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Enumerative tropical algebraic geometry in R2

2003; Cornell University; Linguagem: Inglês

10.48550/arxiv.math/0312530

Grigory Mikhalkin,

Algebraic Geometry and Number Theory

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 http://arxiv.org/abs/math.AG/0209253. 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 Euclidean n-space and holomorphic curves with certain piecewise-linear graphs there.