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

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.