New perspectives on categorical Torelli theorems for del Pezzo threefolds

Abstract

Let Yd be a del Pezzo threefold of Picard rank one and degree d≥2. In this paper, we apply two different viewpoints to study Yd via a particular admissible subcategory of its bounded derived category, called the Kuznetsov component: (i) Brill–Noether reconstruction. We show that Yd can be uniquely recovered as a Brill–Noether locus of Bridgeland stable objects in its Kuznetsov component. (ii) Exact equivalences. We prove that up to composing with an explicit auto-equivalence, any Fourier–Mukai type equivalence of Kuznetsov components of two del Pezzo threefolds of degree 2≤d≤4 can be lifted to an equivalence of their bounded derived categories. As a result, we obtain a complete description of the group of Fourier–Mukai type auto-equivalences of the Kuznetsov component of Yd. In an appendix, we classify instanton sheaves on quartic double solids, generalizing a result of Druel. © 2024 Elsevier Masson SAS