Derived categories of flips and cubic hypersurfaces. Issue 6 (3rd October 2022) - Belmans, Pieter; Fu, Lie; Raedschelders, Theo
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.