Derived categories of flips and cubic hypersurfaces

TL;DR

This paper advances the understanding of derived categories in birational geometry by completing semiorthogonal decompositions related to flips and cubic hypersurfaces, revealing new Fano and hyperk"ahler structures.

Contribution

It completes the semiorthogonal decomposition for derived categories of flips and cubic hypersurfaces, and lifts the quadratic Fano correspondence to this framework.

Findings

Hilbert square of a cubic hypersurface of dimension ≥3 is Fano

Fano variety of lines on a cubic hypersurface is a Fano visitor

First higher-dimensional hyperk"ahler variety as a Fano visitor

Abstract

A classical result of Bondal-Orlov states that a standard flip in birational geometry gives rise to a fully faithful functor between derived categories of coherent sheaves. We complete their embedding into a semiorthogonal decomposition by describing the complement. As an application, we can lift the "quadratic Fano correspondence" (due to Galkin-Shinder) in the Grothendieck ring of varieties between a smooth cubic hypersurface, its Fano variety of lines, and its Hilbert square, to a semiorthogonal decomposition. We also show that the Hilbert square of a cubic hypersurface of dimension at least 3 is again a Fano variety, so in particular the Fano variety of lines on a cubic hypersurface is a Fano visitor. The most interesting case is that of a cubic fourfold, where this exhibits the first higher-dimensional hyperk\"ahler variety as a Fano visitor.