Enumerative tropical algebraic geometry in R2

TL;DR

This paper introduces a formula for counting algebraic curves of any genus in toric surfaces using tropical geometry, which simplifies complex algebraic problems to combinatorial lattice path calculations.

Contribution

It provides a new enumerative formula for curves in toric surfaces based on tropical algebraic geometry, extending previous results to arbitrary genus.

Findings

Derived a formula for counting curves of arbitrary genus in toric surfaces.

Connected tropical geometry with classical enumerative algebraic geometry.

Validated the formula through combinatorial lattice path methods.

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 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.