The desingularization of the theta divisor of a cubic threefold as a moduli space
Abstract
We show that the moduli space ¯Mₓ(v) of Gieseker stable sheaves on a smooth cubic threefold X with Chern character v = (3, − H, − 1/2 H², 1/6 H³) 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 ⊂ ¯Mₓ (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, ie that X can be recovered from its intermediate Jacobian. Show more
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https://doi.org/10.3929/ethz-b-000668704Publication status
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Journal / series
Geometry & TopologyVolume
Pages / Article No.
Publisher
Mathematical Sciences PublishersSubject
cubic threefolds; derived categories; stability conditionsMore
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