Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T23:44:43.541Z Has data issue: false hasContentIssue false
  • English
  • Français

Stable Rationality of Cyclic Covers of Projective Spaces

Part of: Birational geometry Cycles and subschemes Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  11 January 2019

Takuzo Okada
Takuzo Okada*
Affiliation:
Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

The main aim of this paper is to show that a cyclic cover of ℙn branched along a very general divisor of degree d is not stably rational, provided that n ≥ 3 and d ≥ n + 1. This generalizes the result of Colliot-Thélène and Pirutka. Generalizations for cyclic covers over complete intersections and applications to suitable Fano manifolds are also discussed.

Keywords

stable rationalitycyclic cover

MSC classification

Primary: 14E08: Rationality questions
Secondary: 14C15: (Equivariant) Chow groups and rings; motives 14G17: Positive characteristic ground fields
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 3 , August 2019 , pp. 667 - 682
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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.)

References

1.Beauville, A., A very general sextic double solid is not stable rational, Bull. Lond. Math. Soc. 48(2) (2016), 321–324.Google Scholar
2.Brown, G. and Kasprzyk, A., Four-dimensional projective orbifold hypersurfaces, Experiment. Math. 25(2) (2016), 176–193.Google Scholar
3.Chatzistamatiou, A. and Levine, M., Torsion orders of complete intersections, Algebra Number Theory 11(8) (2017), 1779–1835.Google Scholar
4.Cheltsov, I., Birationally super-rigid cyclic triple spaces, Izv. Math. 68(6) (2004), 1229–1275.Google Scholar
5.Cheltsov, I., Double spaces with singularities, Sb. Math. 199(1–2) (2008), 291–306.Google Scholar
6.Colliot-Thélène, J.-L. and Pirutka, A., Hypersurfaces quartiques de dimension 3: non rationalité stable, Ann. Sci. Éc. Norm. Supér. (4) 49(2) (2016), 371–397.Google Scholar
7.Colliot-Thélène, J.-L. and Pirutka, A., Cyclic covers that are not stably rational, Izv. Ross. Akad. Nauk Ser. Mat. 80(4) (2016), 35–48.Google Scholar
8.Hassett, B. and Tschinkel, Y., On stable rationality of Fano threefolds and del Pezzo fibration, J. Reine Angew. Math. (2016), DOI:10.1515/crelle-2016-0058.Google Scholar
9.Hassett, B., Kresch, A. and Tschinkel, Y., Stable rationality and conic bundles, Math. Ann. 365(3–4) (2016), 1201–1217.Google Scholar
10.Hassett, B., Pirutka, A. and Tschinkel, Y., Stable rationality of quadric surface bundles over surfaces, Acta Math. 2 (2018), 341–365.Google Scholar
11.Iano–Fletcher, A. R., Working with weighted complete intersections, Explicit birational geometry of 3-folds, London Mathematical Society Lecture Note Series, Volume 281 (Cambridge University Press, Cambridge, 2000).Google Scholar
12.Kollár, J., Nonrational hypersurfaces, J. Amer. Math. Soc. 8(1) (1995), 241–249.Google Scholar
13.Kollár, J., Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 32 (Springer-Verlag, Berlin, 1996).Google Scholar
14.Pirutka, A., Varieties that are not stably rational, zero-cycles and unramified cohomology, in Algebraic geometry: Salt Lake City, 2015, Proceedings of the Symposia in Pure Mathematics, Volume 97, part 2, pp. 459–484 (American Mathematical Society, 2018).Google Scholar
15.Pukhlikov, A. V., Birational automorphisms of double spaces with singularities, J. Math. Sci. (N.Y.) 85(4) (1997), 2128–2141.Google Scholar
16.Totaro, B., Hypersurfaces that are not stably rational, J. Amer. Math. Soc. 29(3) (2016), 883–891.Google Scholar
17.Voisin, C., Unirational threefolds with no universal codimension 2 cycle, Invent. Math. 201(1) (2015), 207–237.Google Scholar