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Embeddings and immersions of tropical curves

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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.

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Tennessee, 227 Ayres Hall, Knoxville, 37996, TN, USA

    Dustin Cartwright

  2. Department of Mathematics, University of California, 970 Evans Hall, Berkeley, 94720, CA, USA

    Andrew Dudzik & Madhusudan Manjunath

  3. Department of Mathematics, University of Texas, 1 University Station C1200, Austin, 78712, TX, USA

    Yuan Yao

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  1. Dustin Cartwright
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  2. Andrew Dudzik
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  3. Madhusudan Manjunath
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  4. Yuan Yao
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Correspondence to Dustin Cartwright.

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Cartwright, D., Dudzik, A., Manjunath, M. et al. Embeddings and immersions of tropical curves. Collect. Math. 67, 1–19 (2016). https://doi.org/10.1007/s13348-015-0149-8

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