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

Research output: Working paper/Preprint › Preprint

Authors

  • Arend Bayer
  • Sjoerd Beentjes
  • Soheyla Feyzbakhsh
  • Georg Hein
  • Diletta Martinelli
  • Fatemeh Rezaee
  • Benjamin Schmidt

Research Organisations

External Research Organisations

  • University of Edinburgh
  • Imperial College London
  • University of Duisburg-Essen
  • University of Amsterdam
View graph of relations

Details

Original languageEnglish
Number of pages25
Publication statusE-pub ahead of print - 24 Nov 2020

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.

Cite this

The desingularization of the theta divisor of a cubic threefold as a moduli space. / Bayer, Arend; Beentjes, Sjoerd; Feyzbakhsh, Soheyla et al.
2020.

Research output: Working paper/Preprint › Preprint

Bayer, A, Beentjes, S, Feyzbakhsh, S, Hein, G, Martinelli, D, Rezaee, F & Schmidt, B 2020 'The desingularization of the theta divisor of a cubic threefold as a moduli space'. <https://arxiv.org/abs/2011.12240>
Bayer, A., Beentjes, S., Feyzbakhsh, S., Hein, G., Martinelli, D., Rezaee, F., & Schmidt, B. (2020). The desingularization of the theta divisor of a cubic threefold as a moduli space. Advance online publication. https://arxiv.org/abs/2011.12240
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. 2020 Nov 24. Epub 2020 Nov 24.
Bayer, Arend ; Beentjes, Sjoerd ; Feyzbakhsh, Soheyla et al. / The desingularization of the theta divisor of a cubic threefold as a moduli space. 2020.
Download
@techreport{5f522b8ad394471ca9854fa4141c5c96,
title = "The desingularization of the theta divisor of a cubic threefold as a moduli space",
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. ",
author = "Arend Bayer and Sjoerd Beentjes and Soheyla Feyzbakhsh and Georg Hein and Diletta Martinelli and Fatemeh Rezaee and Benjamin Schmidt",
note = "25 pages",
year = "2020",
month = nov,
day = "24",
language = "English",
type = "WorkingPaper",

}

Download

TY - UNPB

T1 - The desingularization of the theta divisor of a cubic threefold as a moduli space

AU - Bayer, Arend

AU - Beentjes, Sjoerd

AU - Feyzbakhsh, Soheyla

AU - Hein, Georg

AU - Martinelli, Diletta

AU - Rezaee, Fatemeh

AU - Schmidt, Benjamin

N1 - 25 pages

PY - 2020/11/24

Y1 - 2020/11/24

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.

AB - 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.

M3 - Preprint

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

ER -