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
On lattice polarizable cubic fourfolds
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 forExpand
Hodge theory and derived categories of cubic fourfolds
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
Variety of power sums and divisors in the moduli space of cubic fourfolds
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 cubicsExpand
Some non-special cubic fourfolds
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-LefschetzExpand
Rational cubic fourfolds with associated singular K3 surfaces
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
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 showExpand
Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields
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 anExpand
Cubic fourfolds containing a plane and K3 surfaces of Picard rank two
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
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 toExpand
Maximally algebraic potentially irrational cubic fourfolds
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 theExpand
...
1
2
3
...

References

SHOWING 1-10 OF 51 REFERENCES
Fano varieties of cubic fourfolds containing a plane
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
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
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
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
...
1
2
3
4
5
...