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Vol. 52 No. 2 (2026)
Serdica Mathematical Journal

Articles

A guide to tropical modifications

https://doi.org/10.55630/serdica.2026.52.109-156

Published 24-06-2026

  • Nikita Kalinin

Nikita Kalinin
Guangdong Technion Israel Institute of Technology and Technion-Israel Institute of Technology

Abstract

This paper surveys tropical modifications, a notion that has become part of the folklore of tropical geometry. Tropical modifications are used in tropical intersection theory, tropical Hodge theory, and the study of singularities. They admit interpretations in several contexts, including hyperbolic geometry, Berkovich spaces, and non-standard analysis.

The goal of the paper is to present several points of view, give references, and illustrate the usefulness of tropical modifications. We assume that the reader has already encountered tropical modifications and wants to understand them better.

The paper also contains two new contributions: an obstruction to the realizability of non-transversal intersections and a tropical version of Weil's reciprocity law.

2020 Mathematics Subject Classification:

14T10, 14T20, 14T90, 14B10, 05C99

Keywords

tropical geometry, Weil reciprocity, tropical modifications, singularity, Hodge theory

Full text

Author Details

Nikita Kalinin

Guangdong Technion Israel Institute of Technology (GTIIT)
241 Daxue Road, Shantou, Guangdong Province 515603, P.R. China
and
Technion-Israel Institute of Technology
Haifa, 32000, Haifa district, Israel

