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

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Autorschaft

  • Arend Bayer
  • Sjoerd Beentjes
  • Soheyla Feyzbakhsh
  • Georg Hein
  • Benjamin Schmidt

Organisationseinheiten

Externe Organisationen

  • University of Edinburgh
  • Imperial College London
  • Universität Duisburg-Essen
  • Universiteit van Amsterdam (UvA)
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  • Citations
    • Citation Indexes: 2
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Details

OriginalspracheEnglisch
Seiten (von - bis)127–160
Seitenumfang34
FachzeitschriftGeometry and Topology
Jahrgang28
Ausgabenummer1
PublikationsstatusVeröffentlicht - 27 Feb. 2024

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,−H2/2,H3/6) 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)⊂Db(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, i.e., that X can be recovered from its intermediate Jacobian.

ASJC Scopus Sachgebiete

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The desingularization of the theta divisor of a cubic threefold as a moduli space. / Bayer, Arend; Beentjes, Sjoerd; Feyzbakhsh, Soheyla et al.
in: Geometry and Topology, Jahrgang 28, Nr. 1, 27.02.2024, S. 127–160.

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Bayer, A, Beentjes, S, Feyzbakhsh, S, Hein, G, Martinelli, D, Rezaee, F & Schmidt, B 2024, 'The desingularization of the theta divisor of a cubic threefold as a moduli space', Geometry and Topology, Jg. 28, Nr. 1, S. 127–160. https://doi.org/10.2140/gt.2024.28.127
Bayer, A., Beentjes, S., Feyzbakhsh, S., Hein, G., Martinelli, D., Rezaee, F., & Schmidt, B. (2024). The desingularization of the theta divisor of a cubic threefold as a moduli space. Geometry and Topology, 28(1), 127–160. https://doi.org/10.2140/gt.2024.28.127
Bayer A, Beentjes S, Feyzbakhsh S, Hein G, Martinelli D, Rezaee F et al. The desingularization of the theta divisor of a cubic threefold as a moduli space. Geometry and Topology. 2024 Feb 27;28(1):127–160. doi: 10.2140/gt.2024.28.127
Bayer, Arend ; Beentjes, Sjoerd ; Feyzbakhsh, Soheyla et al. / The desingularization of the theta divisor of a cubic threefold as a moduli space. in: Geometry and Topology. 2024 ; Jahrgang 28, Nr. 1. S. 127–160.
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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,−H2/2,H3/6) 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)⊂Db(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, i.e., that X can be recovered from its intermediate Jacobian. ",
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AU - Bayer, Arend

AU - Beentjes, Sjoerd

AU - Feyzbakhsh, Soheyla

AU - Hein, Georg

AU - Martinelli, Diletta

AU - Rezaee, Fatemeh

AU - Schmidt, Benjamin

N1 - Publisher Copyright: © 2024 Mathematical Sciences Publishers.

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Y1 - 2024/2/27

N2 - 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,−H2/2,H3/6) 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)⊂Db(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, i.e., that X can be recovered from its intermediate Jacobian.

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