Skip to main content Accessibility help
×
Home

Jordan property for groups of birational selfmaps

  • Yuri Prokhorov (a1) (a2) (a3) and Constantin Shramov (a1) (a2)

Abstract

Assuming a particular case of the Borisov–Alexeev–Borisov conjecture, we prove that finite subgroups of the automorphism group of a finitely generated field over  $\mathbb{Q}$ have bounded orders. Further, we investigate which algebraic varieties have groups of birational selfmaps satisfying the Jordan property. Unless explicitly stated otherwise, all varieties are assumed to be algebraic, geometrically irreducible and defined over an arbitrary field $\Bbbk$ of characteristic zero.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Jordan property for groups of birational selfmaps
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Jordan property for groups of birational selfmaps
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Jordan property for groups of birational selfmaps
      Available formats
      ×

Copyright

References

Hide All
[Bir12]Birkar, C., Existence of log canonical flips and a special LMMP, Publ. Math. Inst. Hautes Études Sci. 115 (2012), 325368.
[BCHM10]Birkar, C., Cascini, P., Hacon, C. D. and McKernan, J., Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405468.
[BM08]Bierstone, E. and Milman, P. D., Functoriality in resolution of singularities, Publ. Res. Inst. Math. Sci. 44 (2008), 609639.
[Bor96]Borisov, A., Boundedness theorem for Fano log-threefolds, J. Algebraic Geom. 5 (1996), 119133.
[CR62]Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, vol. XI (Wiley-Interscience, New York, 1962).
[Edi10]Problems for the workshop ‘Subgroups of Cremona groups: classification’, 29–30 March 2010,ICMS, Edinburgh, available at http://www.mi.ras.ru/∼prokhoro/preprints/edi.pdf.
[Fak03]Fakhruddin, N., Questions on self maps of algebraic varieties, J. Ramanujan Math. Soc. 18 (2003), 109122.
[GHS03]Graber, T., Harris, J. and Starr, J., Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), 5767.
[Han87]Hanamura, M., On the birational automorphism groups of algebraic varieties, Compositio Math. 63 (1987), 123142.
[Han88]Hanamura, M., Structure of birational automorphism groups. I. Nonuniruled varieties, Invent. Math. 93 (1988), 383403.
[HMX10]Hacon, C., McKernan, J. and Xu, C., On the birational automorphisms of varieties of general type, Ann. of Math. (2) 177 (2013), 10771111.
[KMM87]Kawamata, Y., Matsuda, K. and Matsuki, K., Introduction to the minimal model problem, in Algebraic geometry, Sendai, 1985, Advanced Studies in Pure Mathematics, vol. 10 (North-Holland, Amsterdam, 1987), 283360.
[Kol92]Kollár, J. (ed.), Flips and abundance for algebraic threefolds: A summer seminar at the University of Utah (Salt Lake City, 1991), Astérisque, vol. 211 (Société Mathématique de France, 1992).
[Kol96]Kollár, J., Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32 (Springer, Berlin, 1996).
[KMMT00]Kollár, J., Miyaoka, Y., Mori, S. and Takagi, H., Boundedness of canonical Q-Fano 3-folds, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), 7377.
[KM98]Kollár, J. and Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998), with the collaboration of C. H. Clemens and A. Corti, translated from the 1998 Japanese original.
[Lan83]Lang, S., Fundamentals of Diophantine geometry (Springer, New York, 1983).
[Mat02]Matsuki, K., Introduction to the Mori program, Universitext (Springer, New York, 2002).
[Mer96]Merel, L., Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), 437449.
[MM86]Miyaoka, Y. and Mori, S., A numerical criterion for uniruledness, Ann. of Math. (2) 124 (1986), 6569.
[MS94]Morton, P. and Silverman, J. H., Rational periodic points of rational functions, Int. Math. Res. Not. IMRN 2 (1994), 97110.
[Poo12]Poonen, B., Uniform boundedness of rational points and preperiodic points, Preprint (2012), arXiv:1206.7104.
[Pop11]Popov, V. L., On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties, in Peter Russell’s Festschrift, Proceedings of the conference on Affine Algebraic Geometry held in Professor Russell’s honour, 1–5 June 2009, McGill Univ., Montreal, CRM Proceedings and Lecture Notes, vol. 54 (Centre de Recherches Mathématiques, Montréal, 2011), 289311.
[PS15]Prokhorov, Y. and Shramov, C., Jordan property for Cremona groups, Amer. J. Math. (2015), to appear, arXiv:1211.3563.
[Ser07]Serre, J.-P., Bounds for the orders of the finite subgroups of G (k), in Group representation theory (EPFL Press, Lausanne, 2007), 405450.
[Ser09]Serre, J.-P., A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field, Mosc. Math. J. 9 (2009), 193208.
[Zar10]Zarhin, Y. G., Theta groups and products of abelian and rational varieties, Proc. Edinb. Math. Soc. (2) 57 (2014), 299304.
[Zha06]Zhang, Q., Rational connectedness of log Q-Fano varieties, J. Reine Angew. Math. 590 (2006), 131142.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Related content

Powered by UNSILO

Jordan property for groups of birational selfmaps

  • Yuri Prokhorov (a1) (a2) (a3) and Constantin Shramov (a1) (a2)

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.