Skip to main content
SpringerLink
Log in

A virtually ample field that is not ample

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

A field K is called ample if for every geometrically integral K-variety V with a smooth K-point, V (K) is Zariski-dense in V. A field K is virtually ample if some finite extension of K is ample. We prove that there exists a virtually ample field that is not ample.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. Bary-Soroker and A. Fehm, Open problems in the theory of ample fields, in Geometric and Differential Galois Theories, Séminaires et Congrès, Vol. 27, Société Mathématique de France, Paris, 2013, pp. 1–11.

    MathSciNet  MATH  Google Scholar 

  2. G. Faltings, G. Wüstholz, F. Grunewald, N. Schappacher and U. Stuhler, Rational Points, Aspects of Mathematics, Vol. E6, Friedr. Vieweg & Sohn, Braunschweig, 1992.

  3. M. D. Fried and M. Jarden, Field Arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 11, Springer, Berlin, 2008.

  4. W.-D. Geyer and M. Jarden, Non-PAC fields whose Henselian closures are separably closed, Mathematical Research Letters 8 (2001), 509–519.

    Google Scholar 

  5. M. Jarden, Algebraic Patching, Springer Monographs in Mathematics, Springer, Heidelberg, 2011.

    Book  Google Scholar 

  6. M. Jarden and B. Poonen, Galois points on varieties, Journal of the Ramanujan Mathematical Society 31 (2016), 189–194.

    MathSciNet  MATH  Google Scholar 

  7. F. Pop, Embedding problems over large fields, Annals of Mathematics 144 (1996), 1–34.

    Google Scholar 

  8. P. Samuel, Compléments à un article de Hans Grauert sur la conjecture de Mordell, Institut des Hautes Études Scientifiques. Publications Mathématiques 29 (1966), 55–62.

    Google Scholar 

Download references

Acknowledgements

I would like to thank my advisor, Bjorn Poonen, for suggesting this problem to me, for several helpful discussions, and for his suggestions for improving the exposition. I also thank Moshe Jarden for helpful comments.

Author information

Authors and Affiliations

  1. School of Mathematics, University of Georgia, 452 Boyd Graduate Studies, 1023 D. W. Brooks Drive, Athens, GA, 30602, USA

    Padmavathi Srinivasan

Authors
  1. Padmavathi Srinivasan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Padmavathi Srinivasan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Srinivasan, P. A virtually ample field that is not ample. Isr. J. Math. 234, 769–776 (2019). https://doi.org/10.1007/s11856-019-1934-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-019-1934-y

Search

Navigation