Correspondence theoren in tropical geometry

Abstract

Tropical varieties are piecewise linear co-dimension 1 subsets of Rn. For 1- dimensional tropical varieties, we will de ne an enumerative problem analogous to the classical enumerative problem in (C )2, which counts curves of degree d and genus g, passing through 3d + g {u100000} 1 points in general position. Combinatorial nature of tropical geometry will lead us to an algortihm, which solves the tropical enumerative problem. By the correspondence theorem in tropical geometry, which basically says that the two problems coincide, we will indeed have an algorithm that solves the classical enumerative problem. Throughout this work, we will follow a paper [1] of Grigory Mikhalkin, where he introduces and proves the correspondence theorem in tropical geometry.