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Faithful tropicalization of the Grassmannian of planes

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Abstract

We show that the tropical projective Grassmannian of planes is homeomorphic to a closed subset of the analytic Grassmannian in Berkovich’s sense by constructing a continuous section to the tropicalization map. Our main tool is an explicit description of the algebraic coordinate rings of the toric strata of the Grassmannian. We determine the fibers of the tropicalization map and compute the initial degenerations of all the toric strata. As a consequence, we prove that the tropical multiplicities of all points in the tropical projective Grassmannian are equal to one. Finally, we determine a piecewise linear structure on the image of our section that corresponds to the polyhedral structure on the tropical projective Grassmannian.

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Acknowledgments

We wish to thank Antoine Ducros and Sam Payne for very fruitful conversations and Josephine Yu for helping us with some computations using Macaulay2 [15] and Gfan [20]. We would like to express our gratitude to the anonymous referee for his/her careful reading and detailed comments. The first author was supported by an Alexander von Humboldt Postdoctoral Research Fellowship.

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Authors and Affiliations

  1. Mathematics Department, Columbia University, 2990 Broadway, New York, NY, 10027, USA

    Maria Angelica Cueto

  2. Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 6-8, 60325 , Frankfurt am Main, Germany

    Mathias Häbich & Annette Werner

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  1. Maria Angelica Cueto
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  2. Mathias Häbich
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  3. Annette Werner
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Correspondence to Annette Werner.

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Cueto, M.A., Häbich, M. & Werner, A. Faithful tropicalization of the Grassmannian of planes. Math. Ann. 360, 391–437 (2014). https://doi.org/10.1007/s00208-014-1037-3

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  • DOI: https://doi.org/10.1007/s00208-014-1037-3

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