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,−1/2 H², 1/6H³)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 versio n of the Torelli theorem for cubic threefolds, which says that X can be recovered from its Kuznetsov component Ku(X) ⊂ D^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, ie that X can be recovered from its intermediate Jacobian.