Skip to main content
SpringerLink
Log in

2-Elementary Subgroups of the Space Cremona Group

  • Conference paper
  • First Online:
Book cover Automorphisms in Birational and Affine Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 79))

Abstract

We give a sharp bound for orders of elementary abelian two-groups of birational automorphisms of rationally connected threefolds.

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 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover 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. A. Beauville, Variétés de Prym et jacobiennes intermédiaires. Ann. Sci. École Norm. Sup. (4) 10(3), 309–391 (1977)

    Google Scholar 

  2. A. Beauville, in Surfaces algébriques complexes. Astérisque, vol. 54 (Société Mathématique de France, Paris, 1978)

    Google Scholar 

  3. A. Beauville, p-elementary subgroups of the Cremona group. J. Algebra 314(2), 553–564 (2007)

    Google Scholar 

  4. A.B. Coble, in Algebraic Geometry and Theta Functions. American Mathematical Society Colloquium Publications, vol. 10 (American Mathematical Society, New York, 1929)

    Google Scholar 

  5. I. Dolgachev, V. Iskovskikh, Finite subgroups of the plane Cremona group, in Algebra, Arithmetic, and Geometry: In Honor of Yu.I. Manin. Vol. I. Progress in Mathematics, vol. 269 (Birkhäuser, Boston, 2009), pp. 443–548

    Google Scholar 

  6. V.A. Iskovskikh, Anticanonical models of three-dimensional algebraic varieties. J. Sov. Math. 13, 745–814 (1980)

    Article  MATH  Google Scholar 

  7. V. Iskovskikh, Y. Prokhorov, in Fano Varieties. Algebraic Geometry V.. Encyclopaedia of Mathematical Sciences, vol. 47 (Springer, Berlin, 1999)

    Google Scholar 

  8. Y. Kawamata, Boundedness of Q-Fano threefolds, in Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989). Contemporary Mathematics, vol. 131 (American Mathematical Society, Providence, 1992), pp. 439–445

    Google Scholar 

  9. J. Kollár, Y. Miyaoka, S. Mori, H. Takagi, Boundedness of canonical Q-Fano 3-folds. Proc. Jpn. Acad. Ser. A Math. Sci. 76(5), 73–77 (2000)

    Article  MATH  Google Scholar 

  10. S. Mori, On 3-dimensional terminal singularities. Nagoya Math. J. 98, 43–66 (1985)

    MATH  MathSciNet  Google Scholar 

  11. S. Mori, Y. Prokhorov, Multiple fibers of del Pezzo fibrations. Proc. Steklov Inst. Math. 264(1), 131–145 (2009)

    Article  MathSciNet  Google Scholar 

  12. Y. Namikawa, Smoothing Fano 3-folds. J. Algebr. Geom. 6(2), 307–324 (1997)

    MATH  MathSciNet  Google Scholar 

  13. V. Nikulin, Finite automorphism groups of Kähler K3 surfaces. Trans. Mosc. Math. Soc. 2, 71–135 (1980)

    MATH  Google Scholar 

  14. Y. Prokhorov, p-elementary subgroups of the Cremona group of rank 3, in Classification of Algebraic Varieties. EMS Series of Congress Reports (European Mathematical Society, Zürich, 2011), pp. 327–338

    Google Scholar 

  15. Y. Prokhorov, Simple finite subgroups of the Cremona group of rank 3. J. Algebr. Geom. 21, 563–600 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Y. Prokhorov, G-Fano threefolds, I. Adv. Geom. 13(3), 389–418 (2013)

    MATH  MathSciNet  Google Scholar 

  17. Y. Prokhorov, G-Fano threefolds, II. Adv. Geom. 13(3), 419–434 (2013)

    MATH  MathSciNet  Google Scholar 

  18. Y. Prokhorov, On birational involutions of P 3. Izvestiya Math. Russ. Acad. Sci. 77(3), 627–648 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  19. M. Reid, Young person’s guide to canonical singularities, in Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985). Proceedings of Symposia in Pure Mathematics, vol. 46 (American Mathematical Society, Providence, 1987), pp. 345–414

    Google Scholar 

  20. J.-P. Serre, Bounds for the orders of the finite subgroups of G(k), in Group Representation Theory (EPFL Press, Lausanne, 2007), pp. 405–450

    Google Scholar 

  21. K.-H. Shin, 3-dimensional Fano varieties with canonical singularities. Tokyo J. Math. 12(2), 375–385 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  22. V. Shokurov, 3-fold log flips. Russ. Acad. Sci. Izv. Math. 40(1), 95–202 (1993)

    MathSciNet  Google Scholar 

  23. R. Varley, Weddle’s surfaces, Humbert’s curves, and a certain 4-dimensional abelian variety. Am. J. Math. 108(4), 931–952 (1986)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The work was completed during the author’s stay at the International Centre for Theoretical Physics, Trieste. The author would like to thank ICTP for hospitality and support.

I acknowledge partial supports by RFBR grants No. 11-01-00336-a, the grant of Leading Scientific Schools No. 4713.2010.1, Simons-IUM fellowship, and AG Laboratory SU-HSE, RF government grant ag. 11.G34.31.0023.

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute, 8 Gubkina str., Moscow, 119991, Russia

    Yuri Prokhorov

  2. Laboratory of Algebraic Geometry, SU-HSE, 7 Vavilova str., Moscow, 117312, Russia

    Yuri Prokhorov

  3. Faculty of Mathematics, Department of Algebra, Moscow State University, Vorobievy Gory, Moscow, 119991, Russia

    Yuri Prokhorov

Authors
  1. Yuri Prokhorov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuri Prokhorov .

Editor information

Editors and Affiliations

  1. School of Mathematics, University of Edinburgh, Edinburgh, United Kingdom

    Ivan Cheltsov

  2. Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

    Ciro Ciliberto

  3. Faculty of Mathematics, Ruhr University Bochum, Bochum, Germany

    Hubert Flenner

  4. Department of Mathematics, University of California San Diego, La Jolla, California, USA

    James McKernan

  5. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

    Yuri G. Prokhorov

  6. Institute Fourier, UMR 5582, CNRS-UJF, University of Grenoble I, Saint Martin D’Hères, France

    Mikhail Zaidenberg

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Prokhorov, Y. (2014). 2-Elementary Subgroups of the Space Cremona Group. In: Cheltsov, I., Ciliberto, C., Flenner, H., McKernan, J., Prokhorov, Y., Zaidenberg, M. (eds) Automorphisms in Birational and Affine Geometry. Springer Proceedings in Mathematics & Statistics, vol 79. Springer, Cham. https://doi.org/10.1007/978-3-319-05681-4_12

Download citation

Publish with us

Policies and ethics

Search

Navigation