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

DOI: 10.2140/ant.2020.14.1381

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 $ℳ_{g} \to 𝒜_{g}$ 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

Vol. 14, No. 6, 2020