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 D 3; H; 1 2 H 2 ; 1 6 H 3 is smooth and of dimension four.Moreover, the Abel-Jacobi map to the intermediate Jacobian of X maps it birationally onto the theta divisor ‚, contracting only a copy of X M X .v/ to the singular point 0 2 ‚.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 / 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.14D20; 14F08, 14J30, 14J45 1. Introduction 128 2. Cubic threefolds and intermediate Jacobians 131 3. Divisors on hyperplane sections 133 4. Notions of stability 134 5. Construction of sheaves 142 6. Variation of stability 143 7. Proof of the main theorem 150 8. Kuznetsov component