Arend Bayer (March 5, 2021):  Fano varieties: from derived categories to geometry via stability
Channel: Algebraic Geometry at Stanford
Date: 2021-04-20

A Fano variety $X$ can be reconstructed from its bounded derived category $D^b(X)$. How to use this fact to extract concrete geometric information from $D^b(X)$?  In this talk, I will survey one such approach, via certain subcategories of $D^b(X)$ called Kuznetsov components, and stability conditions. Via moduli spaces of stable objects inside Kuznetsov components, this naturally leads to the reconstruction of many natural moduli spaces classically associated to $X$. In addition to results by a number of authors for Fano threefolds, I will also discuss work in progress (joint with Bertram, Macri, Perry) for cubic fourfolds. Combined with studying Brill-Noether loci, this leads to the construction of special surfaces on an infinite sequence of Hassett-special cubic fourfolds. In some cases, this leads to a natural reinterpretation of recent proofs of rationality of such cubic fourfolds via wall-crossing.