TY - JOUR
T1 - A tropical approach to enumerative geometry
AU - Shustin, E.
PY - 2006
Y1 - 2006
N2 - 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.
AB - 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.
KW - Singular curves
KW - Toric surfaces
KW - Tropicalization
UR - http://www.scopus.com/inward/record.url?scp=85009813236&partnerID=8YFLogxK
U2 - 10.1090/S1061-0022-06-00908-3
DO - 10.1090/S1061-0022-06-00908-3
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AN - SCOPUS:85009813236
SN - 1061-0022
VL - 17
SP - 343
EP - 375
JO - St. Petersburg Mathematical Journal
JF - St. Petersburg Mathematical Journal
IS - 2
ER -