Skip to main content
SpringerLink
Log in

Smooth tropical surfaces with infinitely many tropical lines

  • Published:
Arkiv för Matematik

Abstract

We study the tropical lines contained in smooth tropical surfaces in ℝ3. On smooth tropical quadric surfaces we find two one-dimensional families of tropical lines, like in classical algebraic geometry. Unlike the classical case, however, there exist smooth tropical surfaces of any degree with infinitely many tropical lines.

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. Franz, M., Convex—a maple package for convex geometry, Version 1.1, 2006. http://www-fourier.ujf-grenoble.fr/~franz/convex/

  2. Gathmann, A., Tropical algebraic geometry, Jahresber. Deutsch. Math.-Verein.108 (2006), 3–32.

    MATH  MathSciNet  Google Scholar 

  3. Gathmann, A. and Markwig, H., The numbers of tropical plane curves through points in general position, J. Reine Angew. Math.602 (2007), 155–177.

    MATH  MathSciNet  Google Scholar 

  4. Gelfand, I. M., Kapranov, M. M. and Zelevinsky, A. V., Discriminants, Resultants, and Multidimensional Determinants, Mathematics: Theory & Applications, Birkhäuser, Boston, MA, 1994.

    Book  MATH  Google Scholar 

  5. Katz, E., Markwig, H. and Markwig, T., The j-invariant of a plane tropical cubic, J. Algebra320 (2008), 3832–3848.

    Article  MATH  MathSciNet  Google Scholar 

  6. Maple 10 software, copyright by Waterloo Maple Inc.

  7. Mikhalkin, G., Enumerative tropical algebraic geometry in ℝ2, J. Amer. Math. Soc.18 (2005), 313–377.

    Article  MATH  MathSciNet  Google Scholar 

  8. Mikhalkin, G., Moduli spaces of rational tropical curves, Preprint, 2007. arXiv:0704.0839

  9. Mikhalkin, G. and Zharkov, I., Tropical curves, their Jacobians and Theta functions, in Curves and Abelian Varieties, Contemporary Mathematics 465, pp. 203–230, Amer. Math. Soc., Providence, RI, 2008.

    Google Scholar 

  10. Speyer, D. and Sturmfels, B., The tropical Grassmannian, Adv. Geom.4 (2004), 389–411.

    Article  MATH  MathSciNet  Google Scholar 

  11. Sturmfels, B., Richter-Gebert, J. and Theobald, T., First steps in tropical geometry, in Idempotent Mathematics and Mathematical Physics, Contemporary Mathematics 377, pp. 289–317, Amer. Math. Soc., Providence, RI, 2005.

    Google Scholar 

  12. Vigeland, M. D., Tropical lines on smooth tropical surfaces, Preprint, 2007. arXiv:0708.3847

  13. Vigeland, M. D., The group law on a tropical elliptic curve, Math. Scand.104 (2009), 188–204.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Oslo, NO-0316, Oslo, Norway

    Magnus Dehli Vigeland

Authors
  1. Magnus Dehli Vigeland
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Magnus Dehli Vigeland.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vigeland, M.D. Smooth tropical surfaces with infinitely many tropical lines. Ark Mat 48, 177–206 (2010). https://doi.org/10.1007/s11512-009-0116-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11512-009-0116-2

Keywords

Search

Navigation