General hyperplane sections of a Fano threefold $Y$ of index 2 and Picard rank 1 are del Pezzo surfaces, and their Picard group is related to a root system. To the corresponding roots, we associate objects in the Kuznetsov component of $Y$ and investigate their moduli spaces, using the stability condition constructed by Bayer, Lahoz, Macr\`i, and Stellari, and the Abel--Jacobi map. We identify a subvariety of the moduli space isomorphic to $Y$ itself, and as an application we prove a (refined) categorical Torelli theorem for general quartic double solids.

This note describes moduli spaces of objects in the Kuznetsov component of a Veronese double cone, and some related constructions. We consider classes in the numerical Grothendieck group which are minimal with respect to the Euler form, and show that the corresponding moduli spaces are isomorphic. They are a blow-down of a moduli of tilt-stable objects. We interpret the latter space as a generalized Hilbert scheme of lines and give a wall-crossing interpretation of the blow-down morphism.

No abstract