Global spherical tropicalization via toric embeddings

  • Evan D. NashEmail author


The first steps in defining tropicalization for spherical varieties have been taken in the last few years. There are two parts to this theory: tropicalizing subvarieties of homogeneous spaces and tropicalizing their closures in spherical embeddings. In this paper, we obtain a new description of spherical tropicalization that is equivalent to the other theories. This works by embedding in a toric variety, tropicalizing there, and then applying a particular piecewise projection map. We use this theory to prove that taking closures commutes with the spherical tropicalization operation.



The author thanks Gary Kennedy for suggesting this direction of research and numerous discussions. Thanks also to Giuliano Gagliardi for providing input on several points.


  1. 1.
    A’Campo-Neuen, A., Hausen, J.: Quotients of toric varieties by the action of a subtorus. Tohoku Math. J. (2) 51(1), 1–12 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Allermann, L., Rau, J.: First steps in tropical intersection theory. Math. Z. 264(3), 633–670 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arzhantsev, I., Derenthal, U., Hausen, J., Laface, A.: Cox rings, Cambridge Studies in Advanced Mathematics, vol. 144. Cambridge University Press, Cambridge (2015)zbMATHGoogle Scholar
  4. 4.
    Brion, M.: Sur la géométrie des variétés sphériques. Comment. Math. Helv. 66(2), 237–262 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brion, M.: The total coordinate ring of a wonderful variety. J. Algebra 313(1), 61–99 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cox, D.A.: The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4(1), 17–50 (1995)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cox, D.: Erratum to: The homogeneous coordinate ring of a toric variety [MR1299003]. J. Algebraic Geom. 23(2), 393–398 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cox, D.A., Little, J.B., Schenck, H.K.: Toric varieties, vol. 124. American Mathematical Society, Providence, RI. Graduate Studies in Mathematics (2011)Google Scholar
  9. 9.
    Gagliardi, G.: The Cox ring of a spherical embedding. J. Algebra 397, 548–569 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gagliardi, G.: Spherical varieties with the \(A_k\)-property. Math. Res. Lett. 24(4), 1043–1065 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hartshorne, R.: Algebraic geometry. Springer, New York (1977). Graduate Texts in Mathematics, No. 52CrossRefzbMATHGoogle Scholar
  12. 12.
    Hu, Yi, Keel, Sean: Mori dream spaces and GIT. Michigan Math. J. 48, 331–348 (2000). Dedicated to William Fulton on the occasion of his 60th birthdayMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kajiwara, T.: Tropical toric geometry, Toric topology, Contemp. Math., vol. 460, pp. 197–207. Am. Math. Soc., Providence, RI (2008)Google Scholar
  14. 14.
    Kaveh, K., Manon, C.: Gröbner theory and tropical geometry on spherical varieties, ArXiv e-prints (2016). arXiv:1611.01841
  15. 15.
    Knop, F.: The Luna-Vust theory of spherical embeddings. In: Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), pp. 225–249. Manoj Prakashan, Madras (1991)Google Scholar
  16. 16.
    Knop, F., Kraft, H., Vust, T.: The Picard group of a \(G\)-variety, Algebraische Transformationsgruppen und Invariantentheorie, DMV Sem., vol. 13, pp. 77–87. Birkhäuser, Basel (1989)Google Scholar
  17. 17.
    Luna, D., Vust, Th: Plongements d’espaces homogènes. Comment. Math. Helv. 58(2), 186–245 (1983). (French)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Maclagan, D., Sturmfels, B.: Introduction to tropical geometry, vol. 161. American Mathematical Society, Providence, RI (2015). Graduate Studies in MathematicsGoogle Scholar
  19. 19.
    Mikhalkin, Grigory: Enumerative tropical algebraic geometry in \({\mathbb{R}}^2\). J. Am. Math. Soc. 18(2), 313–377 (2005)CrossRefzbMATHGoogle Scholar
  20. 20.
    Mumford, D.: The red book of varieties and schemes, Second, expanded edition, Lecture Notes in Mathematics, vol. 1358, Springer-Verlag, Berlin (1999). Includes the Michigan lectures (1974) on curves and their Jacobians; With contributions by Enrico ArbarelloGoogle Scholar
  21. 21.
    Nash, E.D.: Tropicalizing Spherical Embeddings, ArXiv e-prints (2016). arXiv:1609.07455
  22. 22.
    Nishinou, Takeo, Siebert, Bernd: Toric degenerations of toric varieties and tropical curves. Duke Math. J. 135(1), 1–51 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pasquier, B.: Introduction to spherical varieties and description of special classes of spherical varieties (2009). Lecture notes.
  24. 24.
    Payne, S.: Analytification is the limit of all tropicalizations. Math. Res. Lett. 16(3), 543–556 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Perrin, N.: On the geometry of spherical varieties. Transform. Groups 19(1), 171–223 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Święcicka, Joanna: Quotients of toric varieties by actions of subtori. Colloq. Math. 82(1), 105–116 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Tevelev, Jenia: Compactifications of subvarieties of tori. Am. J. Math. 129(4), 1087–1104 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Vogiannou, T.: Spherical Tropicalization, thesis, University of Massachusetts Amherst (2015). arXiv:1511.02203
  29. 29.
    Włodarczyk, Jarosław: Embeddings in toric varieties and prevarieties. J. Algebraic Geom. 2(4), 705–726 (1993)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA

Personalised recommendations