Fano varieties: from derived categories to geometry via stability

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.

The discussion for Arend Bayer’s talk is taking place not in zoom-chat, but at https://tinyurl.com/2021-03-05-ab  (and will be deleted after ~3-7 days).