Skip to main content
SpringerLink
Log in

Non rationalité stable sur les corps quelconques

  • Chapter
  • First Online:
Birational Geometry of Hypersurfaces

Part of the book series: Lecture Notes of the Unione Matematica Italiana ((UMILN,volume 26))

Abstract

This is a survey on (lack of) stable rationality over arbitrary fields (including algebraically closed fields). Topics addressed include: Rationality and unirationality, R-equivalence on rational points, Chow groups of zero-cycles, Galois action on the Picard group, Brauer group, higher unramified cohomology, global differentials, specialisation method (via R-equivalence), geometrically rational surfaces, cubic hypersurfaces.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. M. Artin, D. Mumford, Some elementary examples of unirational varieties which are not rational. Proc. Lond. Math. Soc. (3) 25, 75–95 (1972)

    Google Scholar 

  2. A. Auel, J.-L. Colliot-Thélène, R. Parimala, Universal unramified cohomology of cubic fourfolds containing a plane, in Brauer Groups and Obstruction Problems: Moduli Spaces and Arithmetic (Palo Alto, 2013). Progress in Mathematics, vol. 320 (Birkhäuser, Basel, 2017), pp. 29–56

    Google Scholar 

  3. A. Auel, A. Bigazzi, C. Böhning, H.-G. Graf von Bothmer, Universal triviality of the Chow group of 0-cycles and the Brauer group (2018), https://arxiv.org/abs/1806.02676

  4. A. Beauville, J.-L. Colliot-Thélène, J.-J. Sansuc, Sir Peter Swinnerton-Dyer, Variétés stablement rationnelles non rationnelles. Ann. Math. (2) 121(2), 283–318 (1985)

    Google Scholar 

  5. S. Bloch, V. Srinivas, Remarks on correspondences and algebraic cycles. Am. J. Math. 105, 1235–1253 (1983)

    Article  MathSciNet  Google Scholar 

  6. Y. Cao, Troisième groupe de cohomologie non ramifiée des torseurs universels sur les surfaces rationnelles. Épijournal de Géométrie Algébrique 2 (2018). Article Nr. 12

    Google Scholar 

  7. A. Chatzistamatiou, M. Levine, Torsion orders of complete intersections. Algebra Number Theory 11(8), 1779–1835 (2017)

    Article  MathSciNet  Google Scholar 

  8. J.-L. Colliot-Thélène, Hilbert’s theorem 90 for K 2, with application to the Chow groups of rational surfaces. Invent. math. 71, 1–20 (1983)

    Article  MathSciNet  Google Scholar 

  9. J.-L. Colliot-Thélène, Birational invariants, purity and the Gersten conjecture, in K-theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras (Santa Barbara, CA, 1992). Proceedings of Symposia in Pure Mathematics, vol. 58.1 (American Mathematical Society, Providence, RI, 1995), pp. 1–64

    Google Scholar 

  10. J.-L. Colliot-Thélène, 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, 589–606 (2005)

    Article  MathSciNet  Google Scholar 

  11. J.-L. Colliot-Thélène, Descente galoisienne sur le second groupe de Chow: mise au point et applications. Documenta mathematica, extra volume: Alexander S. Merkurjev’s Sixtieth Birthday (2015), pp. 195–220

    Google Scholar 

  12. J.-L. Colliot-Thélène, CH 0-trivialité universelle d’hypersurfaces cubiques presque diagonales. Algebr. Geom. 4(5), 597–602 (2017)

    Article  MathSciNet  Google Scholar 

  13. J.-L. Colliot-Thélène, Non rationalité stable d’hypersurfaces cubiques sur des corps non algébriquement clos, in Proceedings of the International Colloquium on K-Theory, TIFR, 2016, vol. 19 (Tata Institute of Fundamental Research Publications, Mumbai, 2018)

    Google Scholar 

  14. J.-L. Colliot-Thélène, Surfaces stablement rationnelles sur un corps quasi-fini, à paraître dans Izvestija RAN Ser. Mat. Tom 83 (2019), volume dédié à V. A. Iskovskikh, https://arxiv.org/abs/1711.09595

  15. J.-L. Colliot-Thélène, Introduction to work of Hassett-Pirutka-Tschinkel and Schreieder, in Birational Geometry of Hypersurfaces, ed. by A. Hochenegger et al. Lecture Notes of the Unione Matematica Italiana, vol. 26 (Springer, cham, 2019). https://doi.org/10.1007/978-3-030-18638-8_3

