Some remarks on Fano threefolds of index two and stability conditions

We prove that ideal sheaves of lines in a Fano threefold $X$ of Picard rank one and index two are stable objects in the Kuznetsov component $\mathsf{K}\mathsf{u}(X)$, with respect to the stability conditions constructed by Bayer, Lahoz, Macr`i and Stellari, giving a modular description to the Hilbert scheme of lines in $X$. When $X$ is a cubic threefold, we show that the Serre functor of $\mathsf{K}\mathsf{u}(X)$ preserves these stability conditions. As an application, we obtain the smoothness of non-empty moduli spaces of stable objects in $\mathsf{K}\mathsf{u}(X)$. When $X$ is a quartic double solid, we describe a connected component of the stability manifold parametrizing stability conditions on $\mathsf{K}\mathsf{u}(X)$.