e-mail: nikaanspb@gmail.com

References

  1. O. Amini, M. Baker, E. Brugallé, J. Rabinoff. Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta. Res. Math. Sci. 2 (2015), Art. 7, 67.
  2. O. Amini, M. Baker, E. Brugallé, J. Rabinoff. Lifting harmonic morphisms II: Tropical curves and metrized complexes. Algebra Number Theory 9, 2 (2015), 267–315.
  3. O. Amini, M. Piquerez. Hodge theory for tropical varieties. arXiv:2007.07826 [math.AG], 2020, https://doi.org/10.48550/arXiv.2007.07826.
  4. O. Amini, M. Piquerez. Homology of tropical fans. arXiv:2105.01504 [math.AG], 2020, https://doi.org/10.48550/arXiv.2105.01504.
  5. M. Baker. An introduction to Berkovich analytic spaces and non-archimedean potential theory on curves. p-adic Geometry (Lectures from the 2007 Arizona Winter School), 2008, https://swc-math.github.io/aws/2007/BakerNotesMarch21.pdf.
  6. M. Baker, S. Payne, J. Rabinoff. Nonarchimedean geometry, tropicalization, and metrics on curves. Algebr. Geom, 3, 1 (2016), 63–105.
  7. B. Bertrand, E. Brugallé, L. López De Medrano. Planar tropical cubic curves of any genus, and higher dimensional generalisations. Enseign. Math. 64, 3–4 (2018), 415–457.
  8. B. Bertrand, E. Brugallé, G. Mikhalkin. Tropical open Hurwitz numbers. Rend. Semin. Mat. Univ. Padova 125 (2011), 157–171.
  9. B. H. Bowditch, D. B. A. Epstein. Natural triangulations associated to a surface. Topology 27, 1 (1988), 91–117.
  10. E. Brugallé, I. Itenberg, G. Mikhalkin, K. Shaw. Brief introduction to tropical geometry. In: Proceedings of the G¨okova Geometry-Topology Conference 2014 (Eds S. Akbulut, D. Auroux, T. Önder), Gökova Geometry/Topology Conference (GGT), Gökova. Somerville, MA, International Press, 2015, 1–75.
  11. E. Brugallé, K. Shaw. A bit of tropical geometry. Amer. Math. Monthly 121, 7 (2014), 563–589.
  12. E. A. Brugallé, L. M. López de Medrano. Inflection points of real and tropical plane curves. J. Singul. 4 (2012), 74–103.
  13. P. Buser. The collar theorem and examples. Manuscripta Math. 25, 4 (1978), 349–357.
  14. M. Coppens. A metric graph satisfying w14 = 1 that cannot be lifted to a curve satisfying dim(W14) = 1. Open Math. 14, 1 (2016), 1–12.
  15. M. A. Cueto, H. Markwig. How to repair tropicalizations of plane curves using modifications. Exp. Math. 25, 2 (2016), 130–164.
  16. M. A. Cueto, H. Markwig. Tropical geometry of genus two curves. J. Algebra 517 (2019), 457–512.
  17. M. A. Cueto, H. Markwig. Combinatorics and real lifts of bitangents to tropical quartic curves. Discrete Comput. Geom. 69, 3 (2023), 597–658.
  18. L. L. de Medrano, F. Rincón, K. Shaw. Chern–Schwartz–MacPherson cycles of matroids. Proc. Lond. Math. Soc. (3) 120, 1 (2020), 1–27.
  19. A. M. M. del Campo, S. Carpentier. Tropical embeddings of metric graphs. arXiv:1604.06176 [math.AG], 2016, https://doi.org/10.48550/arXiv.1604.06176.
  20. H. Dërvodeli. Construction of tropical morphisms from tropical modifications of nonhyperelliptic genus 3 metric graphs with tree gonality 3 to metric trees. arXiv:2301.01989 [math.AG], 2023, https://doi.org/10.48550/arXiv.2301.01989.
  21. N. Do. Intersection theory on moduli spaces of curves via hyperbolic geometry. Ph. D. Thesis, University of Melbourne, 2008.
  22. J. Draisma, A. Vargas. Catalan-many tropical morphisms to trees; Part I: Constructions. J. Symbolic Comput. 104 (2021), 580–629.
  23. B. El Hilany. Constructing polynomial systems with many positive solutions using tropical geometry. Rev. Mat. Complut. 31, 2 (2018), 525–544.
  24. Y. Ganor. Enumerating Cuspidal Curves on Toric Surfaces. arXiv:1306.3514 [math.AG], 2013, https://doi.org/10.48550/arXiv.1306.3514.
  25. Y. Ganor, E. Shustin. Enumeration of unicuspidal curves of any degree and genus on toric surfaces. Int. Math. Res. Not. IMRN, 21 (2022), 16464–16523.
