Moduli spaces on the Kuznetsov component of Fano threefolds of index 2

TL;DR

This paper explores the moduli spaces of objects in the Kuznetsov component of Fano threefolds of index 2, revealing geometric structures and proving a categorical Torelli theorem for quartic double solids.

Contribution

It introduces a novel connection between roots of del Pezzo surfaces and objects in the Kuznetsov component, and identifies a subvariety of the moduli space isomorphic to the threefold.

Findings

Identifies a subvariety of the moduli space isomorphic to the threefold

Establishes a refined categorical Torelli theorem for quartic double solids

Connects root systems of del Pezzo surfaces to objects in the Kuznetsov component

Abstract

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.