Comptes Rendus
Algebraic Geometry, Hodge Structure
Rational cubic fourfolds in Hassett divisors
Comptes Rendus. Mathématique, Volume 358 (2020) no. 2, pp. 129-137.

We prove that every Hassett’s Noether–Lefschetz divisor of special cubic fourfolds contains a union of three subvarieties parametrizing rational cubic fourfolds, of codimension-two in the moduli space of smooth cubic fourfolds.

Nous prouvons que chaque diviseur de Hassett–Noether–Lefschetz de cubiques spéciales de dimension 4 contient une union de trois sous-variétés paramétrant des cubiques rationnelles de dimension 4, de codimension deux dans l’espace de modules des cubiques lisses de dimension 4.

Received:
Accepted:
Published online:
DOI: 10.5802/crmath.4
Classification: 14C30, 14E08, 14M20

Song Yang 1; Xun Yu 1

1 Center for Applied Mathematics, Tianjin University, Weijin Road 92, Tianjin 300072, China
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Song Yang; Xun Yu. Rational cubic fourfolds in Hassett divisors. Comptes Rendus. Mathématique, Volume 358 (2020) no. 2, pp. 129-137. doi : 10.5802/crmath.4. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.4/

[1] Nicolas Addington; Brendan Hassett; Yuri Tschinkel; Anthony Várilly-Alvarado Cubic fourfolds fibered in sextic del Pezzo surfaces, Am. J. Math., Volume 141 (2019) no. 6, pp. 1479-1500 | DOI | MR | Zbl

[2] Nicolas Addington; Richard Thomas Hodge theory and derived categories of cubic fourfolds, Duke Math. J., Volume 163 (2014) no. 10, pp. 1885-1927 | DOI | MR | Zbl

[3] Asher Auel; Marcello Bernardara; Michele Bolognesi; Anthony Várilly-Alvarado Cubic fourfolds containing a plane and a quintic del Pezzo surface, Algebr. Geom., Volume 1 (2014) no. 2, pp. 181-193 | DOI | MR | Zbl

[4] Arend Bayer; Martí Lahoz; Emanuele Macrì; Howard Nuer; Alexander Perry; Paolo Stellari Stability conditions in families (2019) (https://arxiv.org/abs/1902.08184v1)

[5] Arend Bayer; Martí Lahoz; Emanuele Macrì; Paolo Stellari Stability conditions on Kuznetsov components (2019) (https://arxiv.org/abs/1703.10839v2)

[6] Arnaud Beauville; Ron Donagi La variété des droites d’une hypersurface cubique de dimension 4, C. R. Math. Acad. Sci. Paris, Volume 301 (1985), pp. 703-706 | Zbl

[7] Michele Bolognesi; Francesco Russo; Giovanni Staglianò Some loci of rational cubic fourfolds, Math. Ann., Volume 373 (2019) no. 1-2, pp. 165-190 | DOI | MR | Zbl

[8] C. Herbert Clemens; Phillip Griffiths The intermediate Jacobian of the cubic threefold, Ann. Math., Volume 95 (1972), pp. 281-356 | DOI | MR | Zbl

[9] Gino Fano Sulle forme cubiche dello spazio a cinque dimensioni contenenti rigate razionali del 4 0 ordine, Comment. Math. Helv., Volume 15 (1943), pp. 71-80 | DOI | MR | Zbl

[10] Brendan Hassett Some rational cubic fourfolds, J. Algebr. Geom., Volume 8 (1999) no. 1, pp. 103-114 | MR | Zbl

[11] Brendan Hassett Special cubic fourfolds, Compos. Math., Volume 120 (2000) no. 1, pp. 1-23 | DOI | MR | Zbl

[12] Brendan Hassett Cubic fourfolds, K3 surfaces, and rationality questions, Rationality problems in algebraic geometry (Lecture Notes in Mathematics), Volume 2172, Springer, 2016, pp. 29-66 | DOI | MR | Zbl

[13] Daniel Huybrechts The K3 category of a cubic fourfold, Compos. Math., Volume 153 (2017) no. 3, pp. 586-620 | DOI | MR | Zbl

[14] Daniel Huybrechts Hodge theory of cubic fourfolds, their Fano varieties, and associated K3 categories, Birational Geometry of Hypersurfaces (Lecture Notes of the Unione Matematica Italiana), Volume 26, Springer, 2019, pp. 165-198 | DOI

[15] Maxim Kontsevich; Yuri Tschinkel Specialization of birational types, Invent. Math., Volume 217 (2019) no. 2, pp. 415-432 | DOI | MR | Zbl

[16] Alexander Kuznetsov Derived categories of cubic fourfolds, Cohomological and geometric approaches to rationality problems. New Perspectives (Progress in Mathematics), Volume 282, Birkhäuser, 2010, pp. 219-243 | DOI | MR | Zbl

[17] Radu Laza The moduli space of cubic fourfolds via the period map, Ann. Math., Volume 172 (2010) no. 1, pp. 673-711 | DOI | MR | Zbl

[18] Eduard Looijenga The period map for cubic fourfolds, Invent. Math., Volume 177 (2009) no. 1, pp. 213-233 | DOI | MR | Zbl

[19] Viacheslav V. Nikulin Integral symmetric bilinear forms and some of their applications, Math. USSR, Izv., Volume 14 (1980), pp. 103-167 | DOI | Zbl

[20] Francesco Russo; Giovanni Staglianò Explicit rationality of some cubic fourfolds (2018) (https://arxiv.org/abs/1811.03502v1)

[21] Francesco Russo; Giovanni Staglianò Congruences of 5-secant conics and the rationality of some admissible cubic fourfolds, Duke Math. J., Volume 168 (2019) no. 5, pp. 849-865 | DOI | MR | Zbl

[22] Francesco Russo; Giovanni Staglianò Trisecant Flops, their associated K3 surfaces and the rationality of some Fano fourfolds (2019) (https://arxiv.org/abs/1909.01263v2)

[23] Jean-Pierre Serre A course in arithmetic, Graduate Texts in Mathematics, 7, Springer, 1973 | MR | Zbl

[24] Claire Voisin Théorème de Torelli pour les cubiques de 5 , Invent. Math., Volume 86 (1986), pp. 577-601 | DOI | Zbl

[25] Claire Voisin Some aspects of the Hodge conjecture, Jpn. J. Math., Volume 2 (2007) no. 2, pp. 261-296 | DOI | MR | Zbl

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