Fano varieties of cubic fourfolds containing a plane

TL;DR

This paper establishes a deep connection between the Fano varieties of lines on certain cubic fourfolds and moduli spaces of twisted stable complexes and sheaves on K3 surfaces, revealing birational equivalences and wall-crossing phenomena.

Contribution

It demonstrates that Fano varieties of cubic fourfolds containing a plane are isomorphic to moduli spaces of twisted stable complexes and are birational to moduli spaces of twisted stable sheaves on K3 surfaces, linking geometric and derived category perspectives.

Findings

Fano varieties are isomorphic to moduli spaces of twisted stable complexes.

Fano varieties are birational to moduli spaces of twisted stable sheaves.

Wall-crossing relates complexes and sheaves in the derived category.

Abstract

We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that the Fano varieties are always birational to moduli spaces of twisted stable coherent sheaves on a K3 surface. The moduli spaces of complexes and of sheaves are related by wall-crossing in the derived category of twisted sheaves on the corresponding K3 surface.