The derived category of a Fano threefold Y of Picard rank 1 and index 2 admits a semiorthogonal decomposition. This defines a non-trivial subcategory Ku(Y) called the Kuznetsov component, which encodes most of the geometry of Y.

I will present a joint work with M. Altavilla and M. Petkovic, in which we describe certain moduli spaces of Bridgeland-stable objects in Ku(Y), via the stability conditions constructed by Bayer, Macrì, Lahoz and Stellari. Furthermore, in our work we study the behavior of the Abel-Jacobi map on these moduli space. As an application in the case of degree d = 2, we prove a strengthening of a categorical Torelli Theorem by Bernardara and Tabuada.