# Stable rationality of quadric and cubic surface bundle fourfolds

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## Abstract

We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the specialization method for the Chow group of 0-cycles. Our main result is that a very general hypersurface *X* of bidegree (2, 3) in Open image in new window is not stably rational. Via projections onto the two factors, Open image in new window is a cubic surface bundle and Open image in new window is a conic bundle, and we analyze the stable rationality problem from both these points of view. Also, we introduce, for any \(n\geqslant 4\), new quadric surface bundle fourfolds Open image in new window with discriminant curve Open image in new window of degree 2*n*, such that \(X_n\) has nontrivial unramified Brauer group and admits a universally \(\mathrm {CH}_0\)-trivial resolution.

## Keywords

Stable rationality Brauer group Quadric bundles Cubic surface bundles Fano fourfolds## Mathematics Subject Classification

14C35 14D06 14E05 14E08 14F22 14J20 14J26## References

- 1.Ahmadinezhad, H., Okada, T.: Stable rationality of higher dimensional conic bundles (2016). arXiv:1612.04206
- 2.Auel, A., Colliot-Thélène, J.L., Parimala, R.: Universal unramified cohomology of cubic fourfolds containing a plane. In: Auel, A., Hassett, B., Várilly-Alvarado, A., Viray, B. (eds.) Brauer Groups and Obstruction Problems. Progress in Mathematics, vol. 320, pp. 29–56. Birkhäuser, Basel (2017)CrossRefGoogle Scholar
- 3.Auel, A., Böhning, Chr., Graf von Bothmer, H.-Chr., Pirutka, A.: Conic bundles with nontrivial unramified Brauer group over threefolds (2016). arXiv:1610.04995
- 4.Beauville, A.: A very general sextic double solid is not stably rational. Bull. London Math. Soc
**48**(2), 321–324 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Beklemishev, N.D.: Invariants of cubic forms of four variables. Moscow Univ. Math. Bull.
**37**, 54–62 (1982)zbMATHGoogle Scholar - 6.Bloch, S., Ogus, A.: Gersten’s conjecture and the homology of schemes. Ann. Sci. Éc. Norm. Supér.
**7**, 181–201 (1974)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Böhning, Chr., Graf von Bothmer, H.-Chr.: On stable rationality of some conic bundles and moduli spaces of Prym curves. Comment. Math. Helv.
**93**(1), 133–155 (2018)Google Scholar - 8.Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput.
**24**(3–4), 235–265 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Chatzistamatiou, A., Levine, M.: Torsion orders of complete intersections. Algebra Number Theory
**11**(8), 1779–1835 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Clebsch, A.: Ueber eine Transformation der homogenen Functionen dritter Ordnung mit vier Veränderlichen. J. Reine Angew. Math.
**58**, 109–126 (1861)MathSciNetCrossRefGoogle Scholar - 11.Clebsch, A.: Zur Theorie der algebraischer Flächen. J. Reine Angew. Math.
**58**, 93–108 (1861)MathSciNetCrossRefGoogle Scholar - 12.Colliot-Thélène, J.-L.: Birational invariants, purity and the Gersten conjecture. In: Jacob, W., Rosenberg, A. (eds.) \(K\)-Theory and Algebraic Geometry. Proceedings of Symposia in Pure Mathematics, vol. 58.1, pp. 1–64. American Mathematical Society, Providence (1995)Google Scholar
- 13.Colliot-Thélène, J.-L.: Un théorème de finitude pour le groupe de Chow des zéro-cycles d’un groupe algébrique linéaire sur un corps \(p\)-adique. Invent. Math.
