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Bitangents of tropical plane quartic curves

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Abstract

We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we show that every such curve admits either infinitely many or exactly 7 bitangent lines. We also prove that a smooth tropical plane quartic curve cannot be hyperelliptic.

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Notes

  1. Since X is not hyperelliptic, one can in fact remove the phrase “linear equivalence classes” here.

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Acknowledgments

This paper arose from the first author’s project group at the 2013 AMS Math Research Communities workshop on Tropical and Non-Archimedean Geometry in Snowbird, Utah. The authors are grateful to the AMS for their support of this program. Thanks to Bernd Sturmfels for his encouragement of this project, and to Johannes Rau for helpful comments. We also thank the referees for their insightful remarks. The first author was supported in part by the NSF Grant DMS-1201473.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    Matthew Baker

  2. Fachrichtung Mathematik, Universität des Saarlandes, Postfach 151150, 66041, Saarbrücken, Germany

    Yoav Len

  3. Department of Mathematics, KTH, 100 44, Stockholm, Sweden

    Ralph Morrison

  4. Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA, 02138, USA

    Nathan Pflueger

  5. Google Inc., 1600 Amphitheatre Parkway, Mountain View, CA, 94043, USA

    Qingchun Ren

Authors
  1. Matthew Baker
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  2. Yoav Len
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  3. Ralph Morrison
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  4. Nathan Pflueger
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Corresponding author

Correspondence to Yoav Len.

Appendix: Theta characteristics for genus 3 graphs

Appendix: Theta characteristics for genus 3 graphs

In the proof of Proposition 3.6, we made use of knowledge about the 7 theta characteristics of each of the four graphs arising as a skeleton of a smooth plane quartic curve. In Fig. 9, we illustrate all the theta characteristics for the four types of genus 3 graphs relevant to us. Each graph has the support of a nonzero \(({\mathbf Z}/2{\mathbf Z})\)-flow in bold, and in most cases the corresponding theta characteristic is illustrated as a pair of circular points. In the cases where edge lengths might change the combinatorial position of the points of the theta characteristic, the other possibility is illustrated by a pair of crosses. There are degenerate cases where one of these moves to a vertex, but this will not affect our arguments. We have also taken advantage of Theorem 4.3, which allows us to assume asymmetry for the middle cycle in the second, third, and fourth columns.

Fig. 9
figure 9

The 7 theta characteristics for each type of graph

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Labeling the columns 1, 2, 3, and 4 and the rows A, B, C, D, E, F, and G, we have the following classification of these 28 theta characteristics by cases (i), (ii), and (iii) of Proposition 3.6:

  1. (i)

    The 20 not falling into cases (ii) or (iii).

  2. (ii)

    The 6 theta characteristics 3E, 3F, 3G, 4D, 4F, 4G.

  3. (iii)

    The 2 theta characteristics 2E, 4E.

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Baker, M., Len, Y., Morrison, R. et al. Bitangents of tropical plane quartic curves. Math. Z. 282, 1017–1031 (2016). https://doi.org/10.1007/s00209-015-1576-7

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