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Stably irrational hypersurfaces of small slopes. (English) Zbl 1442.14138
A projective variety $$X$$ defined over a field $$k$$ is called stably rational if and only if $$X \times \mathbb{P}^n_k$$ is birational to $$\mathbb{P}^m_k$$, for some integers $$n,m$$. Stable rationality is a generalization of rationality. A rational variety is always stably rational but the converse is not always true.
Let $$X_d \subset \mathbb{P}_k^n$$ be a hypersurface of degree $$d$$. A classical problem in algebraic geometry if to determine for which values of $$d$$ $$X$$ is rational or nonrational.
The main result of this paper is the following.
Let $$k$$ be an uncountable field of characteristic different from two. let $$N\geq 3$$ be an integer and write $$N=n+r$$, where $$2^{n-1}-2 \leq r \leq 2^n-2$$. Then a very general hypersurface $$X \subset \mathbb{P}^{N+1}_K$$ of degree $$d\geq n+2$$ is not stably rational over the algebraic closure of $$k$$. In particular it is not rational.
A consequence of the previous result is that there exists hypersurfaces $$X$$ such that the slope $\frac{\deg(X)}{\dim X+1},$ is arbitrary small.

##### MSC:
 14J70 Hypersurfaces and algebraic geometry 14E08 Rationality questions in algebraic geometry 14M20 Rational and unirational varieties 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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##### References:
 [1] Artin, M.; Mumford, D., Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. (3), 25, 75-95, (1972) · Zbl 0244.14017 [2] Asok, Aravind, Rationality problems and conjectures of Milnor and Bloch-Kato, Compos. Math., 149, 8, 1312-1326, (2013) · Zbl 1279.14063 [3] Beauville, Arnaud, A very general sextic double solid is not stably rational, Bull. Lond. Math. Soc., 48, 2, 321-324, (2016) · Zbl 1386.14184 [4] Clemens, C. Herbert; Griffiths, Phillip A., The intermediate Jacobian of the cubic threefold, Ann. of Math. (2), 95, 281-356, (1972) · Zbl 0214.48302 [5] Colliot-Th\'el\ene, J.-L., Birational invariants, purity and the Gersten conjecture.$$K$$-theory and algebraic geometry: connections with quadratic forms and division algebras, Santa Barbara, CA, 1992, Proc. Sympos. Pure Math. 58, 1-64, (1995), Amer. Math. Soc., Providence, RI · Zbl 0834.14009 [6] [CT2]CT2 J.-L. Colliot-Th\'el\ene, \em Introduction to work of Hassett-Pirutka-Tschinkel and Schreieder, preprint, 2018, https://www.math.u-psud.fr/\textasciitilde colliot/HPT+Srevisited2Feb18.pdf. [7] Colliot-Th\'el\‘ene, Jean-Louis; Ojanguren, Manuel, Vari\'’et\'es unirationnelles non rationnelles: au-del\‘a de l”exemple d’Artin et Mumford, Invent. Math., 97, 1, 141-158, (1989) · Zbl 0686.14050 [8] Colliot-Th\'el\‘ene, Jean-Louis; Skorobogatov, Alexei N., Groupe de Chow des z\'’ero-cycles sur les fibr\'es en quadriques $$, K$$-Theory, 7, 5, 477-500, (1993) · Zbl 0837.14002 [9] Colliot-Th\'el\‘ene, Jean-Louis; Pirutka, Alena, Hypersurfaces quartiques de dimension 3: non-rationalit\'’e stable, Ann. Sci. \'Ec. Norm. Sup\'er. (4), 49, 2, 371-397, (2016) · Zbl 1371.14028 [10] Kol\textprime\"e-Tel\en, Zh.-L.; Piryutko, E. V., Cyclic covers that are not stably rational, Izv. Ross. Akad. Nauk Ser. Mat.. Izv. Math., 80 80, 4, 665-677, (2016) · Zbl 1375.14053 [11] Colliot-Th\'el\‘ene, Jean-Louis; Voisin, Claire, Cohomologie non ramifi\'’ee et conjecture de Hodge enti\`ere, Duke Math. J., 161, 5, 735-801, (2012) · Zbl 1244.14010 [12] 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., 142, 89-96 (2009), (2008) · Zbl 1200.14087 [13] de Fernex, Tommaso, Birationally rigid hypersurfaces, Invent. Math., 192, 3, 533-566, (2013) · Zbl 1279.14019 [14] de Fernex, Tommaso, Erratum to: Birationally rigid hypersurfaces [MR3049929], Invent. Math., 203, 2, 675-680, (2016) · Zbl 1441.14045 [15] de Jong, A. J., Smoothness, semi-stability and alterations, Inst. Hautes \'Etudes Sci. Publ. Math., 83, 51-93, (1996) · Zbl 0916.14005 [16] Elman, Richard; Lam, T. Y., Pfister forms and $$K$$-theory of fields, J. Algebra, 23, 