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Tropical images of intersection points

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Abstract

A key issue in tropical geometry is the lifting of intersection points to a non-Archimedean field. Here, we ask: where can classical intersection points of planar curves tropicalize to? An answer should have two parts: first, identifying constraints on the images of classical intersections, and, second, showing that all tropical configurations satisfying these constraints can be achieved. This paper provides the first part: images of intersection points must be linearly equivalent to the stable tropical intersection by a suitable rational function. Several examples provide evidence for the conjecture that our constraints may suffice for part two.

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References

  1. Baker, M., Norine, S.: Riemann–Roch and Abel–Jacobi theory on a finite graph. Adv. Math. 215, 766–788 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Baker, M., Payne, S., Rabinoff, J.: Nonarchimedean geometry, tropicalization, and metrics on curves (2012). Preprint: http://arxiv.org/abs/1104.0320

  3. Brugallé, E.A., López de Medrano, L.M.: Inflection points of real and tropical plane curves. J. Singul. 4, 74–103 (2012)

    MATH  MathSciNet  Google Scholar 

  4. Gathmann, A., Kerber, M.: A Riemann–Roch theorem in tropical geometry. Mathematische Zeitschrift 259, 217–230 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Gubler, W.: A guide to tropicalizations. In: Algebraic and combinatorial aspects of tropical geometry. Contemp. Math., , vol. 589, pp. 125–189. Amer. Math. Soc., Providence (2013)

  6. Hasse, C., Musiker, G., Yu, J.: Linear systems on tropical curves. Mathematische Zeitschrift 270(3–4), 1111–1140 (2012)

    Article  MathSciNet  Google Scholar 

  7. Luo, Y.: Tropical Convexity and Canonical Projections (2013). Preprint: http://arxiv.org/abs/1304.7963

  8. Maclagan, D., Sturmfels, B.: Introduction to tropical geometry (2014). Preprint: http://homepages.warwick.ac.uk/staff/D.Maclagan/papers/TropicalBook.html

  9. Mikhalkin, G.: Tropical Geometry and its applications. In: International Congress of Mathematicians, vol. II, pp. 827–852. Eur. Math. Soc., Zürich (2006)

  10. Osserman, B., Payne, S.: Lifting tropical intersections. Doc. Math. 18, 121–175 (2013)

    MATH  MathSciNet  Google Scholar 

  11. Osserman, B.; Rabinoff J.: Lifting non-proper tropical intersections. Contemp. Math. (2014, to appear)

Download references

Acknowledgments

The author would like to thank Matt Baker and Bernd Sturmfels for introducing him to these questions in tropical geometry. The author would also like to thank Sarah Brodsky, Melody Chan, Nikita Kalinin, Kristin Shaw, and Josephine Yu for helpful conversations and insights. The author was supported by the NSF through grant DMS-0968882 and a graduate research fellowship, and by the Max Planck Institute for Mathematics in Bonn.

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  1. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Ralph Morrison

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  1. Ralph Morrison
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Correspondence to Ralph Morrison.

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Morrison, R. Tropical images of intersection points. Collect. Math. 66, 273–283 (2015). https://doi.org/10.1007/s13348-014-0118-7

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  • DOI: https://doi.org/10.1007/s13348-014-0118-7

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