Tropical and Algebraic Curves with Multiple Points

Part of the Progress in Mathematics book series (PM, volume 296)


Patchworking theorems serve as a basic element of the correspondence between tropical and algebraic curves, which is a core area of tropical enumerative geometry. We present a new version of a patchworking theorem that relates plane tropical curves to complex and real algebraic curves having prescribed multiple points. It can be used to compute Welschinger invariants of nontoxic del Pezzo surfaces.


Patchworking Tropical curve Multiple point Welschinger invariant Del Pezzo surface 



The author was supported by the grant 465/04 from the Israel Science Foundation, a grant from the Higher Council for Scientific Cooperation between France and Israel, and a grant from Tel Aviv University. This work was completed during the author’s stay at the Centre Interfacultaire Bernoulli, École Polytechnique Fédérale da Lausanne and at Laboratoire Émile Picard, Université Paul Sabatier, Toulouse. The author thanks the CIB-EPFL and UPS for the hospitality and excellent working conditions. Special thanks are due to I. Itenberg, who pointed out a mistake in the preliminary version of Theorem 3. Finally, I express my gratitude to the unknown referee for numerous remarks, corrections, and suggestions.


  1. 1.
    Gathmann, A., and Markwig, H.: The numbers of tropical plane curves through points in general position. J. reine angew. Math. 602 (2007), 155–177.Google Scholar
  2. 2.
    Greuel, G.-M., and Karras, U.: Families of varieties with prescribed singularities. Compos. Math. 69 (1989), no. 1, 83–110.Google Scholar
  3. 3.
    Greuel, G.-M., and Lossen, C.: Equianalytic and equisingular families of curves on surfaces. Manuscripta Math. 91 (1996), no. 3, 323–342.Google Scholar
  4. 4.
    Itenberg, I. V., Kharlamov, V. M., and Shustin, E. I.: Logarithmic equivalence of Welschinger and Gromov–Witten invariants. Russian Math. Surveys 59 (2004), no. 6, 1093–1116.Google Scholar
  5. 5.
    Itenberg, I. V., Kharlamov, V. M., and Shustin, E. I.: New cases of logarithmic equivalence of Welschinger and Gromov–Witten invariants. Proc. Steklov Math. Inst. 258 (2007), 65–73.Google Scholar
  6. 6.
    Itenberg, I., Kharlamov, V., and Shustin, E.: Welschinger invariants of small non-toric del Pezzo surfaces. Preprint at arXiv:1002.1399.Google Scholar
  7. 7.
    Itenberg, I., Mikhalkin, G., and Shustin, E.: Tropical algebraic geometry Oberwolfach seminars, vol. 35. Birkhäuser, 2007.Google Scholar
  8. 8.
    Mikhalkin, G.: Decomposition into pairs-of-pants for complex algebraic hypersurfaces. Topology 43 (2004), 1035–1065.Google Scholar
  9. 9.
    Mikhalkin, G.: Enumerative tropical algebraic geometry in \({\mathbb{R}}^{2}\). J. Amer. Math. Soc. 18 (2005), 313–377.Google Scholar
  10. 10.
    Nishinou, T., and Siebert, B.: Toric degenerations of toric varieties and tropical curves. Duke Math. J. 135 (2006), no. 1, 1–51.Google Scholar
  11. 11.
    Orevkov, S., and Shustin, E.: Pseudoholomorphic, algebraically unrealizable curves. Moscow Math. J. 3 (2003), no. 3, 1053–1083.Google Scholar
  12. 12.
    Richter-Gebert, J., Sturmfels, B., and Theobald, T.: First steps in tropical geometry. Idempotent mathematics and mathematical physics. Contemp. Math. 377, Amer. Math. Soc., Providence, RI, 2005, pp. 289–317.Google Scholar
  13. 13.
    Shustin, E.: A tropical approach to enumerative geometry. Algebra i Analiz 17 (2005), no. 2, 170–214 (English translation: St. Petersburg Math. J. 17 (2006), 343–375).Google Scholar
  14. 14.
    Shustin, E.: On manifolds of singular algebraic curves. Selecta Math. Sov. 10, no. 1, 27–37 (1991).Google Scholar
  15. 15.
    Shustin, E.: A tropical calculation of the Welschinger invariants of real toric del Pezzo surfaces. J. Alg. Geom. 15 (2006), no. 2, 285–322 (corrected version at arXiv:math/0406099).Google Scholar
  16. 16.
    Shustin, E.: Patchworking construction in the tropical enumerative geometry. Singularities and Computer Algebra, C. Lossen and G. Pfister, eds., Lond. Math. Soc. Lec. Notes Ser. 324, Proceedings of Conference dedicated to the 60th birthday of G.-M. Greuel, Cambridge University Press, Cambridge, MA, 2006, pp. 273–300.Google Scholar
  17. 17.
    Shustin, E.: Welschinger invariants of toric del Pezzo surfaces with non-standard real structures. Proc. Steklov Math. Inst. 258 (2007), 219–247.Google Scholar
  18. 18.
    Shustin, E.: New enumerative invariants and correspondence theorems for plane tropical curves, Preprint, 2011.Google Scholar
  19. 19.
    Viro, O. Ya.: Gluing of plane real algebraic curves and construction of curves of degrees 6 and 7. Lect. Notes Math. 1060, Springer, Berlin, 1984, pp. 187–200.Google Scholar
  20. 20.
    Viro, O. Ya.: Real algebraic plane curves: constructions with controlled topology. Leningrad Math. J. 1 (1990), 1059–1134.Google Scholar
  21. 21.
    Viro, O. Ya.: Patchworking Real Algebraic Varieties. Preprint at arXiv:math/0611382.Google Scholar
  22. 22.
    Viro, O.: Dequantization of real algebraic geometry on a logarithmic paper. Proceedings of the 3rd European Congress of Mathematicians, Birkhäuser, Progress in Math. 201 (2001), 135–146.Google Scholar
  23. 23.
    Welschinger, J.-Y.: Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry. Invent. Math. 162 (2005), no. 1, 195–234.Google Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael

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