Some remarks on Fano threefolds of index two and stability conditions

Abstract

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 $\mathsf{Ku}(X)$, 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 threefold, we show that the Serre functor of $\mathsf{Ku}(X)$ preserves these stability conditions. As an application, we obtain the smoothness of non-empty moduli spaces of stable objects in $\mathsf{Ku}(X)$. When $X$ is a quartic double solid, we describe a connected component of the stability manifold parametrizing stability conditions on $\mathsf{Ku}(X)$.

arXiv comment31 pages, 5 figures. Minor modifications according to the referees suggestions. To appear in IMRN

References in corpus (12)

• Markov convexity and local rigidity of distorted metrics
• A generalized Bogomolov-Gieseker inequality for the three-dimensional projective space
• Instanton bundles on Fano threefolds
• Stability conditions on Kuznetsov components
• Stability conditions in families
• Birational models of moduli spaces of coherent sheaves on the projective plane
• Derived categories of cubic and V14 threefolds
• Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai varieties
• A remark on the Torelli theorem for cubic fourfolds
• Moduli spaces on the Kuznetsov component of Fano threefolds of index 2
• Generalized Bogomolov-Gieseker type inequalities on Fano 3-folds
• Stability conditions on product varieties

Cited by in corpus (8)

• The desingularization of the theta divisor of a cubic threefold as a moduli space
• Moduli spaces on the Kuznetsov component of Fano threefolds of index 2
• A note on the Kuznetsov component of the Veronese double cone
• Categorical Torelli theorems for Gushel-Mukai threefolds
• Serre-invariant stability conditions and Ulrich bundles on cubic threefolds
• Bridgeland Moduli spaces for Gushel-Mukai threefolds and Kuznetsov's Fano threefold conjecture
• Bridgeland stability of minimal instanton bundles on Fano threefolds
• Serre functors and dimensions of residual categories