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

Abstract

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 $\Theta$, contracting only a copy of $X\subset {\overline{M}}_X(v)$ to the singular point $0\in \Theta$. 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 $\mathrm{Ku}(X)\subset {\mathrm{D}}^{\mathrm{b}}(X)$. 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.

26 pages, improvements based on referee comments

References in corpus (2)

• Some remarks on Fano threefolds of index two and stability conditions
• A new model for the theta divisor of the cubic threefold

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• Bridgeland Moduli spaces for Gushel-Mukai threefolds and Kuznetsov's Fano threefold conjecture
• Serre functors and dimensions of residual categories
• Hilbert Scheme of a Pair of Skew Lines on Cubic Threefolds