  16. J.-L. Colliot-Thélène, F. Ischebeck, L’équivalence rationnelle sur les cycles de dimension zéro des variétés algébriques réelles. C. R. Acad. Sc. Paris, t. 292, 723–725 (1981)

    Google Scholar 

  17. J.-L. Colliot-Thélène, M. Ojanguren, Variétés unirationnelles non rationnelles: au-delà de l’exemple d’Artin et Mumford. Invent. math. 97, 141–158 (1989)

    Article  MathSciNet  Google Scholar 

  18. J.-L. Colliot-Thélène, R. Parimala, Real components of algebraic varieties and étale cohomology. Invent. math. 101, 81–99 (1990)

    Article  MathSciNet  Google Scholar 

  19. J.-L. Colliot-Thélène, A. Pirutka, Hypersurfaces quartiques de dimension 3: non rationalité stable. Ann. Scient. Éc. Norm. Sup. 4e série, 49(2), 371–397 (2016)

    Google Scholar 

  20. J.-L. Colliot-Thélène, A. Pirutka, Revêtements cycliques non stablement rationnels (en russe). Izvestija RAN Ser. Mat., Ser. Math. Tom 80(4), 35–48 (2016)

    Google Scholar 

  21. J.-L. Colliot-Thélène, J.-J. Sansuc, La R-équivalence sur les tores. Ann. Sc. Éc. Norm. Sup. 10, 175–229 (1977)

    Article  Google Scholar 

  22. J.-L. Colliot-Thélène, J.-J. Sansuc, La descente sur les variétés rationnelles, II. Duke Math. J. 54(2), 375–492 (1987)

    Article  Google Scholar 

  23. J.-L. Colliot-Thélène, C. Voisin, Cohomologie non ramifiée et conjecture de Hodge entière. Duke Math. J. 161, 735–801 (2012)

    Article  MathSciNet  Google Scholar 

  24. J.-L. Colliot-Thélène, B. Kahn, R. Hoobler, The Bloch-Ogus-Gabber theorem, in Algebraic K-theory, Proceedings of the Great Lakes K-Theory Conference (Toronto 1996), ed. R. Jardine, V. Snaith. The Fields Institute for Research in Mathematical Sciences Communications Series, vol. 16 (American Mathematical Society, Providence, RI, 1997), pp. 31–94

    Google Scholar 

  25. W. Fulton, Intersection Theory. Ergebnisse der Math. und ihrer Grenzg. 3. Folge, Bd. 2 (Springer, Berlin, 1984/1998)

    Google Scholar 

  26. O. Gabber, Q. Liu, D. Lorenzini, The index of an algebraic variety. Invent. math. 192(3), 567–626 (2013)

    Article  MathSciNet  Google Scholar 

  27. P. Gille, Le problème de Kneser-Tits, exposé Bourbaki no. 983, Astérisque 326, 39–81 (2009)

    Google Scholar 

  28. S. Gille, Permutation modules and Chow motives of geometrically rational surfaces. J. Algebra 440, 443–463 (2015)

    Article  MathSciNet  Google Scholar 

  29. P. Gille, T. Szamuely, Central Simple Algebras and Galois Cohomology, 2nd edn. Cambridge Studies in Advances Mathematics, vol. 165 (Cambridge University Press, Cambridge, 2017)

    Google Scholar 

  30. A. Grothendieck, Le groupe de Brauer I, II, III, in Dix exposés sur la cohomologie des schémas (Masson et North Holland, Paris, 1968)

    Google Scholar 

  31. B. Hassett, Cubic fourfolds, K3 surfaces, and rationality questions, in Rationality Problems in Algebraic Geometry (Levico Terme 2015). Lecture Notes in Mathematics, vol. 2172, Fond. CIME/CIME Found. Subser. (Springer, Cham, 2016), pp. 29–66

    Google Scholar 

  32. B. Hassett, A. Pirutka, Y. Tschinkel, Stable rationality of quadric surface bundles over surfaces. Acta Math. 220(2), 341–365 (2018)

    Article  MathSciNet  Google Scholar 

  33. B. Kahn, R. Sujatha, Birational geometry and localisation of categories. With appendices by Jean-Louis Colliot-Thélène and by Ofer Gabber. Doc. Math. (2015), Extra vol.: Alexander S. Merkurjev’s sixtieth birthday, pp. 277–334