**159**(3), 589–606 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Colliot-Thélène, J.-L., Ojanguren, M.: Variétés unirationnelles non rationnelles: au-delà de l’exemple d’Artin et Mumford. Invent. Math.
**97**(1), 141–158 (1989)MathSciNetCrossRefzbMATHGoogle Scholar - 15.Colliot-Thélène, J.-L., Pirutka, A.: Cyclic covers that are not stably rational. Izv. Math.
**80**(4), 665–677 (2016) (in Russian)Google Scholar - 16.Colliot-Thélène, J.-L., Pirutka, A.: Hypersurfaces quartiques de dimension 3: non-rationalité stable. Ann. Sci. Éc. Norm. Supér.
**49**(2), 371–397 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Coombes, K.R.: Every rational surface is separably split. Comment. Math. Helv.
**63**(2), 305–311 (1988)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Dolgachev, I.: Lectures on Invariant Theory. London Mathematical Society Lecture Note Series, vol. 296. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
- 19.Edge, W.L.: The discriminant of a cubic surface. Proc. R. Irish Acad. Sect. A
**80**(1), 75–78 (1980)MathSciNetzbMATHGoogle Scholar - 20.Elsenhans, A.-S., Jahnel, J.: The discriminant of a cubic surface. Geom. Dedicata
**159**, 29–40 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 21.Gabber, O., Liu, Q., Lorenzini, D.: The index of an algebraic variety. Invent. Math.
**192**(3), 567–626 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Goursat, É.: Étude des surfaces qui admettent tous les plans de symétrie d’un polyèdre régulier. Ann. Sci. Éc. Norm. Supér.
**4**, 159–200 (1887)CrossRefzbMATHGoogle Scholar - 23.Hassett, B., Kresch, A., Tschinkel, Yu.: Stable rationality and conic bundles. Math. Ann.
**365**(3–4), 1201–1217 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Hassett, B., Pirutka, A., Tschinkel, Yu.: Stable rationality of quadric surface bundles over surfaces (2016). arXiv:1603.09262
- 25.Hassett, B., Pirutka, A., Tschinkel, Yu.: A very general quartic double fourfold is not stably rational (2016). arXiv:1605.03220
- 26.Hassett, B., Pirutka, A., Tschinkel, Yu.: Intersections of three quadrics in \(\mathbb{P}^7\) (2017). arXiv:1706.01371
- 27.Hassett, B., Tschinkel, Yu.: On stable rationality of Fano threefolds and del Pezzo fibrations. J. Reine Angew. Math. https://doi.org/10.1515/crelle-2016-0058
- 28.Kollár, J.: Nonrational hypersurfaces. J. Amer. Math. Soc.
**8**(1), 241–249 (1995)MathSciNetCrossRefzbMATHGoogle Scholar - 29.Krylov, I., Okada, T.: Stable rationality of del Pezzo fibrations of low degree over projective spaces (2017). arXiv:1701.08372
- 30.Liu, Q.: Algebraic Geometry and Arithmetic Curves. Oxford Graduate Texts in Mathematics, vol. 6. Oxford University Press, Oxford (2002). Translated from the French by Reinie ErnéGoogle Scholar
- 31.Okada, T.: Stable rationality of cyclic covers of projective spaces (2016). arXiv:1604.08417
- 32.Okada, T.: Smooth weighted hypersurfaces that are not stably rational (2017). arXiv:1709.07748
- 33.Okada, T.: Stable rationality of index one Fano hypersurfaces containing a linear space (2017). arXiv:1709.07757
- 34.Pirutka, A.: Varieties that are not stably rational, zero-cycles and unramified cohomology (2016). arXiv:1603.09261
- 35.Salmon, G.: On quaternary cubics. Philos. Trans. R. Soc. Lond.
**150**, 229–239 (1860)CrossRefGoogle Scholar - 36.Schreieder, S.: Quadric surface bundles over surfaces and stable rationality (2017). arXiv:1706.01358 (to appear in Algebra Number Theory)
- 37.Schreieder, S.: On the rationality problem for quadric bundles (2017). arXiv:1706.01356
- 38.Sir Swinnerton-Dyer, P.: The Brauer group of cubic surfaces. Math. Proc. Cambridge Philos. Soc.
**113**(3), 449–460 (1993)MathSciNetCrossRefzbMATHGoogle Scholar - 39.Totaro, B.: Hypersurfaces that are not stably rational. J. Amer. Math. Soc.
**29**(3), 883–891 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 40.Voisin, C.: Unirational threefolds with no universal codimension \(2\) cycle. Invent. Math.
**201**(1), 207–237 (2015)MathSciNetCrossRefzbMATHGoogle Scholar