Hostname: page-component-669899f699-vbsjw Total loading time: 0 Render date: 2025-04-25T19:22:58.950Z Has data issue: false hasContentIssue false
  • English
  • Français

RATIONALITY OVER NONCLOSED FIELDS OF FANO THREEFOLDS WITH HIGHER GEOMETRIC PICARD RANK

Part of: Surfaces and higher-dimensional varieties Birational geometry

Published online by Cambridge University Press:  05 August 2022

Alexander Kuznetsov  and
Yuri Prokhorov Open the ORCID record for Yuri Prokhorov [Opens in a new window]
Alexander Kuznetsov
Affiliation:
Steklov Mathematical Institute of the Russian Academy of Sciences, 8 Gubkin Street, Moscow 119991, Russia and Laboratory of Algebraic Geometry, National Research University Higher School of Economics, 6 Usachev Street, Moscow, 119048, Russia ([email protected])
Yuri Prokhorov*
Affiliation:
Steklov Mathematical Institute of the Russian Academy of Sciences, 8 Gubkin Street, Moscow 119991, Russia; Laboratory of Algebraic Geometry, National Research University Higher School of Economics, 6 Usachev Street, Moscow, 119048, Russia and Department of Algebra, Moscow State University, Moscow 119992, Russia
*
[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove rationality criteria over nonclosed fields of characteristic $0$ for five out of six types of geometrically rational Fano threefolds of Picard number $1$ and geometric Picard number bigger than $1$. For the last type of such threefolds, we provide a unirationality criterion and construct examples of unirational but not stably rational varieties of this type.

Keywords

Fano varietiesrationalityMori contractionsHilbert schemes

MSC classification

Primary: 14E08: Rationality questions
Secondary: 14J45: Fano varieties 14E30: Minimal model program (Mori theory, extremal rays)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 23 , Issue 1 , January 2024 , pp. 207 - 247
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Alzati, A. and Bertolini, M., Sulla razionalità delle $3$ -varietà di Fano con ${B}_2\ge2$ , Matematiche. 47(1) (1992), 63–74.Google Scholar
Benveniste, X., Sur le cone des $1$ -cycles effectifs en dimension $3$ , Math. Ann. 272 (1985), 257–265.CrossRefGoogle Scholar
Benoist, O. and Wittenberg, O., The Clemens–Griffiths method over non-closed fields,Algebraic Geometry. 7(6) (2020),696–721.CrossRefGoogle Scholar
Colliot-Thélène, J.-L. and Sansuc, J.-J., La descente sur les variétés rationnelles, II. Duke Math. J. 54(2) (1987), 375–492.CrossRefGoogle Scholar
Cutkosky, S., Elementary contractions of Gorensteinthreefolds, Math. Ann. 280(3) (1988),521–525.CrossRefGoogle Scholar
Iskovskikh, V. A., Anticanonical models of three-dimensionalalgebraic varieties, J. Sov. Math. 13 (1980),745–814.CrossRefGoogle Scholar
Kollár, J. and Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics. Volume 134 (Cambridge University Press, Cambridge, 1998). With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998Japanese original.Google Scholar
Kollár, J., Flops, Nagoya Math.J. 113 (1989), 15–36.CrossRefGoogle Scholar
Kollár, J., Unirationality of cubichypersurfaces, J. Inst. Math. Jussieu 1(3) (2002), 467–476.CrossRefGoogle Scholar
Kuznetsov, A. and Prokhorov, Y., Rationality of Fano threefolds over non-closed fields, 2019, arXiv e-print, 1911.08949. to appear in Amer. J. Math.Google Scholar
Kuznetsov, A. and Prokhorov, Y., On higher-dimensional del Pezzo varieties, 2022, arXive-print, 2206.01549.Google Scholar
Kuznetsov, A., Prokhorov, Y. and Shramov, C., Hilbert schemes of lines andconics and automorphism groups of Fano threefolds, Japanese J. Math. 13(1) (2018), 109–185.CrossRefGoogle Scholar
Kuznetsov, A., Derived categories of families of sextic del Pezzosurfaces, Int. Math. Res. Not. IMRN 12 (2021),9262–9339.CrossRefGoogle Scholar
Kuznetsov, A., Derived categories of families of Fanothreefolds, 2022, arXiv e-print, 2202.12345.Google Scholar
Mori, S. and Mukai, S., Classification of Fano $3$ -folds with ${B}_2\ge2$ , Manuscripta Math. 36(2) (1981/82), 147–162. Erratum:Manuscripta Math. 110 (2003), 407.CrossRefGoogle Scholar
Mori, S., Threefolds whose canonical bundles are not numericallyeffective, Ann. Math. 115 (1982),133–176.CrossRefGoogle Scholar
Mori, S. and Prokhorov, Y., Threefold extremal curve germs with one non-Gorenstein point, Izv. Math. 83(3) (2019), 565–612.CrossRefGoogle Scholar
Nishimura, H., Some remark on rational points,Mem. College Sci. Univ. Kyoto Ser. A Math. 29(2) (1955),189–192.Google Scholar
Nicaise, J. and Shinder, E., The motivic nearby fiber and degeneration of stable rationality, Invent. Math. 217(2) (2019), 377–413.CrossRefGoogle Scholar
Prokhorov, Y., G-Fano threefolds, I. Adv.Geom. 13(3) (2013), 389–418.CrossRefGoogle Scholar
Prokhorov, Y., G-Fano threefolds, II. Adv.Geom. 13(3)(2013), 419–434.CrossRefGoogle Scholar
Voskresenskiĭ, V. E., Algebraic groups and their birationalinvariants, Translations of Mathematical Monographs. Volume 179. (American MathematicalSociety, Providence, RI, 1998). Translated from the Russian manuscript by BorisKunyavski.Google Scholar
Zak, N. F., Quasi-triviality of forms of Segrevarieties, Russ. Math. Surv. 62(5) (2007),1018–1020.CrossRefGoogle Scholar