# 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

#### 28 Citations

On lattice polarizable cubic fourfolds

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We extend non-emtpyness and irreducibility of Hassett divisors to the moduli spaces of M -polarizable cubic fourfolds for higher rank lattices M , which in turn provides a systematic approach for… Expand

Hodge theory and derived categories of cubic fourfolds

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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 with… Expand

Variety of power sums and divisors in the moduli space of cubic fourfolds

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We show that a cubic fourfold F that is apolar to a Veronese surface has the property that its variety of power sums VSP(F,10) is singular along a K3 surface of genus 20. We prove that these cubics… Expand

Some non-special cubic fourfolds

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In [1309.1899], Ranestad and Voisin showed, quite surprisingly, that the divisor in the moduli space of cubic fourfolds consisting of cubics "apolar to a Veronese surface" is not a Noether-Lefschetz… Expand

Rational cubic fourfolds with associated singular K3 surfaces

- Mathematics
- 2020

Generalizing a recent construction of Yang and Yu, we study to what extent one can intersect Hassett's Noether-Lefschetz divisors $\mathcal{C}_d$ in the moduli space of cubic fourfolds $\mathcal{C}$.… Expand

The derived category of a non generic cubic fourfold containing a plane

- Mathematics
- 2016

We describe an Azumaya algebra on the resolution of singularities of the double cover of a plane ramified along a nodal sextic associated to a non generic cubic fourfold containing a plane. We show… Expand

Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

- Mathematics
- 2015

We study the birational properties of geometrically rational surfaces from a derived categorical perspective. In particular, we give a criterion for the rationality of a del Pezzo surface S over an… Expand

Cubic fourfolds containing a plane and K3 surfaces of Picard rank two

- Mathematics
- 2013

We present some new examples of families of cubic hypersurfaces in $$\mathbb {P}^5 (\mathbb {C})$$P5(C) containing a plane whose associated quadric bundle does not have a rational section.

Brauer Groups on K3 Surfaces and Arithmetic Applications

- Mathematics
- 2017

For a prime p, we study subgroups of order p of the Brauer group Br(S) of a general complex polarized K3 surface of degree 2d, generalizing earlier work of van Geemen. These groups correspond to… Expand

Maximally algebraic potentially irrational cubic fourfolds

- Mathematics
- 2018

A well known conjecture asserts that a cubic fourfold $X$ whose transcendental cohomology $T_X$ can not be realized as the transcendental cohomology of a $K3$ surface is irrational. Since the… Expand

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