General hyperplane sections of a Fano threefold $Y$ of index 2 and Picard rank 1 are del Pezzo surfaces, and their Picard group is related to a root system. To the corresponding roots, we associate objects in the Kuznetsov component of $Y$ and investigate their moduli spaces, using the stability condition constructed by Bayer, Lahoz, Macr\`i, and Stellari, and the Abel--Jacobi map. We identify a subvariety of the moduli space isomorphic to $Y$ itself, and as an application we prove a (refined) categorical Torelli theorem for general quartic double solids.

This note describes moduli spaces of objects in the Kuznetsov component of a Veronese double cone, and some related constructions. We consider classes in the numerical Grothendieck group which are minimal with respect to the Euler form, and show that the corresponding moduli spaces are isomorphic. They are a blow-down of a moduli of tilt-stable objects. We interpret the latter space as a generalized Hilbert scheme of lines and give a wall-crossing interpretation of the blow-down morphism.

We construct Bridgeland stability conditions on the derived category of smooth quasiprojective Deligne-Mumford surfaces whose coarse moduli spaces have ADE singularities. This unifies the construction for smooth surfaces and Bridgeland's work on Kleinian singularities. The construction hinges on an orbifold version of the Bogomolov-Gieseker inequality for slope semistable sheaves on the stack, and makes use of the Toën-Hirzebruch-Riemann-Roch theorem.

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A. For an Enriques surface S, the non-degeneracy invariant nd(S) retains information on the elliptic brations of S and its polarizations. In the current paper, we introduce a combinatorial version of the non-degeneracy invariant which depends on S together with a con guration of smooth rational curves, and gives a lower bound for nd(S). We provide a SageMath code that computes such combinatorial invariant and we apply it in several examples. First we identify a new family of nodal Enriques surfaces satisfying nd(S) = 10 which are not general and with in nite automorphism group. We obtain lower bounds on nd(S) for the Enriques surfaces with eight disjoint smooth rational curves studied by Mendes Lopes-Pardini. Finally, we recover Dolgachev and Kondō's computation of the non-degeneracy invariant of the Enriques surfaces with nite automorphism group and provide additional information on the geometry of their elliptic brations.