  26. W. Gubler, J. Rabinoff, A. Werner. Skeletons and tropicalizations. Adv. Math., 294 (2016), 150–215.
  27. W. Gubler, J. Rabinoff, A. Werner. Tropical skeletons. Ann. Inst. Fourier (Grenoble) 67, 5 (2017), 1905–1961.
  28. M. H. Gunturkun, A. U. O. Kisisel. Using tropical degenerations for proving the nonexistence of certain nets. arXiv:1107.5530 [math.AG], 2011, https://doi.org/10.48550/arXiv.1107.5530.
  29. M. A. Hahn, H. Markwig, Y. Ren, I. Tyomkin. Tropicalized quartics and canonical embeddings for tropical curves of genus 3. Int. Math. Res. Not. IMRN 12 (2021), 89468976.
  30. X. He. A generalization of lifting non-proper tropical intersections. J. Pure Appl. Algebra 223, 2 (2019), 794–817.
  31. I. Itenberg, G. Mikhalkin, E. Shustin. Tropical algebraic geometry, 2nd edn. Oberwolfach Semin., vol. 35. Basel, Birkhäuser Verlag, 2009.
  32. P. Jell, V. Wanner. Poincaré duality for the tropical Dolbeault cohomology of non-archimedean Mumford curves. J. Number Theory 187 (2018), 344–371.
  33. Y. Kageyama. Divisorial condition for the stable gonality of tropical curves. arXiv:1801.07405 [math.AG], 2018, https://doi.org/10.48550/arXiv.1801.07405.
  34. N. Kalinin. Tropical geometry for Nagata’s conjecture and Legendrian curves. Doctoral Thesis, Université de Genève, 2015, https://archive-ouverte.unige.ch/unige:80308.
  35. N. Kalinin. The Newton polygon of a planar singular curve and its subdivision. J. Combin. Theory Ser. A 137 (2016), 226–256.
  36. N. Kalinin. Tropical approach to Nagata’s conjecture in positive characteristic. Discrete Comput. Geom. 58, 1 (2017), 158–179.
  37. N. Kalinin, M. Magin. Tropical Weil’s reciprocity law and Weil’s pairing. J. Math. Sci., 2025, https://doi.org/10.1007/s10958-025-07754-9.
  38. H. J. Keisler. Foundations of infinitesimal calculus. Boston, Massachusetts, Prindle, Weber & Schmidt, 1976.
  39. M. Kerber, H. Markwig. Intersecting Psi-classes on tropical M0,n. Int. Math. Res. Not. IMRN 2 (2009), 221–240.
  40. A. Khovanskii. Logarithmic functional and the Weil reciprocity laws. In: Proceedings of the Waterloo Workshop on Computer Algebra (eds I. Kotsireas, E. Zima). World Scientific Press, 2007, 85–108, https://www.math.toronto.edu/askold/Khovanskii-zima.pdf.
  41. A. Khovanskii. Logarithmic functional and reciprocity laws. Contemp. Math., vol. 460. Providence, RI, American Mathematical Society, 2008, 221–229.
  42. A. G. Khovanskii. Newton polytopes, curves on toric surfaces, and inversion of Weil’s theorem. Uspekhi Mat. Nauk 52, 6(318) (1997), 113–142 (in Russian); English translation in Russian Math. Surveys 52, 6 (1997), 1251–1279.
  43. M. Kontsevich. Intersection theory on the moduli space of curves and the matrix Airy function. Comm. Math. Phys. 147, 1 (1992), 1–23.
  44. L. Lang. Geometry of tropical curves and application to real algebraic geometry. Doctoral Thesis, Université de Genève, 2014, https://doi.org/10.13097/archive-ouverte/unige:43180.
  45. Y. Len, H. Markwig. Lifting tropical bitangents. J. Symbolic Comput. 96 (2020), 122–152.
  46. Y. Len, M. Satriano. Lifting tropical self intersections. J. Combin. Theory Ser. A 170 (2020), 105138, 21 pp.
  47. H. Markwig, T. Markwig, E. Shustin. Tropical curves with a singularity in a fixed point. Manuscripta Math. 137, 3–4 (2012), 383–418.
  48. H. Markwig, L. Ristau, V. Schleis. Faithful tropicalization of hyperelliptic curves. In: The computer algebra system OSCARalgorithms and examples (eds W. Decker, C. Eder, C. Fieker, M. Horn, M. Joswig). Algorithms Comput. Math., vol. 32. Cham, Springer, 2025, 403–428.
  49. M. Mazin. Geometric theory of Parshin residues. C. R. Math. Acad. Sci. Soc. R. Can. 32, 3 (2010), 81–96.
  50. M. Melo, A. Zheng. Tropical trigonal curves. Algebr. Comb. 9, 1 (2026), 95–130.
