Skip to main content
SpringerLink
Log in

Numerical Trivial Automorphisms of Enriques Surfaces in Arbitrary Characteristic

  • Chapter
  • First Online:
Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds

Part of the book series: Fields Institute Communications ((FIC,volume 67))

Abstract

We extend to arbitrary characteristic some known results on automorphisms of complex Enriques surfaces that act identically on the cohomology or the cohomology modulo torsion.

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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

Notes

  1. 1.

    The assertion is not true for non-classical Enriques surfaces. The analysis of this case reveals a missing case in [5] : X 3 σ may consist of an isolated fixed point and a connected curve.

References

  1. M. Artin, On isolated rational singularities of surfaces. Am. J. Math. 88, 129–136 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Artin, Wildly ramified Z/2 actions in dimension two. Proc. Am. Math. Soc. 52, 60–64 (1975)

    MathSciNet  MATH  Google Scholar 

  3. F. Cossec, I. Dolgachev, Enriques Surfaces. I. Progress in Mathematics, vol. 76 (Birkhäuser, Boston, 1989)

    Google Scholar 

  4. I. Dolgachev, On automorphisms of Enriques surfaces. Invent. Math. 76, 163–177 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  5. I. Dolgachev, J. Keum, Wild p-cyclic actions on K3-surfaces. J. Algebr. Geom. 10, 101–131 (2001)

    MathSciNet  MATH  Google Scholar 

  6. E. Horikawa, On the periods of Enriques surfaces. II. Math. Ann. 235, 217–246 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  7. L. Illusie, in Formule de Lefschetz, par A. Grothendieck. Cohomologie l-adique et fonctions L. Séminaire de Géometrie Algébrique du Bois-Marie 1965–1966 (SGA 5). Edité par Luc Illusie. Lecture Notes in Mathematics, vol. 589 (Springer, Berlin, 1977), pp. 73–137

    Google Scholar 

  8. B. Iversen, A fixed point formula for action of tori on algebraic varieties. Invent. Math. 16, 229–236 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  9. K. Kato, S. Saito, T. Saito, Artin characters for algebraic surfaces. Am. J. Math. 110, 49–75 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Keum, Wild p-cyclic actions on smooth projective surfaces with \(p_{g} = q = 0\). J. Algebra 244, 45–58 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Kondō, Enriques surfaces with finite automorphism groups. Jpn. J. Math. (N.S.) 12, 191–282 (1986)

    Google Scholar 

  12. S. Mukai, Numerically trivial involutions of Kummer type of an Enriques surface. Kyoto J. Math. 50, 889–902 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Mukai, Y. Namikawa, Automorphisms of Enriques surfaces which act trivially on the cohomology groups. Invent. Math. 77, 383–397 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  14. P. Salomonsson, Equations for some very special Enriques surfaces in characteristic two [math.AG/0309210]

    Google Scholar 

  15. J.-P. Serre, Le groupe de Cremona et ses sous-groupes finis. Séminaire Bourbaki. Volume 2008/2009. Exposés 997–1011. Astérisque No. 332 (2010), Exp. No. 1000, pp. 75–100

    Google Scholar 

  16. J. Silverman, The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 106 (Springer, New York, 1992)

    Google Scholar 

  17. E. Vinberg, in Some Arithmetical Discrete Groups in Lobachevsky Spaces. Discrete Subgroups of Lie Groups and Applications to Moduli, International Colloquium, Bombay, 1973 (Oxford University Press, Bombay, 1975), pp. 323–348

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 525 E. University Avenue, Ann Arbor, MI, 49109, USA

    Igor V. Dolgachev

Authors
  1. Igor V. Dolgachev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Igor V. Dolgachev .

Editor information

Editors and Affiliations

  1. , Mathematics Department, Stony Brook University, N/A, Stony Brook, 11794-3651, New York, USA

    Radu Laza

  2. , Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Matthias Schütt

  3. , Department of Math & Stats, Queen's University, University Ave., Kingston, K7L 3N6, Ontario, Canada

    Noriko Yui

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Dolgachev, I.V. (2013). Numerical Trivial Automorphisms of Enriques Surfaces in Arbitrary Characteristic. In: Laza, R., Schütt, M., Yui, N. (eds) Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds. Fields Institute Communications, vol 67. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6403-7_8

Download citation

Publish with us

Policies and ethics

Search

Navigation