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.References
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Bibliographic Information
- E. Shustin
- Affiliation: School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
- MR Author ID: 193452
- Email: shustin@post.tau.ac.il
- Received by editor(s): June 20, 2003
- Published electronically: February 20, 2006
- Additional Notes: The author was supported by the GermanâIsraeli Foundation for Research and Development (grant no. Gâ616â15.6/99), by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University, and by the Bessel research award from the Alexander von Humboldt Foundation
- © Copyright 2006 American Mathematical Society
- Journal: St. Petersburg Math. J. 17 (2006), 343-375
- MSC (2000): Primary 14H15; Secondary 12J25, 14H20, 14M25, 14N10
- DOI: https://doi.org/10.1090/S1061-0022-06-00908-3
- MathSciNet review: 2159589