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

Abstract

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 ${\operatorname{\mathsf{Ku}}}(X)$, 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 ${\operatorname{\mathsf{Ku}}}(X)$.

A refined derived Torelli theorem for Enriques surfaces

“…Such a component is meant to play the role of intermediate Jacobians or middle cohomologies and the additional data is the analogue of principal polarizations or special Hodge structures. In the case of Fano manifolds, many papers were devoted to this kind of problems including [1,2,3,14,23,26]. All these results heavily rely on the existence of Bridgeland stability conditions on the Kuznetsov component as proved in [2].…”

A note on Bridgeland moduli spaces and moduli spaces of sheaves on $X_{14}$ and $Y_3$

“…In the present note, we mainly study various classical moduli spaces on them from a modern point of view. We apply the techniques developed in [BMMS12], [PY21], [APR21] and [Zha20] to show the following results in this note.…”

“…A possible solution is as follows: It's not hard to show that E is stable with respect to some stability conditions on A X14 constructed in [BLMS17]. If one can show that these stability conditions on A X14 are Serre-invariant as in [PY21], then we have Ext 2 (E, E) = 0 by Lemma 3.12. We will address this question in subsequent work.…”

“…The first example of stability conditions constructed in the Kuznetsov component of a cubic threefold is given in [BMMS12]. After a direct construction of stability conditions in the Kuznetsov components of a series Fano threefolds in [BLMS17], the various Bridgeland moduli space of stable objects in the Kuznetsov components of index two and three Fano threefolds are studied in [PY21], [APR21], [PR20] [BBF + 20], [BF21]. In [PY21], the Fano surface of lines on Y d , d = 1 is identified with moduli space of stable objects with (−1)-class 1 − L in Ku(Y d ), where d = 1 case is treated in [PR20].…”

“…After a direct construction of stability conditions in the Kuznetsov components of a series Fano threefolds in [BLMS17], the various Bridgeland moduli space of stable objects in the Kuznetsov components of index two and three Fano threefolds are studied in [PY21], [APR21], [PR20] [BBF + 20], [BF21]. In [PY21], the Fano surface of lines on Y d , d = 1 is identified with moduli space of stable objects with (−1)-class 1 − L in Ku(Y d ), where d = 1 case is treated in [PR20]. In [Zha20] and [JLZ21], the authors studied Bridgeland moduli spaces for Gushel-Mukai threefolds and used them to study several conjectures on them.…”