Tropical cycles and Chow polytopes

  • Alex FinkEmail author
Original Paper


The Chow polytope of an algebraic cycle in a torus depends only on its tropicalisation. Generalising this, we associate a Chow polytope to any abstract tropical variety X in a tropicalised toric variety: the normal tropical hypersurface to this Chow polytope is the Minkowski sum of X and a reflected skeleton of the fan of the toric variety. Several significant polyhedra associated to tropical varieties are special cases of our Chow polytope. We show also that Chow polytope subdivisions do not fully distinguish the combinatorial types of tropical variety, and record a proof of the equivalence of two standard definitions of tropical linear spaces.


Chow polytope Chow variety Polytope subdivision Tropical variety Tropical intersection theory Tropical linear space 

Mathematics Subject Classification (2010)

14T05 52B20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alessandrini, D., Nesci, M.: On the tropicalization of the Hilbert scheme. Collectanea Mathamatica (2011), 1–21. doi: 10.1007/s13348-011-0055-7; Preprint arXiv:0912.0082
  2. Allermann, L.: Tropical intersection products on smooth varieties. Preprint arXiv:0904.2693 (2009)Google Scholar
  3. Allermann L., Rau J.: First steps in tropical intersection theory. Math. Z. 264(3), 633–670 (2010) arXiv:0709.3705v3MathSciNetzbMATHCrossRefGoogle Scholar
  4. Ardila F., Klivans C.J.: The Bergman complex of a matroid and phylogenetic trees. J. Combin. Theory Ser. B 96(1), 38–49 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  5. Barvinok A.: Lattice points, polyhedra, and complexity. Geom. Combin. IAS/Park City Mathematics Series 13, 19–62 (2007)MathSciNetGoogle Scholar
  6. Brion M.: The structure of the polytope algebra. Tōhoku Math. J. 49, 1–32 (1997)MathSciNetzbMATHGoogle Scholar
  7. Chow W.-L., Waerden B.L.: Zur algebraischen Geometrie. IX. Über zugeordnete Formen und algebraische Systeme von algebraischen Mannigfaltkeiten. Math. Ann. 113, 692–704 (1937)MathSciNetzbMATHCrossRefGoogle Scholar
  8. Cueto, M.A., Morton, J., Sturmfels, B.: Geometry of the restricted Boltzmann machine. Preprint arXiv:0908.4425Google Scholar
  9. Dalbec J., Sturmfels B.: Introduction to Chow forms. In: White, N. (ed.) Invariant Methods in Discrete and Computational Geometry, pp. 37–58. Springer, Berlin (1995)CrossRefGoogle Scholar
  10. Develin M., Sturmfels B.: Tropical convexity. Documenta Math. 9, 1–27 (2004)MathSciNetzbMATHGoogle Scholar
  11. Dickenstein A., Feichtner E.M., Sturmfels B.: Tropical discriminants. J. Am. Math. Soc. 20, 1111–1133 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  12. Edmonds J.: Submodular functions, matroids, and certain polyhedra. In: Guy, R., Hanani, H., Sauer, N., Schonheim, J. (eds.) Combinatorial Structures and their Applications, pp. 69–87. Gordon and Breach, New York (1970)Google Scholar
  13. Fulton W., Sturmfels B.: Intersection theory on toric varieties. Topology 36(2), 335–353 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  14. Gathmann A., Kerber M., Markwig H.: Tropical fans and the moduli spaces of tropical curves. Comput. Math. 145, 173–195 (2009)MathSciNetzbMATHGoogle Scholar
  15. Gelfand I., Goresky R., MacPherson R., Serganova V.: Combinatorial geometries, convex polyhedra, and Schubert cells. Adv. Math. 63, 301–316 (1987)MathSciNetCrossRefGoogle Scholar
  16. Gelfand I., Kapranov M., Zelevinsky A.: Discriminants, Resultants, and Multidimensional Determinants. Birkhäuser, Boston (2008)zbMATHGoogle Scholar
  17. Huh, J., Katz, E.: Log-concavity of characteristic polynomials and the Bergman fan of matroids. Math. Ann. doi: 10.1007/s00208-011-0777-6 ; arXiv:1104.2519 (2012)
  18. Itenberg I., Mikhalkin G., Shustin E.: Tropical Algebraic Geometry, 2nd edn. Birkhäuser, Boston (2009)zbMATHCrossRefGoogle Scholar
  19. Kapranov M., Sturmfels B., Zelevinsky A.: Chow polytopes and general resultants. Duke Math. J. 67(1), 189–218 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  20. Katz E.: A tropical toolkit. Expo. Math. 27, 1–36 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  21. Katz, E.: Tropical intersection theory from toric varieties, revision of 24 Feb. 2010. Earlier Preprint, arXiv:0907.2488v1Google Scholar
  22. Katz, E., Payne, S.: Realization spaces for tropical fans. Combinatorial Aspects of Commutative Algebra and Algebraic Geometry. Abel Symp. 6, pp. 73–88. Springer, Berlin (2011). arXiv:0909.4582Google Scholar
  23. Kollár, J.: Rational curves on algebraic variety. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 32. Springer, Berlin (1996)Google Scholar
  24. Maclagan, D.: Notes from the AARMS Tropical Geometry summer school.
  25. McMullen P.: Polytope algebra. Adv. Math. 78, 76–130 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  26. McMullen P.: Weights on polytopes. Discrete Comput. Geom. 15, 363–388 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  27. Mikhalkin, G.: Tropical geometry and its application. In: Proceedings of the ICM 2006 Madrid (2006). arXiv:math.AG/0601041Google Scholar
  28. Richter-Gebert, J., Sturmfels, B., Theobald, T.: First steps in tropical geometry. In: Litvinov, G.L., Maslov, V.P. (eds.) Proceedings of Conference on Idempotent Mathematics and Mathematical Physics, Vienna 2003. Contemporary Mathematics. AMSGoogle Scholar
  29. Speyer D.: Tropical linear spaces. SIAM J. Discrete Math. 22(4), 1527–1558 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  30. Stembridge J.: Some permutation representations of Weyl groups associated with cohomology of toric varieties. Adv. Math. 106, 244–301 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  31. Sturmfels B.: Asymptotic analysis of toric ideals. Mem. Fac. Sci. Kyushu Univ. Ser. A 46, 217–228 (1992)MathSciNetzbMATHGoogle Scholar
  32. Sturmfels B., Tevelev J., Yu J.: The Newton polytope of the implicit equation. Moscow Math. J. 7(2), 327–346 (2007)MathSciNetzbMATHGoogle Scholar
  33. Ziegler G.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, Berlin (1995)CrossRefGoogle Scholar

Copyright information

© The Managing Editors 2012

Authors and Affiliations

  1. 1.North Carolina State UniversityRaleighUSA

Personalised recommendations