Derived categories of flips and cubic hypersurfaces

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