Hypersurfaces that are not stably rational

Totaro, Burt¹

¹ Mathematics Department, UCLA, Box 951555, Los Angeles, California 90095-1555

Journal of the American Mathematical Society, Tome 29 (2016) no. 3, pp. 883-891

Résumé

We show that a wide class of hypersurfaces in all dimensions are not stably rational. Namely, for all $d\geq 2\lceil (n+2)/3\rceil$ and $n\geq 3$, a very general complex hypersurface of degree $d$ in $\textbf {P}^{n+1}$ is not stably rational. The statement generalizes Colliot-Thélène and Pirutka’s theorem that very general quartic 3-folds are not stably rational. The result covers all the degrees in which Kollár proved that a very general hypersurface is non-rational, and a bit more. For example, very general quartic 4-folds are not stably rational, whereas it was not even known whether these varieties are rational.

Détail

DOI : 10.1090/jams/840

Bibliographie

[1] Beauville, A. A very general sextic double solid is not stably rational

[2] Beauville, A. A very general quartic double fourfold or fivefold is not stably rational

[3] Bourbaki, N. Éléments de mathématique. Algèbre commutative. Chapitres 8 et 9 2006

[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, Jean-Louis, Coray, Daniel L’équivalence rationnelle sur les points fermés des surfaces rationnelles fibrées en coniques Compositio Math. 1979 301 332

[6] Colliot-Thã©Lã¨Ne, J.-L., Pirutka, A. Hypersurfaces quartiques de dimension 3: non rationalité stable

[7] De Fernex, Tommaso, Fusi, Davide Rationality in families of threefolds Rend. Circ. Mat. Palermo (2) 2013 127 135

[8] Fulton, William Intersection theory 1998

[9] Gros, Michel Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique Mém. Soc. Math. France (N.S.) 1985 87

[10] Hacon, Christopher D., Mckernan, James On Shokurov’s rational connectedness conjecture Duke Math. J. 2007 119 136

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

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

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

[14] Kollã¡R, Jã¡Nos Nonrational covers of 𝐶𝑃^{𝑚}×𝐶𝑃ⁿ 2000 51 71

[15] Kollã¡R, Jã¡Nos Singularities of the minimal model program 2013

[16] Kollã¡R, Jã¡Nos, Smith, Karen E., Corti, Alessio Rational and nearly rational varieties 2004

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

[18] Mori, Shigefumi On a generalization of complete intersections J. Math. Kyoto Univ. 1975 619 646

[19] Pukhlikov, Aleksandr Birationally rigid varieties 2013

[20] Serre, Jean-Pierre Local fields 1979

[21] Totaro, B. The motive of a classifying space

[22] Voisin, Claire Chow rings, decomposition of the diagonal, and the topology of families 2014

[23] Voisin, Claire Unirational varieties with no universal codimension 2 cycle Invent. Math., 2015