A Tropical Approach to Enumerative Geometry
نویسنده
چکیده
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.
منابع مشابه
Tropical Geometry
Tropical geometry still is a rather young field of mathematics which nevertheless over the past couple of years has proved to be rather powerful for tackling questions in enumerative geometry (see e.g. [Mik05], [GM08], [IKS04]). It also has been used to provide alternative algorithmic means to eliminate variables (see [StY08]). People from applied mathematics such as optimisation or control the...
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