# Cubic fourfolds containing a plane and a quintic del Pezzo surface

@article{Auel2014CubicFC,
title={Cubic fourfolds containing a plane and a quintic del Pezzo surface},
author={Asher Auel and Marcello Bernardara and Michele Bolognesi and Anthony V{\'a}rilly-Alvarado},
journal={arXiv: Algebraic Geometry},
year={2014},
volume={1},
pages={181-193}
}
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 of the even Cliord algebra over the K3 surface S of degree 2 arising from X. Specically, we show that in the moduli space of cubic fourfolds, the intersection of divisorsC8\C14 has ve irreducible components. In the component corresponding to the existence of a tangent conic, we prove that… Expand
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#### References

SHOWING 1-10 OF 51 REFERENCES
Fano varieties of cubic fourfolds containing a plane
• Mathematics
• 2009
We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that theExpand
Hodge theory and derived categories of cubic fourfolds
• Mathematics
• 2014
Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics - conjecturally, the ones that are rational - have specific K3s associated to them geometrically. Hassett has studied cubics withExpand
Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems
• Mathematics
• 2014
Abstract Let X → Y be a fibration whose fibers are complete intersections of r quadrics. We develop new categorical and algebraic tools—a theory of relative homological projective duality and theExpand
The period map for cubic fourfolds
The period map for cubic fourfolds takes values in a locally symmetric variety of orthogonal type of dimension 20. We determine the image of this period map (thus confirming a conjecture of Hassett)Expand
Special Cubic Fourfolds
AbstractA cubic fourfold is a smooth cubic hypersurface of dimension four; it is special if it contains a surface not homologous to a complete intersection. Special cubic fourfolds form a countablyExpand
Homological projective duality for Grassmannians of lines
We show that homologically projectively dual varieties for Grassmannians Gr(2,6) and Gr(2,7) are given by certain noncommutative resolutions of singularities of the corresponding Pfaffian varieties.Expand
On the hodge conjecture for unirational fourfolds
1. In a recent paper S. Zucker has proved the Hodge conjecture for cubic fourfolds [4]. His proof uses the method of normal functions. Zucker’s paper contains also an alternate proof, due to Clemens,Expand
The moduli space of cubic fourfolds via the period map
We characterize the image of the period map for cubic fourfolds with at worst simple singularities as the complement of an arrangement of hyperplanes in the period space. It follows then that the GITExpand
K3 surfaces with Picard number one and infinitely many rational points
In general, not much is known about the arithmetic of K3 surfaces. Once the geometric Picard number, which is the rank of the Neron-Severi group over an algebraic closure of the base field, is highExpand
Derived categories and rationality of conic bundles
• Mathematics
• Compositio Mathematica
• 2013
Abstract We show that a standard conic bundle over a minimal rational surface is rational and its Jacobian splits as the direct sum of Jacobians of curves if and only if its derived category admits aExpand