Duke Mathematical Journal

On the rationality problem for quadric bundles

Stefan Schreieder

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We classify all positive integers n and r such that (stably) nonrational complex r-fold quadric bundles over rational n-folds exist. We show in particular that, for any n and r, a wide class of smooth r-fold quadric bundles over PCn are not stably rational if r[2n11,2n2]. In our proofs we introduce a generalization of the specialization method of Voisin and of Colliot-Thélène and Pirutka which avoids universally CH0-trivial resolutions of singularities.

Article information

Duke Math. J., Volume 168, Number 2 (2019), 187-223.

Received: 19 October 2017
Revised: 12 July 2018
First available in Project Euclid: 8 November 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14E08: Rationality questions [See also 14M20]
Secondary: 14D06: Fibrations, degenerations 14M20: Rational and unirational varieties [See also 14E08] 14J35: $4$-folds

rationality problem stable rationality decomposition of the diagonal unramified cohomology Lüroth problem


Schreieder, Stefan. On the rationality problem for quadric bundles. Duke Math. J. 168 (2019), no. 2, 187--223. doi:10.1215/00127094-2018-0041. https://projecteuclid.org/euclid.dmj/1541646041

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  • [1] H. Ahmadinezhad and T. Okada, Stable rationality of higher dimensional conic bundles, Épijournal Geom. Algébrique 2 (2018), Art. ID 5.
  • [2] M. Artin and D. Mumford, Some elementary examples of unirational varieties which are not rational, Proc. Lond. Math. Soc. (3) 25 (1972),75–95.
  • [3] A. Asok, Rationality problems and conjectures of Milnor and Bloch-Kato, Compos. Math. 149 (2013), 1312–1326.
  • [4] A. Auel, C. Böhning, H.-C. Graf v. Bothmer, and A. Pirutka, Conic bundles with nontrivial unramified Brauer group over threefolds, preprint, arXiv:1610.04995v2 [math.AG].
  • [5] A. Beauville, A very general sextic double solid is not stably rational, Bull. Lond. Math. Soc. 48 (2016), 312–324.
  • [6] A. Beauville, Erratum: A very general quartic double fourfold or fivefold is not stably rational, Algebr. Geom. 3 (2016), 137.
  • [7] C. Böhning and H.-C. Graf von Bothmer, On stable rationality of some conic bundles and moduli spaces of Prym curves, Comment. Math. Helv. 93 (2018), 133–155.
  • [8] J.-L. Colliot-Thélène, “Birational invariants, purity and the Gersten conjecture” in K-theory and Algebraic Geometry (Santa Barbara, Calif., 1992), Proc. Sympos. Pure Math. 58, Part 1, Amer. Math. Soc., Providence, 1995, 1–64.
  • [9] J.-L. Colliot-Thélène and M. Ojanguren, Variétés unirationnelles non rationnelles: au-delà de l’exemple d’Artin et Mumford, Invent. Math. 97 (1989), 141–158.
  • [10] J.-L. Colliot-Thélène and A. Pirutka, Cyclic covers that are not stably rational, Izv. Ross. Akad. Nauk Ser. Mat. 80 (2016), no. 4, 35–48; English translation in Izv. Math. 80 (2016), no. 4, 665–677.
  • [11] J.-L. Colliot-Thélène and A. Pirutka, Hypersurfaces quartiques de dimension 3: non rationalité stable, Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), 371–397.
  • [12] J.-L. Colliot-Thélène and C. Voisin, Cohomologie non ramifiée et conjecture de Hodge entière, Duke Math. J. 161 (2012), 735–801.
  • [13] R. Elman, N. Karpenko, and A. Merkurjev, Algebraic and Geometric Theory of Quadratic Forms, Amer. Math. Soc. Colloq. Publ. 56, Amer. Math. Soc., Providence, 2008.
  • [14] W. Fulton, Intersection Theory, 2nd ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 1998.
