Rational cubic fourfolds in Hassett divisors

Song Yang; Xun Yu

doi:10.5802/crmath.4

Géométrie algébrique, Structure de Hodge

Rational cubic fourfolds in Hassett divisors
[Cubiques rationnelles de dimension 4 dans les diviseurs de Hassett]

Song Yang ¹ ; Xun Yu ¹

¹ Center for Applied Mathematics, Tianjin University, Weijin Road 92, Tianjin 300072, China

Comptes Rendus. Mathématique, Volume 358 (2020) no. 2, pp. 129-137.

Résumé
Abstract

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.

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.

Reçu le : 2020-01-19
Accepté le : 2020-01-28
Publié le : 2020-06-15

DOI : 10.5802/crmath.4

Classification : 14C30, 14E08, 14M20

Affiliations des auteurs :

Song Yang ¹ ; Xun Yu ¹

¹ Center for Applied Mathematics, Tianjin University, Weijin Road 92, Tianjin 300072, China

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2020__358_2_129_0,
     author = {Song Yang and Xun Yu},
     title = {Rational cubic fourfolds in {Hassett} divisors},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {129--137},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {2},
     year = {2020},
     doi = {10.5802/crmath.4},
     language = {en},
}

TY  - JOUR
AU  - Song Yang
AU  - Xun Yu
TI  - Rational cubic fourfolds in Hassett divisors
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 129
EP  - 137
VL  - 358
IS  - 2
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.4
LA  - en
ID  - CRMATH_2020__358_2_129_0
ER  -

%0 Journal Article
%A Song Yang
%A Xun Yu
%T Rational cubic fourfolds in Hassett divisors
%J Comptes Rendus. Mathématique
%D 2020
%P 129-137
%V 358
%N 2
%I Académie des sciences, Paris
%R 10.5802/crmath.4
%G en
%F CRMATH_2020__358_2_129_0

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/

Bibliographie
Cité par

[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 $ℙ^{5}$ , 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

Cité par Sources :

Commentaires - Politique