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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

A tropical approach to enumerative geometry

Author: E. Shustin
Original publication: Algebra i Analiz, tom 17 (2005), nomer 2.
Journal: St. Petersburg Math. J. 17 (2006), 343-375
MSC (2000): Primary 14H15; Secondary 12J25, 14H20, 14M25, 14N10
Published electronically: February 20, 2006
MathSciNet review: 2159589
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Abstract | References | Similar Articles | Additional Information

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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Additional Information

E. Shustin
Affiliation: School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel

PII: S 1061-0022(06)00908-3
Keywords: Singular curves, toric surfaces, tropicalization
Received by editor(s): June 20, 2003
Published electronically: February 20, 2006
Additional Notes: The author was supported by the German–Israeli Foundation for Research and Development (grant no. G–616–15.6/99), by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University, and by the Bessel research award from the Alexander von Humboldt Foundation
Article copyright: © Copyright 2006 American Mathematical Society