Manuscripta Mathematica

, Volume 149, Issue 3–4, pp 315–338 | Cite as

Faithful tropicalisation and torus actions

  • Jan DraismaEmail author
  • Elisa Postinghel
Open Access


For any affine variety equipped with coordinates, there is a surjective, continuous map from its Berkovich space to its tropicalisation. Exploiting torus actions, we develop techniques for finding an explicit, continuous section of this map. In particular, we prove that such a section exists for linear spaces, Grassmannians of planes (reproving a result due to Cueto, Häbich, and Werner), matrix varieties defined by the vanishing of 3 × 3-minors, and for the hypersurface defined by Cayley’s hyperdeterminant.

Mathematics Subject Classification

14T05 14M15 14M12 14G22 52B40 


  1. 1.
    Ardila F., Klivans C.J.: The Bergman complex of a matroid and phylogenetic trees. J. Comb. Theory Ser. B 96(1), 38–49 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Berkovich, V.G.: Spectral Theory and Analytic Geometry Over Non-archimedean Fields, Vol. 33 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (1990)Google Scholar
  3. 3.
    Bieri R., Groves J.R.J.: The geometry of the set of characters induced by valuations. J. Reine Angew. Math. 347, 168–195 (1984)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bogart T., Jensen A.N., Speyer D.E., Sturmfels B., Thomas R.R.: Computing tropical varieties. J. Symb. Comput. 42(1–2), 54–73 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Baker, M., Payne, S., Rabinoff, J.: Nonarchimedean geometry, tropicalization, and metrics on curves. Algebraic Geometry (2011) To appear, arXiv:1104.0320
  6. 6.
    Baker, M., Rumely, R.: Potential Theory and Dynamics on the Berkovich Projective Line. Mathematical Surveys and Monographs 159. American Mathematical Society (AMS), Providence (2010)Google Scholar
  7. 7.
    Cueto M.A., Häbich M., Werner A.: Faithful tropicalization of the Grassmannian of planes. Math. Ann. 360(1–2), 391–437 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Dickenstein A., Feichtner E.M., Sturmfels B.: Tropical discriminants. J. Am. Math. Soc. 20(4), 1111–1133 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Draisma J.: A tropical approach to secant dimensions. J. Pure Appl. Algebra 212(2), 349–363 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Develin, M., Santos, F., Sturmfels, B.: On the rank of a tropical matrix. In: Combinatorial and Computational Geometry, pp. 213–242. Cambridge University Press, Cambridge (2005)Google Scholar
  11. 11.
    Einsiedler M., Kapranov M., Lind D.: Non-archimedean amoebas and tropical varieties. J. Reine Angew. Math. 601, 139–157 (2006)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory and Applications. Birkhäuser, Boston (1994)Google Scholar
  13. 13.
    Gubler, W., Rabinoff, J., Werner, A.: Skeletons and Tropicalizations. (2014) Preprint, arXiv:1404.7044
  14. 14.
    Huggins P., Sturmfels B., Yu J., Yuster D.S.: The hyperdeterminant and triangulations of the 4-cube. Math. Comput. 77(263), 1653–1679 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Giraldo, B.I.: Dissimilarity vectors of trees are contained in the tropical Grassmannian. Electron. J. Comb. 17(1): research paper n6, 7 (2010)Google Scholar
  16. 16.
    Manon, C.: Dissimilarity maps on trees and the representation theory of \({{\rm GL}_n(\mathbb{C})}\). Electron. J. Comb. 19(3):research paper p. 38, 12 (2012)Google Scholar
  17. 17.
    Mikhalkin, G.: Tropical geometry and its applications. In: Sanz-Solé, M. et al. (eds.), Proceedings of the International Congress of Mathematicians, Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures., Zürich, European Mathematical Society (2006)Google Scholar
  18. 18.
    Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry, Volume 161 of Graduate Studies in Mathematics. American Mathematical Society, Providence 2015Google Scholar
  19. 19.
    Payne S.: Analytification is the limit of all tropicalizations. Math. Res. Lett. 16(2–3), 543–556 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Payne S.: Fibers of tropicalization. Math. Z. 262(2), 301–311 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Richter-Gebert, J., Sturmfels, B., Theobald, T.: First steps in tropical geometry. In: Litvinov, G.L. et al. (eds.) Idempotent Mathematics and Mathematical Physics. Proceedings of the International Workshop, Vienna, Austria, February 3–10, 2003, Vol. 377 of Contemporary Mathematics, pp. 289–317. AMS, Providence (2005)Google Scholar
  22. 22.
    Schrijver, A.: Combinatorial Optimization. Polyhedra and Efficiency. Number 24 in Algorithms and Combinatorics. Springer, Berlin (2003)Google Scholar
  23. 23.
    Speyer D.E., Sturmfels B.: The tropical Grassmannian. Adv. Geom. 4(3), 389–411 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Yu, J., Yuster, D.S.: Representing tropical linear spaces by circuits. In: Proceedings of the 19th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2007), July 2007, Tianjin, China (2007)Google Scholar

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© The Author(s) 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceTechnische Universiteit EindhovenEindhovenThe Netherlands
  2. 2.Vrije UniversiteitAmsterdamThe Netherlands
  3. 3.Centrum voor Wiskunde en InformaticaAmsterdamThe Netherlands
  4. 4.Department of MathematicsKatholieke Universiteit LeuvenLeuvenBelgium

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