181-213, (1972) · Zbl 0246.15029 [17] [FT]FT L. Fu and Z. Tian, \em 2-cycles sur les hypersurfaces cubiques de dimension 5, arXiv:1801.03995, 2018. [18] Fulton, William, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 2, xiv+470 pp., (1998), Springer-Verlag, Berlin · Zbl 0885.14002 [19] Hassett, Brendan; Kresch, Andrew; Tschinkel, Yuri, Stable rationality and conic bundles, Math. Ann., 365, 3-4, 1201-1217, (2016) · Zbl 1353.14019 [20] Hassett, Brendan; Pirutka, Alena; Tschinkel, Yuri, Stable rationality of quadric surface bundles over surfaces, Acta Math., 220, 2, 341-365, (2018) · Zbl 1420.14115 [21] Hassett, Brendan; Pirutka, Alena; Tschinkel, Yuri, A very general quartic double fourfold is not stably rational, Algebr. Geom., 6, 1, 64-75, (2019) · Zbl 07020392 [22] Illusie, Luc; Temkin, Michael, Expos\'e X. Gabber’s modification theorem (log smooth case), Ast\'erisque, 363-364, 167-212, (2014) · Zbl 1327.14072 [23] Iskovskih, V. A.; Manin, Ju. I., Three-dimensional quartics and counterexamples to the L\"uroth problem, Mat. Sb. (N.S.), 86(128), 140-166, (1971) · Zbl 0222.14009 [24] Karpenko, Nikita A.; Merkurjev, Alexander S., On standard norm varieties, Ann. Sci. \'Ec. Norm. Sup\'er. (4), 46, 1, 175-214 (2013), (2013) · Zbl 1275.14006 [25] Kerz, Moritz, The Gersten conjecture for Milnor $$K$$-theory, Invent. Math., 175, 1, 1-33, (2009) · Zbl 1188.19002 [26] Koll\'ar, J\'anos, Nonrational hypersurfaces, J. Amer. Math. Soc., 8, 1, 241-249, (1995) · Zbl 0839.14031 [27] Koll\'ar, J\'anos, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 32, viii+320 pp., (1996), Springer-Verlag, Berlin · Zbl 0877.14012 [28] [KT]KT M. Kontsevich and Yu. Tschinkel, \em Specialization of birational types, arXiv:1708.05699, 2017. [29] Lam, T. Y., Introduction to quadratic forms over fields, Graduate Studies in Mathematics 67, xxii+550 pp., (2005), American Mathematical Society, Providence, RI · Zbl 1068.11023 [30] Merkurjev, Alexander, Unramified elements in cycle modules, J. Lond. Math. Soc. (2), 78, 1, 51-64, (2008) · Zbl 1155.14017 [31] Murre, J. P., Reduction of the proof of the non-rationality of a non-singular cubic threefold to a result of Mumford, Compositio Math., 27, 63-82, (1973) · Zbl 0271.14020 [32] [NS]NS J. Nicaise and E. Shinder, \em The motivic nearby fiber and degeneration of stable rationality, arXiv:1708.027901, 2017. [33] [Oka]oka T. Okada, \em Stable rationality of cyclic covers of projective spaces, arXiv:1604.08417, 2016. [34] Pukhlikov, A. V., Birational isomorphisms of four-dimensional quintics, Invent. Math., 87, 2, 303-329, (1987) · Zbl 0613.14011 [35] Pukhlikov, Aleksandr V., Birational automorphisms of Fano hypersurfaces, Invent. Math., 134, 2, 401-426, (1998) · Zbl 0964.14011 [36] Serre, Jean-Pierre, Galois cohomology, x+210 pp., (1997), Springer-Verlag, Berlin · Zbl 0902.12004 [37] Schreieder, Stefan, On the rationality problem for quadric bundles, Duke Math. J., 168, 2, 187-223, (2019) · Zbl 1409.14030 [38] Schreieder, Stefan, Quadric surface bundles over surfaces and stable rationality, Algebra Number Theory, 12, 2, 479-490, (2018) · Zbl 1397.14026 [39] Totaro, Burt, Hypersurfaces that are not stably rational, J. Amer. Math. Soc., 29, 3, 883-891, (2016) · Zbl 1376.14017 [40] Voevodsky, Vladimir, Motivic cohomology with $$\mathbf{Z}/2$$-coefficients, Publ. Math. Inst. Hautes \'Etudes Sci., 98, 59-104, (2003) · Zbl 1057.14028 [41] Voisin, Claire, On integral Hodge classes on uniruled or Calabi-Yau threefolds. Moduli spaces and arithmetic geometry, Adv. Stud. Pure Math. 45, 43-73, (2006), Math. Soc. Japan, Tokyo · Zbl 1118.14011 [42] Voisin, Claire, Some aspects of the Hodge conjecture, Jpn. J. Math., 2, 2, 261-296, (2007) · Zbl 1159.14005 [43] Voisin, Claire, Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal, J. Algebraic Geom., 22, 1, 141-174, (2013) · Zbl 1259.14006 [44] Voisin, Claire, Unirational threefolds with no universal codimension $$2$$ cycle, Invent. Math., 201, 1, 207-237, (2015) · Zbl 1327.14223
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