Some remarks on Fano threefolds of index two and stability conditions

We prove that ideal sheaves of lines in a Fano three-fold X of Picard rank one and index two are stable objects in the Kuznetsov component Ku(X)⁠, with respect to the stability conditions constructed by Bayer, Lahoz, Macrì, and Stellari, giving a modular description to the Hilbert scheme of lines in X⁠. When X is a cubic three-fold, we show that the Serre functor of Ku(X) preserves these stability conditions. As an application, we obtain the smoothness of nonempty moduli spaces of stable objects in Ku(X)⁠. When X is a quartic double solid, we describe a connected component of the stability manifold parametrizing stability conditions on Ku(X)⁠.