Stably irrational hypersurfaces of small slopes

Schreieder, Stefan¹

¹ Mathematisches Institut, LMU München, Theresienstr. 39, 80333 München, Germany

Journal of the American Mathematical Society, Tome 32 (2019) no. 4, pp. 1171-1199

Résumé

Let $k$ be an uncountable field of characteristic different from two. We show that a very general hypersurface $X\subset \mathbb {P}^{N+1}_k$ of dimension $N\geq 3$ and degree at least $\log _2N +2$ is not stably rational over the algebraic closure of $k$.

Détail

DOI : 10.1090/jams/928

Bibliographie

[1] Artin, M., Mumford, D. Some elementary examples of unirational varieties which are not rational Proc. London Math. Soc. (3) 1972 75 95

[2] Asok, Aravind Rationality problems and conjectures of Milnor and Bloch-Kato Compos. Math. 2013 1312 1326

[3] Beauville, Arnaud A very general sextic double solid is not stably rational Bull. Lond. Math. Soc. 2016 321 324

[4] Clemens, C. Herbert, Griffiths, Phillip A. The intermediate Jacobian of the cubic threefold Ann. of Math. (2) 1972 281 356

[5] Colliot-Thã©Lã¨Ne, J.-L. Birational invariants, purity and the Gersten conjecture 1995 1 64

[6] Colliot-Thã©Lã¨Ne, Jean-Louis, Ojanguren, Manuel Variétés unirationnelles non rationnelles: au-delà de l’exemple d’Artin et Mumford Invent. Math. 1989 141 158

[7] Colliot-Thã©Lã¨Ne, Jean-Louis, Skorobogatov, Alexei N. Groupe de Chow des zéro-cycles sur les fibrés en quadriques 𝐾-Theory 1993 477 500

[8] Colliot-Thã©Lã¨Ne, Jean-Louis, Pirutka, Alena Hypersurfaces quartiques de dimension 3: non-rationalité stable Ann. Sci. Éc. Norm. Supér. (4) 2016 371 397

[9] Kol′ë-Telã¨N, Zh.-L., Piryutko, E. V. Cyclic covers that are not stably rational Izv. Ross. Akad. Nauk Ser. Mat. 2016 35 48

[10] Colliot-Thã©Lã¨Ne, Jean-Louis, Voisin, Claire Cohomologie non ramifiée et conjecture de Hodge entière Duke Math. J. 2012 735 801

[11] Conte, Alberto, Marchisio, Marina, Murre, Jacob P. On the unirationality of the quintic hypersurface containing a 3-dimensional linear space Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 2008

[12] De Fernex, Tommaso Birationally rigid hypersurfaces Invent. Math. 2013 533 566

[13] De Fernex, Tommaso Erratum to: Birationally rigid hypersurfaces [ MR3049929] Invent. Math. 2016 675 680

[14] De Jong, A. J. Smoothness, semi-stability and alterations Inst. Hautes Études Sci. Publ. Math. 1996 51 93

[15] Elman, Richard, Lam, T. Y. Pfister forms and 𝐾-theory of fields J. Algebra 1972 181 213

[16] Fulton, William Intersection theory 1998

[17] Hassett, Brendan, Kresch, Andrew, Tschinkel, Yuri Stable rationality and conic bundles Math. Ann. 2016 1201 1217

[18] Hassett, Brendan, Pirutka, Alena, Tschinkel, Yuri Stable rationality of quadric surface bundles over surfaces Acta Math. 2018 341 365

[19] Hassett, Brendan, Pirutka, Alena, Tschinkel, Yuri A very general quartic double fourfold is not stably rational Algebr. Geom. 2019 64 75

[20] Illusie, Luc, Temkin, Michael Exposé X. Gabber’s modification theorem (log smooth case) Astérisque 2014 167 212

[21] Iskovskih, V. A., Manin, Ju. I. Three-dimensional quartics and counterexamples to the Lüroth problem Mat. Sb. (N.S.) 1971 140 166

[22] Karpenko, Nikita A., Merkurjev, Alexander S. On standard norm varieties Ann. Sci. Éc. Norm. Supér. (4) 2013

[23] Kerz, Moritz The Gersten conjecture for Milnor 𝐾-theory Invent. Math. 2009 1 33

[24] Kollã¡R, Jã¡Nos Nonrational hypersurfaces J. Amer. Math. Soc. 1995 241 249

[25] Kollã¡R, Jã¡Nos Rational curves on algebraic varieties 1996

[26] Lam, T. Y. Introduction to quadratic forms over fields 2005

[27] Merkurjev, Alexander Unramified elements in cycle modules J. Lond. Math. Soc. (2) 2008 51 64

[28] Murre, J. P. Reduction of the proof of the non-rationality of a non-singular cubic threefold to a result of Mumford Compositio Math. 1973 63 82

[29] Pukhlikov, A. V. Birational isomorphisms of four-dimensional quintics Invent. Math. 1987 303 329

[30] Pukhlikov, Aleksandr V. Birational automorphisms of Fano hypersurfaces Invent. Math. 1998 401 426

[31] Serre, Jean-Pierre Galois cohomology 1997

[32] Schreieder, Stefan On the rationality problem for quadric bundles Duke Math. J. 2019 187 223

[33] Schreieder, Stefan Quadric surface bundles over surfaces and stable rationality Algebra Number Theory 2018 479 490

[34] Totaro, Burt Hypersurfaces that are not stably rational J. Amer. Math. Soc. 2016 883 891

[35] Voevodsky, Vladimir Motivic cohomology with 𝑍/2-coefficients Publ. Math. Inst. Hautes Études Sci. 2003 59 104

[36] Voisin, Claire On integral Hodge classes on uniruled or Calabi-Yau threefolds 2006 43 73

[37] Voisin, Claire Some aspects of the Hodge conjecture Jpn. J. Math. 2007 261 296

[38] Voisin, Claire Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal J. Algebraic Geom. 2013 141 174

[39] Voisin, Claire Unirational threefolds with no universal codimension 2 cycle Invent. Math. 2015 207 237