# Some loci of rational cubic fourfolds

@article{Bolognesi2015SomeLO, title={Some loci of rational cubic fourfolds}, author={M. Bolognesi and F. Russo and Giovanni Staglian{\`o}}, journal={Mathematische Annalen}, year={2015}, volume={373}, pages={165-190} }

In this paper we investigate the divisor $${\mathcal {C}}_{14}$$C14 inside the moduli space of smooth cubic hypersurfaces in $${\mathbb {P}}^5$$P5, whose general element is a smooth cubic containing a smooth quartic rational normal scroll. By showing that all degenerations of quartic scrolls in $${\mathbb {P}}^5$$P5 contained in a smooth cubic hypersurface are surfaces with one apparent double point, we prove that every cubic hypersurface contained in $${\mathcal {C}}_{14}$$C14 is rational… Expand

#### 28 Citations

Rational fibered cubic fourfolds with nontrivial Brauer classes

- Mathematics
- 2019

Some classes of cubic fourfolds are birational to fibrations over $P^2$, where the fibers are rational surfaces. This relates strongly the rationality of the cubic with the rationality of these… Expand

Congruences of $5$-secant conics and the rationality of some admissible cubic fourfolds

- Mathematics
- 2017

The works of Hassett and Kuznetsov identify countably many divisors $C_d$ in the open subset of $\mathbb{P}^{55}=\mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^5}(3)))$ parametrizing all cubic 4-folds and… Expand

On the rationality and the finite dimensionality of a cubic fourfold

- Mathematics
- 2017

Let $X$ be a cubic fourfold in $P^5_{C}$. We prove that, assuming the Hodge conjecture for the product $S \times S$, where $S$ is a complex surface, and the finite dimensionality of the Chow motive… 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 transcendental motive of a cubic fourfold

- Mathematics
- 2017

In this note we introduce the transcendental part $t(X)$ of the motive of a cubic fourfold $X$ and prove that it is isomorphic to the (twisted) transcendental part $h_2^{tr}(F(X))$ in a suitable… Expand

Unirationality of certain universal families of cubic fourfolds

- Mathematics
- 2020

The aim of this short note is to define the \it universal cubic fourfold \rm over certain loci of their moduli space. Then, we propose two methods to prove that it is unirational over the Hassett… 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.

Rationality questions and motives of cubic fourfolds

- Mathematics
- 2019

In this note we propose an approach to some questions about the birational geometry of smooth cubic fourfolds through the theory of Chow motives. We introduce the transcendental part t(X) of the… Expand

On lattice polarizable cubic fourfolds

- Mathematics
- 2021

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

BRAUER GROUPS OF INVOLUTION SURFACE BUNDLES

- 2019

A fundamental breakthrough in the study of rationality properties of complex algebraic varieties was the construction, by Artin and Mumford, of examples of projective unirational threefolds with… Expand

#### References

SHOWING 1-10 OF 27 REFERENCES

Congruences of $5$-secant conics and the rationality of some admissible cubic fourfolds

- Mathematics
- 2017

The works of Hassett and Kuznetsov identify countably many divisors $C_d$ in the open subset of $\mathbb{P}^{55}=\mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^5}(3)))$ parametrizing all cubic 4-folds and… Expand

Special Cubic Fourfolds

- Mathematics
- Compositio Mathematica
- 2000

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 countably… Expand

Some extremal contractions between smooth varieties arising from projective geometry

- Mathematics
- 2004

We construct explicit examples of elementary extremal contractions, both birational and of fiber type, from smooth projective $n$-dimensional varieties, with $n \geq 4$, onto smooth projective… Expand

Restricting linear syzygies: algebra and geometry

- Mathematics
- Compositio Mathematica
- 2005

Let $X\subset \mathbb{P}^r$ be a closed scheme in projective space whose homogeneous ideal is generated by quadrics. We say that X (or its ideal IX) satisfies the condition N2,p if the syzygies of IX… Expand

Cubic fourfolds containing a plane and a quintic del Pezzo surface

- Mathematics
- 2014

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… Expand

On a theorem of Severi

- Mathematics
- 2000

P 2r+1, on the projection we will obtain a finite number δ(X) of improper nodes, i.e. non-normal double points whose tangent cones break into two r-planes. The number δ(X) is called the number of… Expand

On the symplectic eightfold associated to a Pfaffian cubic fourfold

- Mathematics
- 2017

We show that the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane is deformation-equivalent to the Hilbert scheme of four points on a K3 surface.… Expand

The period map for cubic fourfolds

- Mathematics
- 2007

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

SOME SPECIAL CREMONA TRANSFORMATIONS

- Mathematics
- 1989

In this paper we continue the study (c. f. [C. -K.], [S. -T. 1, 2]) of special Cremona transformations, i.e. those whose base scheme Y is smooth and connected. Crauder and Katz have completely… Expand

The moduli space of cubic fourfolds via the period map

- Mathematics
- 2007

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 GIT… Expand