Cubic fourfolds containing a plane and a quantic del Pezzo surface

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2014
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Foundation Compositio Mathematica
Abstract

We isolate a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class β of the even Clifford algebra over the K3 surface S of degree 2 arising from X. Specifically, we show that in the moduli space of cubic fourfolds, the intersection of divisors C8 ∩ C14 has five irreducible components. In the component corresponding to the existence of a tangent conic, we prove that the general member is both pfaffian and has β nontrivial. Such cubic fourfolds provide twisted derived equivalences between K3 surfaces of degrees 2 and 14, hence further corroboration of Kuznetsov’s derived categorical conjecture on the rationality of cubic fourfolds.

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Auel, Asher, Bernardara, Marcello, Bolognesi, Michele, et al.. "Cubic fourfolds containing a plane and a quantic del Pezzo surface." Algebraic Geometry, 1, no. 2 (2014) Foundation Compositio Mathematica: 181-193. http://dx.doi.org/10.14231/AG-2014-010.

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