A tropical approach to enumerative geometry

Shustin, E.

doi:10.1090/S1061-0022-06-00908-3

A tropical approach to enumerative geometry
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by E. Shustin

St. Petersburg Math. J. 17 (2006), 343-375

DOI: https://doi.org/10.1090/S1061-0022-06-00908-3

Published electronically: February 20, 2006

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Abstract:

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.

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