    Google Scholar 

  34. J. Kollár, Rational Curves on Algebraic Varieties. Ergebnisse der Math. und ihrer Grenzg. 3. Folge, Bd. 32 (Springer, Berlin, 1996)

    Google Scholar 

  35. J. Kollár, Rationally connected varieties over local fields. Ann. Math. (2) 150(1), 357–367 (1999)

    Google Scholar 

  36. D. Madore, Sur la spécialisation de la R-équivalence, prépublication (2005), http://perso.telecom-paristech.fr/~madore/specialz.pdf

  37. Yu. I. Manin, Formes cubiques : algèbre, géométrie, arithmétique (en russe) (Nauka, Moscou, 1972)

    Google Scholar 

  38. R. Mboro, Remarks on approximate decompositions of the diagonal, https://arxiv.org/abs/1708.02422

  39. A.S. Merkurjev, Unramified elements in cycle modules. J. Lond. Math. Soc. 78, 51–64 (2008)

    Article  MathSciNet  Google Scholar 

  40. J. Milnor, Algebraic K-theory and quadratic forms. Invent. math. 9, 318–344 (1970)

    Article  MathSciNet  Google Scholar 

  41. M. Ojanguren, The Witt Group and the Problem of Lüroth. With an introduction by Inta Bertuccioni. Dottorato di Ricerca in Matematica (ETS Editrice, Pisa, 1990)

    Google Scholar 

  42. D. Orlov, A. Vishik, V. Voevodsky, An exact sequence for \(K^M_{*}/2\) with applications to quadratic forms. Ann. Math. (2) 165(1), 1–13 (2007)

    Google Scholar 

  43. E. Peyre, Progrès en irrationalité [d’après C. Voisin, J.-L. Colliot-Thélène, B. Hassett, A. Kresch, A. Pirutka, B. Totaro, Y. Tschinkel et al.], dans Séminaire Bourbaki 69-ème année, exposé 1123, Astérisque (2016), pp. 1–26

    Google Scholar 

  44. A. Pirutka, Varieties that are not stably rational, zero-cycles and unramified cohomology, in Algebraic Geometry: Salt Lake City 2015. Proceedings of Symposia in Pure Mathematics, vol. 97.2 (American Mathematical Society, Providence, RI, 2018), pp. 459–483

    Google Scholar 

  45. S. Schreieder, On the rationality problem for quadric bundles. Duke Math. J. 168, 187–223 (2019)

    Article  MathSciNet  Google Scholar 

  46. S. Schreieder, Stably irrational hypersurfaces of small slopes. J. Am. Math. Soc. (to appear), https://arxiv.org/abs/1801.05397

  47. B. Segre, Sull’esistenza, sia nel campo razionale che nel campo reale, di involuzioni piane non birazionali. Rend. Acc. Naz. Lincei, Sc. fis. mat. e nat. 10, 94–97 (1951)

    Google Scholar 

  48. H.P.F. Swinnerton-Dyer, The birationality of cubic surfaces over a given field. Michigan Math. J. 17, 289–295 (1970)

    Article  MathSciNet  Google Scholar 

  49. B. Totaro, Hypersurfaces that are not stably rational. J. Am. Math. Soc. 29, 883–891 (2016)

    Article  MathSciNet  Google Scholar 

  50. C. Voisin, Unirational threefolds with no universal codimension 2 cycle. Invent. math. 201, 207–237 (2015)

    Article  MathSciNet  Google Scholar 

  51. C. Voisin, On the universal CH 0-group of cubic hypersurfaces. J. Eur. Math. Soc. 19(6), 1619–1653 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CNRS et Université Paris Sud, Mathématiques, Orsay, France

    Jean-Louis Colliot-Thélène

Authors
  1. Jean-Louis Colliot-Thélène
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Louis Colliot-Thélène .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Milano, Milano, Italy

    Andreas Hochenegger

  2. Institut für Mathematik, Johannes Gutenberg Universität Mainz, Mainz, Germany

    Manfred Lehn

  3. Dipartimento di Matematica, Università degli Studi di Milano, Milano, Italy

    Paolo Stellari

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Colliot-Thélène, JL. (2019). Non rationalité stable sur les corps quelconques. In: Hochenegger, A., Lehn, M., Stellari, P. (eds) Birational Geometry of Hypersurfaces. Lecture Notes of the Unione Matematica Italiana, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-030-18638-8_2

Download citation

Publish with us

Policies and ethics

Search

Navigation