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

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

doi:10.2140/gt.2024.28.127

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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

DOI: 10.2140/gt.2024.28.127

Abstract

We show that the moduli space ${\bar{M}}_{X} (v)$ of Gieseker stable sheaves on a smooth cubic threefold $X$ with Chern character $v = (3, - H, - \frac{1}{2} H^{2}, \frac{1}{6} H^{3})$ 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 $X \subset {\bar{M}}_{X} (v)$ to the singular point $0 \in Θ$ .

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 $Ku (X) \subset D^{b} (X)$ . 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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Volume 28, issue 1 (2024)