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Collectanea Mathematica

, Volume 67, Issue 1, pp 1–19 | Cite as

Embeddings and immersions of tropical curves

  • Dustin CartwrightEmail author
  • Andrew Dudzik
  • Madhusudan Manjunath
  • Yuan Yao
Article

Abstract

We construct immersions of trivalent abstract tropical curves in the Euclidean plane and embeddings of all abstract tropical curves in higher dimensional Euclidean space. Since not all curves have an embedding in the plane, we define the tropical crossing number of an abstract tropical curve to be the minimum number of self-intersections, counted with multiplicity, over all its immersions in the plane. We show that the tropical crossing number is at most quadratic in the number of edges and this bound is sharp. For curves of genus up to two, we systematically compute the crossing number. Finally, we use our immersed tropical curves to construct totally faithful nodal algebraic curves via lifting results of Mikhalkin and Shustin.

Keywords

Tropical curves Metric graphs Embeddings Immersions Crossing number Faithful tropicalizations Newton polygons 

Mathematics Subject Classification

14T05 

Notes

Acknowledgments

Our work was initiated in the Mathematics Research Communities (MRC) program on “Tropical and Non-Archimedean Geometry” held in Snowbird, Utah in Summer 2013. We thank the MRC as well as the organizers of our program, Matt Baker and Sam Payne, for their support and guidance during and after the program. We would also like to thank Mandy Cheung, Lorenzo Fantini, Jennifer Park, and Martin Ulirsch for sharing their results from the same workshop on log deformations of curves [9], which helped to motivate our work. We would also like to acknowledge Melody Chan and Bernd Sturmfels for several interesting discussions. Madhusudan Manjunath was supported by a Feoder-Lynen Fellowship of the Humboldt Foundation and an AMS-Simons Travel Grant during this work.

References

  1. 1.
    Amini, O., Baker, M., Brugallé, E., Rabinoff, J.: Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta. Res. Math. Sci. arXiv:1303.4812 (2015)
  2. 2.
    Ardila, F., Klivans, C.: The Bergman complex of a matroid and phylogenetic trees. J. Comb. Theory Ser. B 96(1), 38–49 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Baker, M.: Specialization of linear systems from curves to graphs. Algebra Number Theory 2(6), 613–653 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Brodsky, S., Joswig, M., Morrison, R., Sturmfels, B.: Moduli of tropical plane curves. Res. Math. Sci. arXiv:1409.4395 (2015)
  5. 5.
    Baker, M., Len, Y., Morrison, R., Pflueger, N., Ren, Q.: Bitangents of tropical plane quartic curves. arXiv:1404.7568 (2014)
  6. 6.
    Baker, M., Payne, S., Rabinoff, J.: Nonarchimedean geometry, tropicalization, and metrics on curves. arXiv:1104.0320 (2011)
  7. 7.
    Castryck, W., Cools, F.: Newton polygons and curve gonalities. J. Algebr. Comb. 35(3), 345–366 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cools, F., Draisma, J., Payne, S., Robeva, E.: A tropical proof of the Brill-Noether theorem. Adv. Math. 230(2), 759–776 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Cheung, M.-W., Fantini, L., Park, J., Ulirsch, M.: Faithful realizability of tropical curves. arXiv:1410.4152 (2014)
  10. 10.
    Fejes-Tóth, L., Makai Jr, E.: On the thinnest non-separable lattice of convex plates. Stud. Sci. Math. Hung. 9, 191–193 (1974)Google Scholar
  11. 11.
    Gathmann, A., Kerber, M.: A Riemann-Roch theorem in tropical geometry. Math. Z. 259(1), 217–230 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Gubler, W., Rabinoff, J., Werner, A.: Skeletons and tropicalizations. arXiv:1404.7044 (2014)
  13. 13.
    Hartshorne, R.: Algebraic Geometry, volume 52 of Graduate Texts in Mathematics. Springer, Berlin (1977)Google Scholar
  14. 14.
    Haase, C., Schicho, J.: Lattice polygons and the number \(2i+7\). Am. Math. Mon. 116(2), 151–165 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Jensen, D., Payne, S.: Tropical independence I: shapes of divisors and a proof of the Giesker–Petri theorem. Algebra Number Theory 8(9), 2043–2066 (2014)Google Scholar
  16. 16.
    Mikhalkin, G.: Enumerative tropical algebraic geometry in \({\mathbb{R}}^2\). J. Am. Math. Soc. 18, 313–377 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Maclagan, D., Sturmfels, B.: Introduction to tropical geometry. Graduate studies in mathematics, vol. 161. American Mathematical Society, Providence, RIGoogle Scholar
  18. 18.
    Mikhalkin, G., Zharkov, I.: Tropical curves, their Jacobians and theta functions. In: Curves and Abelian Varieties, vol. 465 of Contemp. Math., pp. 203–230. Am. Math. Soc. (2008)Google Scholar
  19. 19.
    Mikhalkin, G., Zharkov, I.: Tropical eigenwave and intermediate Jacobians. In: Homological Mirror Symmetry and Tropical Geometry, vol. 15 of Lecture Notes of the Unione Matematica Italiana, pp. 309–349. Springer, Berlin (2014)Google Scholar
  20. 20.
    Richter-Gebert, J., Sturmfels, B., Theobald, T.: First steps in tropical geometry. In: Idempotent Mathematics and Mathematical Physics, vol. 377 of Contemp. Math. Am. Math. Soc. (2005)Google Scholar
  21. 21.
    Richter, R.B., Salazar, G.: Crossing numbers. In: Beineke, L.W., Wilson, R. (eds.) Topics in Topological Graph Theory, vol. 128 of Encyclopedia Math. Appl., pp. 133–150. Cambridge University Press, Cambridge (2009)Google Scholar
  22. 22.
    Scott, P.R.: On convex lattice polygons. Bull. Aust. Math. Soc. 15(3), 395–399 (1976)zbMATHCrossRefGoogle Scholar
  23. 23.
    Shustin, E.: A tropical approach to enumerative geometry. Algebra i analiz 17(2), 170–214 (2005). English translation. St. Petersburg Math. JMathSciNetGoogle Scholar
  24. 24.
    Smith, G.: Brill-Noether theory of curves on toric surfaces. J. Pure Appl. Algebra 219(7), 2629–2636 (2015)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Universitat de Barcelona 2015

Authors and Affiliations

  • Dustin Cartwright
    • 1
    Email author
  • Andrew Dudzik
    • 2
  • Madhusudan Manjunath
    • 2
  • Yuan Yao
    • 3
  1. 1.Department of MathematicsUniversity of TennesseeKnoxvilleUSA
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  3. 3.Department of MathematicsUniversity of TexasAustinUSA

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