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2003•arXiv (Cornell University)Open access

Enumerative tropical algebraic geometry

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 [17]. The result is established with the help of the so-called tropical algebraic geometry. This geometry allows to replace complex toric varieties with the real space R n and holomorphic curves with certain piecewise-linear graphs there.

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What this paper is about

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 [17]. The result is established with the help of the so-called tropical algebraic geometry. This geometry allows to replace complex toric varieties with the real space 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 [17]. The result is established with the help of the so-called tropical algebraic geometry. This geometry allows to replace complex toric varieties with the real space R n and holomorphic curves with certain piecewise-linear graphs there.

Key concepts: Tropical geometry, Mathematics, Newton polygon, Holomorphic function, Toric variety, Algebraic geometry, Polygon (computer graphics), Algebraic curve

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