The Diagonal of (3,3) fivefolds
[La Diagonale des Intersections Complètes en ℙ 7 de Degré (3,3)]
Lange, Jan1 ; Skauli, Bjørn2
1 Institute of Algebraic Geometry, Leibniz University Hannover, Welfengarten 1, 30167 Hannover (Germany)
2 Department of Mathematics, University of Oslo, Moltke Moes vei 35 0851 Oslo (Norway)
Annales de l'Institut Fourier, Online first, 31 p.

We show that a very general (3,3) complete intersection in ℙ 7 over an uncountable algebraically closed field of characteristic different from 2 admits no decomposition of the diagonal, in particular it is not retract rational. This strengthens Nicaise and Ottem’s result proving stable irrationality of this complete intersection in characteristic 0. The main tool is a Chow-theoretic obstruction which was found by Pavic and Schreieder, who studied quartic fivefolds.

En travaillant sur un corps algébriquement clos non dénombrable de caractéristique différente de 2, nous montrons qu’une intersection complète de bidegré (3,3) et de dimension 5 n’a pas de décomposition de la diagonale. En particulier, une telle intersection complète n’est pas rétracte rationnelle. Cela renforce un résultat de Nicaise et Ottem, qui ont montré qu’en caractéristique 0, une telle intersection complète n’est pas stablement rationnelle. L’outil principal est une obstruction découverte par Pavic et Schreieder, qui ont étudié les hypersurfaces quartiques de dimension cinque.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3735
Classification : 14M10, 14C25, 14E08
Keywords: complete intersections, rational, retract rational
Mots-clés : intersection complète, rationnelle, rétracte rationnelle

Lange, Jan 1 ; Skauli, Bjørn 2

1 Institute of Algebraic Geometry, Leibniz University Hannover, Welfengarten 1, 30167 Hannover (Germany)
2 Department of Mathematics, University of Oslo, Moltke Moes vei 35 0851 Oslo (Norway) 
@unpublished{AIF_0__0_0_A23_0,
     author = {Lange, Jan and Skauli, Bj{\o}rn},
     title = {The {Diagonal} of (3,3) fivefolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2025},
     doi = {10.5802/aif.3735},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Lange, Jan
AU  - Skauli, Bjørn
TI  - The Diagonal of (3,3) fivefolds
JO  - Annales de l'Institut Fourier
PY  - 2025
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3735
LA  - en
ID  - AIF_0__0_0_A23_0
ER  - 
%0 Unpublished Work
%A Lange, Jan
%A Skauli, Bjørn
%T The Diagonal of (3,3) fivefolds
%J Annales de l'Institut Fourier
%D 2025
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3735
%G en
%F AIF_0__0_0_A23_0
Lange, Jan; Skauli, Bjørn. The Diagonal of (3,3) fivefolds. Annales de l'Institut Fourier, Online first, 31 p.

[1] Artin, Michael; Mumford, David Some elementary examples of unirational varieties which are not rational, Proc. Lond. Math. Soc. (3), Volume 25 (1972) no. 3, pp. 75-95 | DOI | Zbl | MR

[2] Beauville, Arnaud; Colliot-Thélène, Jean-Louis; Sansuc, Jean-Jacques; Swinnerton-Dyer, Peter Variétés stablement rationnelles non rationnelles, Ann. Math. (2), Volume 121 (1985) no. 2, pp. 283-318 | DOI | Zbl | MR

[3] Chatzistamatiou, Andre; Levine, Marc Torsion orders of complete intersections, Algebra Number Theory, Volume 11 (2017) no. 8, pp. 1779-1835 | DOI | Zbl | MR

[4] Colliot-Thélène, Jean-Louis; Pirutka, Alena Cyclic covers that are not stably rational, Izv. Math., Volume 80 (2016) no. 4, pp. 665-677 | Zbl | DOI

[5] Colliot-Thélène, Jean-Louis; Pirutka, Alena Hypersurfaces quartiques de dimension 3 : non rationalité stable, Ann. Sci. Éc. Norm. Supér. (4), Volume 49 (2016) no. 2, pp. 371-397 | DOI | Zbl | MR | Numdam

