We show that the moduli space ͞M X(υ) of Gieseker stable sheaves on a smooth cubic threefold X with Chern character υ= (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 ⊂ ͞M X(υ) 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.
Bayer et al. (Mon,) studied this question.