Enumerative tropical algebraic geometry in ℝ²

Mikhalkin, Grigory^1, 2

¹ Department of Mathematics, University of Toronto, 100 St. George St., Toronto, Ontario, M5S 3G3 Canada and St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191011 Russia
² IHES, Le Bois-Marie, 35, route de Chartres, Bures-sur-Yvette, 91440, France

Journal of the American Mathematical Society, Tome 18 (2005) no. 2, pp. 313-377

Résumé

The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in Counting curves via lattice paths in polygons, C. R. Math. Acad. Sci. Paris 336 (2003), no. 8, 629–634. The result is established with the help of the so-called tropical algebraic geometry. This geometry allows one to replace complex toric varieties with the real space $\mathbb {R}^n$ and holomorphic curves with certain piecewise-linear graphs there.

Détail

DOI : 10.1090/S0894-0347-05-00477-7

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