  • [15] B. Hassett, A. Kresch, and Y. Tschinkel, Stable rationality and conic bundles, Math. Ann. 365 (2016), 1201–1217.
  • [16] B. Hassett, A. Pirutka, and Y. Tschinkel, Stable rationality of quadric surface bundles over surfaces, preprint, arXiv:1603.09262v1 [math.AG].
  • [17] B. Hassett, A. Pirutka, and Y. Tschinkel, A very general quartic double fourfold is not stably rational, preprint, arXiv:1605.03220v1 [math.AG].
  • [18] B. Hassett, A. Pirutka, and Y. Tschinkel, Intersections of three quadrics in $\mathbb{P}^{7}$, preprint, arXiv:1706.01371v1 [math.AG].
  • [19] B. Hassett and Y. Tschinkel, On stable rationality of Fano threefolds and del Pezzo fibrations, preprint, arXiv:1601.07074v1 [math.AG].
  • [20] D. W. Hoffmann, Isotropy of quadratic forms over the function field of a quadric, Math. Z. 220 (1995), 461–476.
  • [21] V. A. Iskovskikh, On the rationality problem for conic bundles, Duke Math. J. 54 (1987), 271–294.
  • [22] N. A. Karpenko and A. S. Merkurjev, On standard norm varieties, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), 175–214.
  • [23] M. Kerz, The Gersten conjecture for Milnor $K$-theory, Invent. Math. 175 (2009), 1–33.
  • [24] J. Kollár, Nonrational hypersurfaces, J. Amer. Math. Soc. 8 (1995), 241–249.
  • [25] I. Krylov and T. Okada, Stable rationality of del Pezzo fibrations of low degree over projective spaces, preprint, arXiv:1701.08372v1 [math.AG].
  • [26] S. Lang, On quasi algebraic closure, Ann. of Math. (2) 55 (1952), 373–390.
  • [27] A. S. Merkurjev, Unramified elements in cycle modules, J. Lond. Math. Soc. (2) 78 (2008), 51–64.
  • [28] T. Okada, Stable rationality of cyclic covers of projective spaces, preprint, arXiv:1604.08417v4 [math.AG].
  • [29] D. Orlov, A. Vishik, and V. Voevodsky, An exact sequence for $K^{M}_{\ast}/2$ with applications to quadratic forms, Ann. of Math. (2) 165 (2007), 1–13.
  • [30] E. Peyre, Unramified cohomology and rationality problems, Math. Ann. 296 (1993), 247–268.
  • [31] E. Peyre, Progrès en irrationalité, Séminaire Bourbaki, Exp. 1123, 1–25, available at http://www.bourbaki.ens.fr/TEXTES/1123.pdf (accessed 8 October 2018).
  • [32] A. Pirutka, “Varieties that are not stably rational, zero-cycles and unramified cohomology” in Algebraic Geometry: Salt Lake City 2015, Proc Sympos. Pure Math. 97.2, Amer. Math. Soc., Providence, 2018, 459–483.
  • [33] M. Rost, Chow groups with coefficients, Doc. Math. 1 (1996), 319–393.
  • [34] J.-P. Serre, Galois Cohomology, Springer, Berlin, 1997.
  • [35] S. Schreieder, Quadric surface bundles over surfaces and stable rationality, Algebra Number Theory 12 (2018), 479–490.
  • [36] T. A. Springer, Sur les formes quadratiques d’indice zéro, C. R. Math. Acad. Sci. Paris 234 (1952), 1517–1519.
  • [37] B. Totaro, Birational geometry of quadrics, Bull. Soc. Math. France 137 (2009), 253–276.
  • [38] B. Totaro, Hypersurfaces that are not stably rational, J. Amer. Math. Soc. 29 (2016), 883–891.
  • [39] C. Vial, Algebraic cycles and fibrations, Doc. Math. 18 (2013), 1521–1553.
  • [40] V. Voevodsky, Motivic cohomology with $\mathbb{Z}/2$-coefficients, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 59–104.
  • [41] C. Voisin, “Degree $4$ unramified cohomology with finite coefficients and torsion codimension $3$ cycles” in Geometry and Arithmetic, EMS Ser. Congr. Rep., Eur. Math. Soc., Zurich, 2012, 347–368.
  • [42] C. Voisin, Unirational threefolds with no universal codimension $2$ cycle, Invent. Math. 201 (2015), 207–237.
  • [43] C. Voisin, (Stable) rationality is not deformation invariant, preprint, arXiv:1511.03591v4 [math.AG].