A tropical approach to enumerative geometry

E. Shustin*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


A detailed algebraic-geometric background is presented for the tropical approach to enumeration of singular curves on toric surfaces, which consists of reducing the enumeration of algebraic curves to that of non-Archimedean amoebas, the images of algebraic curves by a real-valued non-Archimedean valuation. This idea was proposed by Kontsevich and recently realized by Mikhalkin, who enumerated the nodal curves on toric surfaces. The main technical tools are a refined tropicalization of one-parametric equisingular families of curves and the patchworking construction of singular algebraic curves. The case of curves with a cusp and the case of real nodal curves are also treated.

Original languageEnglish
Pages (from-to)343-375
Number of pages33
JournalSt. Petersburg Mathematical Journal
Issue number2
StatePublished - 2006


  • Singular curves
  • Toric surfaces
  • Tropicalization


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