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

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 KupXq, 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 KupXq preserves these stability conditions. As an application, we obtain the smoothness of non-empty moduli spaces of stable objects in KupXq. When X is a quartic double solid, we describe a connected component of the stability manifold parametrizing stability conditions on KupXq.

We prove the existence of Bridgeland stability conditions on the Kuznetsov components of Gushel-Mukai varieties, and describe the structure of moduli spaces of Bridgeland semistable objects in these categories in the even-dimensional case. As applications, we construct a new infinite series of unirational locally complete families of polarized hyperkähler varieties of K3 type, and characterize Hodge-theoretically when the Kuznetsov component of an even-dimensional Gushel-Mukai variety is equivalent to the derived category of a K3 surface.

We study moduli spaces of stable objects in Enriques categories by exploiting their relation to moduli spaces of stable objects in associated K3 categories. In particular, we settle the nonemptiness problem for moduli spaces of stable objects in the Kuznetsov components of several interesting classes of Fano varieties, and deduce the nonemptiness of fixed loci of certain antisymplectic involutions on modular hyperkähler varieties.

We prove a general criterion which ensures that a fractional Calabi--Yau category of dimension $\leq 2$ admits a unique Serre-invariant stability condition, up to the action of the universal cover of $\text{GL}^+_2(\mathbb{R})$. We apply this result to the Kuznetsov component $\text{Ku}(X)$ of a cubic threefold $X$. In particular, we show that all the known stability conditions on $\text{Ku}(X)$ are invariant with respect to the action of the Serre functor and thus lie in the same orbit with respect to the action of the universal cover of $\text{GL}^+_2(\mathbb{R})$. As an application, we show that the moduli space of Ulrich bundles of rank $\geq 2$ on $X$ is irreducible, answering a question asked by Lahoz, Macr\`i and Stellari.