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

TL;DR

This paper demonstrates that a specific moduli space of stable sheaves on a smooth cubic threefold is smooth, relates it birationally to the theta divisor, and uses this to provide new proofs of Torelli-type theorems for cubic threefolds.

Contribution

It establishes the smoothness of a moduli space of sheaves on a cubic threefold and links it to the theta divisor, offering new proofs of classical Torelli theorems.

Findings

The moduli space ar{M}_X(v) is smooth and four-dimensional.

The Abel-Jacobi map birationally maps the moduli space onto the theta divisor.

New proofs of Torelli theorems using the moduli space and categorical methods.

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 $\operatorname{Ku}(X) \subset \mathrm{D}^{\mathrm{b}}(X)$. 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.