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The desingularization of the theta divisor of a cubic threefold as a moduli space

Arend Bayer, Sjoerd Viktor Beentjes, Soheyla Feyzbakhsh, Georg Hein, Diletta Martinelli, Fatemeh Rezaee and Benjamin Schmidt

Geometry & Topology 28 (2024) 127–160
Abstract

We show that the moduli space M¯X(v) of Gieseker stable sheaves on a smooth cubic threefold X with Chern character v =(3,H,1 2H2, 1 6H3) is smooth and of dimension four. Moreover, the Abel–Jacobi map to the intermediate Jacobian of X maps it birationally onto the theta divisor Θ, contracting only a copy of X M¯X(v) to the singular point 0 Θ.

We use this result to give a new proof of a categorical version of the Torelli theorem for cubic threefolds, which says that X can be recovered from its Kuznetsov component Ku (X) D b (X). Similarly, this leads to a new proof of the description of the singularity of the theta divisor, and thus of the classical Torelli theorem for cubic threefolds, ie that X can be recovered from its intermediate Jacobian.

Keywords
cubic threefolds, derived categories, stability conditions
Mathematical Subject Classification
Primary: 14D20
Secondary: 14F08, 14J30, 14J45
References
Publication
Received: 21 January 2021
Revised: 19 February 2022
Accepted: 19 March 2022
Published: 27 February 2024
Proposed: Mark Gross
Seconded: Richard P Thomas, Jim Bryan
Authors
Arend Bayer
School of Mathematics and Maxwell Institute
University of Edinburgh
Edinburgh
United Kingdom
Sjoerd Viktor Beentjes
School of Mathematics and Maxwell Institute
University of Edinburgh
Edinburgh
United Kingdom
Soheyla Feyzbakhsh
Department of Mathematics
Imperial College London
London
United Kingdom
Georg Hein
Fakultät für Mathematik
Universität Duisburg-Essen
Essen
Germany
Diletta Martinelli
Korteweg-de Vries Institute for Mathematics
University of Amsterdam
Amsterdam
Netherlands
Fatemeh Rezaee
Centre for Mathematical Sciences
University of Cambridge
Cambridge
United Kingdom
ETH Zürich
Zürich
Switzerland
Benjamin Schmidt
Institut für Algebraische Geometrie
Gottfried Wilhelm Leibniz Universität Hannover
Hannover
Germany

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