Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter October 26, 2016

On stable rationality of Fano threefolds and del Pezzo fibrations

Brendan Hassett EMAIL logo and Yuri Tschinkel

Abstract

We prove that very general non-rational Fano threefolds which are not birational to cubic threefolds are not stably rational.

Award Identifier / Grant number: 1551514

Funding statement: The first author was supported by NSF grant 1551514.

Acknowledgements

We are grateful to Andrew Kresch for his foundational contributions that made this research possible. We also benefitted from conversations with Alena Pirutka.

References

[1] V. A. Alekseev, On conditions for the rationality of three-folds with a pencil of del Pezzo surfaces of degree 4, Mat. Zametki 41 (1987), no. 5, 724–730, 766. Search in Google Scholar

[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. 10.1007/978-1-4757-4265-7_29Search in Google Scholar

[3] A. Auel, J.-L. Colliot-Thélène and R. Parimala, Universal unramified cohomology of cubic fourfolds containing a plane, Brauer groups and obstruction problems: Moduli spaces and arithmetic, Progr. Math. 320, Birkhäuser, Basel (2017). 10.1007/978-3-319-46852-5_4Search in Google Scholar

[4] A. Beauville, Variétés de Prym et jacobiennes intermédiaires, Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), no. 3, 309–391. 10.24033/asens.1329Search in Google Scholar

[5] A. Beauville, Le groupe de monodromie des familles universelles d’hypersurfaces et d’intersections complètes, Complex analysis and algebraic geometry (Göttingen 1985), Lecture Notes in Math. 1194, Springer, Berlin (1986), 8–18. 10.1007/BFb0076991Search in Google Scholar

[6] A. Beauville, A very general sextic double solid is not stably rational, Bull. Lond. Math. Soc. 48 (2016), no. 2, 321–324. 10.1112/blms/bdv098Search in Google Scholar

[7] I. Cheltsov, Nonrational nodal quartic threefolds, Pacific J. Math. 226 (2006), no. 1, 65–81. 10.2140/pjm.2006.226.65Search in Google Scholar

[8] 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. 10.1070/IM8429Search in Google Scholar

[9] J.-L. Colliot-Thélène and A. Pirutka, Hypersurfaces quartiques de dimension 3: Non-rationalité stable, Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), no. 2, 371–397. 10.24033/asens.2285Search in Google Scholar

[10] A. Grothendieck, Séminaire de géométrie algébrique du Bois Marie 1960–1961. Revêtements étales et groupe fondamental (SGA 1), Lecture Notes in Math. 224, Springer, Berlin 1971. 10.1007/BFb0058656Search in Google Scholar

[11] B. Hassett, A. Kresch and Y. Tschinkel, On the moduli of degree 4 Del Pezzo surfaces, Development of moduli theory, Adv. Stud. Pure Math. 69, Mathematical Society of Japan, Tokyo (2016), 349–386. Search in Google Scholar

[12] B. Hassett, A. Kresch and Y. Tschinkel, Stable rationality and conic bundles, Math. Ann. 365 (2016), no. 3–4, 1201–1217. 10.1007/s00208-015-1292-ySearch in Google Scholar

[13] B. Hassett and Y. Tschinkel, Quartic del Pezzo surfaces over function fields of curves, Cent. Eur. J. Math. 12 (2014), no. 3, 395–420. 10.2478/s11533-013-0354-1Search in Google Scholar

[14] B. Hassett and Y. Tschinkel, On stable rationality of Fano threefolds and Del Pezzo fibrations, preprint (2016), http://arxiv.org/abs/1601.07074. 10.1515/crelle-2016-0058Search in Google Scholar

[15] B. Hassett and A. Várilly-Alvarado, Failure of the Hasse principle on general K3 surfaces, J. Inst. Math. Jussieu 12 (2013), no. 4, 853–877. 10.1017/S1474748012000904Search in Google Scholar

[16] A. Iliev and D. Markushevich, The Abel–Jacobi map for a cubic threefold and periods of Fano threefolds of degree 14, Doc. Math. 5 (2000), 23–47. Search in Google Scholar

[17] V. A. Iskovskikh and Y. G. Prokhorov, Fano varieties, Algebraic geometry. V, Encyclopaedia Math. Sci. 47, Springer, Berlin (1999), 1–247. Search in Google Scholar

[18] S. Mori and S. Mukai, Classification of Fano 3-folds with B22, Manuscripta Math. 36 (1981/82), no. 2, 147–162. 10.1007/BF01170131Search in Google Scholar

[19] V. G. Sarkisov, On conic bundle structures, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 2, 371–408, 432. 10.1070/IM1983v020n02ABEH001354Search in Google Scholar

[20] B. Totaro, Hypersurfaces that are not stably rational, J. Amer. Math. Soc. 29 (2016), no. 3, 883–891. 10.1090/jams/840Search in Google Scholar

[21] C. Voisin, Unirational threefolds with no universal codimension 2 cycle, Invent. Math. 201 (2015), no. 1, 207–237. 10.1007/s00222-014-0551-ySearch in Google Scholar

Received: 2016-02-01
Revised: 2016-09-09
Published Online: 2016-10-26
Published in Print: 2019-06-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 1.2.2023 from https://www.degruyter.com/document/doi/10.1515/crelle-2016-0058/html
Scroll Up Arrow