×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.