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

Abstract

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, −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 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) ⊂ 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, i.e., that X can be recovered from its intermediate Jacobian.

A refined derived Torelli theorem for Enriques surfaces

“…As Hom(E, E) = 1≤i≤d Hom(F i , F i ), assumption (b) implies that E ∼ = F i , for a unique i. Thus E is in L i and we get (1).…”

“…ψ − → Hom(F, L i 1 ) is an isomorphism. Since Hom(F [1], L i 1 ) = 0 and Hom(L i 1 , L i 1 [1]) = 0, we get Hom(C, L i 1 ) = 0 and Hom(C, L i 1 [1]) = 0. If we apply RHom(C, −) to (2.14), we get the isomorphisms Hom(C, C) ∼ = Hom(C,…”

“…To proceed in this direction, we first observe that, if we apply RHom(−, F [1]) to (2.14) and we use Serre duality, we get…”

“…Such a component is meant to play the role of intermediate Jacobians or middle cohomologies and the additional data is the analogue of principal polarizations or special Hodge structures. In the case of Fano manifolds, many papers were devoted to this kind of problems including [1,2,3,14,23,26]. All these results heavily rely on the existence of Bridgeland stability conditions on the Kuznetsov component as proved in [2].…”

Stability conditions on Kuznetsov components of Gushel-Mukai threefolds and Serre functor

“…Using this result, the authors further proved that non-empty moduli spaces of stable objects with respect to these stability conditions are smooth. They also gave another proof of the categorical Torelli Theorem for cubic threefolds in [PY20, Theorem 5.17], following the strategy in [BMMS12, Theorem 1.1] where this result was proved for the first time (see also [BBF `20] for a different approach).…”