The desingularization of the theta divisor of a cubic threefold as a moduli space
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arXiv
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| Auteurs principaux: | , , , , , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866916146859474944 |
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| author | Bayer, Arend Beentjes, Sjoerd Feyzbakhsh, Soheyla Hein, Georg Martinelli, Diletta Rezaee, Fatemeh Schmidt, Benjamin |
| author_facet | Bayer, Arend Beentjes, Sjoerd Feyzbakhsh, Soheyla Hein, Georg Martinelli, Diletta Rezaee, Fatemeh Schmidt, Benjamin |
| contents | We show that the moduli space $\overline{M}_X(v)$ of Gieseker stable sheaves on a smooth cubic threefold $X$ with Chern character $v = (3,-H,-H^2/2,H^3/6)$ 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 $X \subset \overline{M}_X(v)$ 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 $\operatorname{Ku}(X) \subset \mathrm{D}^{\mathrm{b}}(X)$. 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, i.e., that $X$ can be recovered from its intermediate Jacobian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_12240 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The desingularization of the theta divisor of a cubic threefold as a moduli space Bayer, Arend Beentjes, Sjoerd Feyzbakhsh, Soheyla Hein, Georg Martinelli, Diletta Rezaee, Fatemeh Schmidt, Benjamin Algebraic Geometry We show that the moduli space $\overline{M}_X(v)$ of Gieseker stable sheaves on a smooth cubic threefold $X$ with Chern character $v = (3,-H,-H^2/2,H^3/6)$ 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 $X \subset \overline{M}_X(v)$ 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 $\operatorname{Ku}(X) \subset \mathrm{D}^{\mathrm{b}}(X)$. 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, i.e., that $X$ can be recovered from its intermediate Jacobian. |
| title | The desingularization of the theta divisor of a cubic threefold as a moduli space |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2011.12240 |