Abstract

Tropical curves are balanced piecewise linear functions from graphs into R^n, with d “infinite legs” going in directions e_1, e_2, and -e_1-e_2. The number d is the degree of the tropical curve, while the first Betti number of the graph is the genus. We can ask enumerative questions: how many tropical curves of genus g and degree d pass through 3d+g-1 points?

It turns out that, in the case of genus 0, this number is precisely the same as the number of algebraic curves of degree d and genus g passing through 3d-1 points. This so-called Mikhalkin’s tropical correspondence has far-reaching consequences. If we want to count algebraic curves, we “simply” have to count tropical curves. I will introduce the notion of tropicalization, sketch the correspondence theorem, and suggest how these results may be generalised.