# 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}
}`
• Published 2015
• Mathematics
• Mathematische Annalen
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
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#### 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 andExpand
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
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 projectiveExpand
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 IXExpand
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 theExpand
On a theorem of Severi
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 ofExpand
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
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 completelyExpand
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