Derived categories of flips and cubic hypersurfaces. Issue 6 (3rd October 2022)

• Record Type: Journal Article
• Authors: Belmans, Pieter; Fu, Lie; Raedschelders, Theo
• Is Part Of: Proceedings of the London Mathematical Society. Volume 125:Issue 6(2022)
• Page Start: 1452
• Page End: 1482
• Publication Date: 2022-10-03
• Subjects: Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1112/plms.12487
• Languages: English
• ISSNs: 0024-6115

Abstract

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 Galkin–Shinder relation 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 hyperkähler variety as a Fano visitor.