# ANNALES DE L'INSTITUT FOURIER

Smooth weighted hypersurfaces that are not stably rational
[Hypersurfaces pondérées lisses qui ne sont pas stablement rationnelles]
Annales de l'Institut Fourier, Online first, 35 p.

Nous prouvons l’absence rationalité stable pour de nombreuses hypersurfaces pondérées lisses de dimension au moins $3$. Il est en particulier prouvé qu’une hypersurface pondérée de Fano très générale lisse de l’indice un n’est pas stablement rationnelle.

We prove the failure of stable rationality for many smooth well formed weighted hypersurfaces of dimension at least $3$. It is in particular proved that a very general smooth well formed Fano weighted hypersurface of index one is not stably rational.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : https://doi.org/10.5802/aif.3390
Classification : 14E08,  14J45,  14J70
Mots clés : variété de Fano, rationalité stable, hypersurface pondérée
@unpublished{AIF_0__0_0_A45_0,
title = {Smooth weighted hypersurfaces  that are not stably rational},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
year = {2021},
doi = {10.5802/aif.3390},
language = {en},
note = {Online first},
}
Okada, Takuzo. Smooth weighted hypersurfaces  that are not stably rational. Annales de l'Institut Fourier, Online first, 35 p.

[1] Beauville, Arnaud A very general sextic double solid is not stably rational, Bull. Lond. Math. Soc., Volume 48 (2016) no. 2, pp. 321-324 | Zbl 1386.14184

[2] Clemens, Herbert; Griffiths, Phillip The intermediate Jacobian of the cubic threefold, Ann. Math., Volume 95 (1972), pp. 281-356

[3] Colliot-Thélène, Jean-Louis; Pirutka, Alena Cyclic covers that are not stably rational, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 80 (2016) no. 4, pp. 35-48

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

[5] Hartshorne, Robin Algebraic Geometry, Graduate Texts in Mathematics, Springer, 1977 no. 52, xvi+496 pages

[6] Hassett, Brendan; Pirutka, Alena; Tschinkel, Yuri A very general quartic double fourfold is not stably rational, Algebr. Geom., Volume 6 (2019) no. 1, pp. 64-75

[7] Hassett, Brendan; Tschinkel, Yuri On stable rationality of Fano threefolds and del Pezzo fibrations, J. Reine Angew. Math., Volume 751 (2019), pp. 275-287

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

[9] Kollár, János Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 32, Springer, 1996, viii+320 pages | Zbl 0877.14012

[10] Mori, Shigefumi On a generalization of complete intersections, J. Math. Kyoto Univ., Volume 15 (1975) no. 3, pp. 619-646

[11] Okada, Takuzo Stable rationality of cyclic covers of projective spaces, Proc. Edinb. Math. Soc., II. Ser., Volume 62 (2019) no. 3, pp. 667-682

[12] Okada, Takuzo Stable rationality of orbifold Fano 3-fold hypersurfaces, J. Algebr. Geom., Volume 28 (2019) no. 1, pp. 99-138

[13] Przyjalkowski, Victor; Shramov, Constantin Bounds for smooth Fano weighted complete intersections (2016) (https://arxiv.org/abs/1611.09556)

[14] Schreieder, Stefan Stably rational hypersurfaces of small slopes, J. Am. Math. Soc., Volume 32 (2019) no. 4, pp. 1171-1199 | Zbl 07121119

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

[16] Voisin, Claire Unirational threefolds with no universal codimension $2$ cycles, Invent. Math., Volume 201 (2015) no. 1, pp. 207-237