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2005•Journal of the American Mathematical SocietyOpen access

Enumerative tropical algebraic geometry in ℝ²

Grigory Mikhalkin

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Abstract

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 R n \mathbb {R}^n and holomorphic curves with certain piecewise-linear graphs there.

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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 R n \mathbb {R}^n and holomorphic curves with certain piecewise-linear graphs there.

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Available abstract

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 R n \mathbb {R}^n and holomorphic curves with certain piecewise-linear graphs there.

Key concepts: Tropical geometry, Mathematics, Holomorphic function, Geometry, Algebraic geometry, Algebraic number, Polygon (computer graphics), Newton polygon

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