[6] Fulton, William Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 2, Springer, 1998 | Zbl | DOI

[7] Hartshorne, Robin Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer, 1977 | Zbl | DOI | MR

[8] Hassett, Brendan; Pirutka, Alena; Tschinkel, Yuri Intersections of three quadrics in ℙ 7 , Surveys in differential geometry 2017. Celebrating the 50th anniversary of the Journal of Differential Geometry (Surveys in Differential Geometry), Volume 22, International Press, 2018, pp. 259-274 | DOI | MR | Zbl

[9] Hassett, Brendan; Pirutka, Alena; Tschinkel, Yuri Stable rationality of quadric surface bundles over surfaces, Acta Math., Volume 220 (2018) no. 2, pp. 341-365 | DOI | Zbl | MR

[10] Hoshi, Akinara; Yamasaki, Aiichi Rationality Problem for Algebraic Tori, Memoirs of the American Mathematical Society, 248, American Mathematical Society, 2017 | DOI | MR | Zbl

[11] Illusie, Luc; Temkin, Michael Exposé X. Gabber’s modification theorem (log smooth case), Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents. Séminaire à l’École Polytechnique 2006–2008 (Astérisque), Volume 363-364, Société Mathématique de France, 2014, pp. 167-212 | Zbl

[12] de Jong, Aise Johan Smoothness, semi-stability and alterations, Publ. Math., Inst. Hautes Étud. Sci., Volume 83 (1996), pp. 51-93 | DOI | Zbl | MR | Numdam

[13] Kollár, János Nonrational hypersurfaces, J. Am. Math. Soc., Volume 8 (1995) no. 1, pp. 241-249 | DOI | Zbl | MR

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

[15] Merkurjev, Alexander Unramified elements in cycle modules, J. Lond. Math. Soc. (2), Volume 78 (2008) no. 1, pp. 51-64 | DOI | Zbl | MR

[16] Nicaise, Johannes; Ottem, John C. Tropical degenerations and stable rationality, Duke Math. J., Volume 171 (2022) no. 15, pp. 3023-3075 | DOI | Zbl | MR

[17] Nicaise, Johannes; Shinder, Evgeny The motivic nearby fiber and degenerations of stable rationality, Invent. Math., Volume 217 (2019) no. 2, pp. 377-413 | DOI | Zbl | MR

[18] Pavic, Nebojsa; Schreieder, Stefan The diagonal of quartic fivefolds, Algebr. Geom., Volume 10 (2023) no. 6, pp. 754-778 | DOI | Zbl | MR

[19] Schreieder, Stefan On the rationality problem for quadric bundles, Duke Math. J., Volume 168 (2019) no. 2, pp. 187-223 | DOI | Zbl | MR

[20] Schreieder, Stefan Stably irrational hypersurfaces of small slopes, J. Am. Math. Soc., Volume 32 (2019) no. 4, pp. 1171-1199 | DOI | Zbl | MR

[21] Schreieder, Stefan Unramified Cohomology, Algebraic Cycles and Rationality, Rationality of Varieties (Farkas, Gavril; van der Geer, Gerard; Shen, Mingmin; Taelman, Lenny, eds.), Springer (2021), pp. 345-388 | DOI | Zbl

[22] Skauli, Bjørn A (2,3)-complete intersection fourfold with no decomposition of the diagonal, Manuscr. Math., Volume 171 (2023), pp. 473-486 | DOI | Zbl

[23] Totaro, Burt Hypersurfaces that are not stably rational, J. Am. Math. Soc., Volume 29 (2016) no. 3, pp. 883-891 | DOI | Zbl

[24] Vial, Charles Algebraic cycles and fibrations, Doc. Math., Volume 18 (2013), pp. 1521-1553 | DOI | Zbl

[25] Voisin, Claire Unirational threefolds with no universal codimension 2 cycle, Invent. Math., Volume 201 (2015) no. 1, pp. 207-237 | DOI | Zbl | MR

Cité par Sources :