Derived categories of flips and cubic hypersurfaces

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 hyperkähler variety as a Fano visitor.

arXiv comment34 pages, nearly identical to published version

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Cited by in corpus (4)

• Derived categories of Quot schemes of locally free quotients
• Decompositions of moduli spaces of vector bundles and graph potentials
• Maximal real varieties from moduli constructions
• A Semi-orthogonal Sequence in the Derived Category of the Hilbert Scheme of Three Points