  51. G. Mikhalkin. Enumerative tropical algebraic geometry in R2. J. Amer. Math. Soc. 18, 2 (2005), 313–377.
  52. G. Mikhalkin. Tropical geometry and its applications. In: International Congress of Mathematicians. Vol. II. Zürich, Eur. Math. Soc., 2006, 827–852.
  53. G. Mikhalkin. Moduli spaces of rational tropical curves. In: Proceedings of Gökova Geometry-Topology Conference 2006. Gökova Geometry/Topology Conference (GGT), Gokova, 2007, 39–51.
  54. G. Mikhalkin. Quantum indices and refined enumeration of real plane curves. Acta Math 219 (2017), 135–180.
  55. G. Mikhalkin, I. Zharkov. Tropical curves, their Jacobians and theta functions. In: Curves and abelian varieties, Contemp. Math. vol. 465. Providence, RI, Amer. Math. Soc., 2008, 203–230.
  56. R. Morrison. Tropical images of intersection points. Collect. Math. 66, 2 (2015), 273–283.
  57. J. Mundinger. The image of a tropical linear space. Electron. J. Combin. 32, 1 (2025), Paper No. 1.45, 18 pp.
  58. B. Osserman, S. Payne. Lifting tropical intersections. Doc. Math. 18 (2013), 121–175.
  59. B. Osserman, J. Rabinoff. Lifting non-proper tropical intersections. In: Tropical and Non-Archimedean Geometry. Contemp. Math., vol. 605 (based on papers from the 2011 Bellairs Workshop in Number Theory). Providence, RI, Amer. Math. Soc., 2013, 15–44.
  60. S. Payne. Analytification is the limit of all tropicalizations. Math. Res. Lett. 16, 3 (2009), 543–556.
  61. J. Rabinoff. Tropical analytic geometry, Newton polygons, and tropical intersections. Adv. Math. 229, 6 (2012), 3192–3255.
  62. Q. Ren, K. Shaw, B. Sturmfels. Tropicalization of Del Pezzo Surfaces. arXiv:1402.5651 [math.AG], 2014, https://doi.org/10.48550/arXiv.1402.5651.
  63. A. Renaudineau. Constructions de surfaces algébriques réelles (Constructions of real algebraic surfaces). PhD thesis, Université Pierre et Marie Curie – Paris VI, 2015, https://doi.org/10.70675/4780fb5cz704ez4b43zafe4zb296426beece.
  64. J. Richter-Gebert, B. Sturmfels, T. Theobald. First steps in tropical geometry. In: Idempotent mathematics and mathematical physics. Contemp. Math., vol. 377. Providence, RI, Amer. Math. Soc., 2005, 289–317.
  65. M. Ruggiero, K. Shaw. Tropical Hopf manifolds and contracting germs. Manuscripta Math. 152, 1–2 (2017), 1–60.
  66. K. Shaw. Tropical surfaces. arXiv:1506.07407 [math.AG], 2015, https://doi.org/10.48550/arXiv.1506.07407.
  67. K. M. Shaw. Tropical Intersection Theory and Surfaces, Doctoral Thesis, Université de Genève, 2011, https://archive-ouverte.unige.ch/unige:22758.
  68. K. M. Shaw. A tropical intersection product in matroidal fans. SIAM J. Discrete Math. 27, 1 (2013), 459–491.
  69. M. Sukenaga. Tropical lifting problem for the intersection of plane curves. Beitr Algebra Geom 65 (2024), 537571, https://doi.org/10.1007/s13366-023-00705-y
  70. T. Tao. Compactness and contradiction. Providence, RI, Amer. Math. Soc., 2013.
  71. M. Temkin. Introduction to Berkovich analytic spaces. In: Berkovich spaces and applications. Lecture Notes in Math., vol. 2119. Cham, Springer, 2015, 3–66.
  72. M. Ulirsch and D. Zakharov. Tropical double ramification loci. arXiv:1910.01499 [math.AG], 2019, https://doi.org/10.48550/arXiv.1910.01499.
  73. O. Ya. Viro. On basic concepts of tropical geometry. Tr. Mat. Inst. Steklova 273 (2011), Sovremennye Problemy Matematiki, 271–303 (in Russian); English translation in Proc. Steklov Inst. Math. 273, 1 (2011), 252–282.
  74. Y. Yamamoto. Periods of tropical Calabi–Yau hypersurfaces. J. Algebraic Geom. 31, 2 (2022), 303–343.
  75. S. Yoshitomi. Jacobian varieties of reduced tropical curves. arXiv:0612810 [math.AG], 2006, https://doi.org/10.48550/arXiv.math/0612810.
  76. D. Zakharov. The trigonal construction and the second moment of the tropical Prym variety. arXiv:2507.06401 [math.AG], 2025, https://doi.org/10.48550/arXiv.2507.06401.