New perspectives on categorical Torelli theorems for del Pezzo threefolds

Let $Y_d$ be a del Pezzo threefold of Picard rank one and degree $d\geq 2$. In this paper, we apply two different viewpoints to study $Y_d$ via a particular admissible subcategory of its bounded derived category, called the Kuznetsov component: (i) Brill-Noether reconstruction. We show that $Y_d$ 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 $2\leq d\leq 4$ 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 $Y_d$. We also describe the group of Fourier-Mukai type auto-equivalences of Kuznetsov components of index one prime Fano threefolds $X_{2g-2}$ of genus $g=6$ and $8$. As an application, first we identify the group of automorphisms of $X_{14}$ and its associated $Y_3$. Then we give a new disproof of Kuznetsov's Fano threefold conjecture by assuming Gushel-Mukai threefolds are general. In an appendix, we classify instanton sheaves on quartic double solids, generalizing a result of Druel.