Some Remarks on Fano Three-Folds of Index Two and Stability Conditions

Abstract

AbstractWe 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 ${\operatorname{\mathsf{Ku}}}(X)$, 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 ${\operatorname{\mathsf{Ku}}}(X)$ preserves these stability conditions. As an application, we obtain the smoothness of nonempty moduli spaces of stable objects in ${\operatorname{\mathsf{Ku}}}(X)$. When $X$ is a quartic double solid, we describe a connected component of the stability manifold parametrizing stability conditions on ${\operatorname{\mathsf{Ku}}}(X)$.