Genus 0 characteristic numbers of the tropical projective plane

TL;DR

This paper translates classical enumerative geometry problems about genus 0 curves in the complex projective plane into tropical geometry, establishing a correspondence and deriving combinatorial formulas for characteristic numbers.

Contribution

It proves a correspondence theorem for genus 0 characteristic numbers in tropical geometry and expresses these numbers via open Hurwitz numbers, linking classical and tropical enumerative problems.

Findings

Tropical problem is well-posed for genus 0.

A correspondence theorem equates tropical and classical characteristic numbers.

Genus 0 characteristic numbers are expressed through open Hurwitz numbers.

Abstract

Finding the so-called characteristic numbers of the complex projective plane ${\mathbb C}P^2$ is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given $d$ and $g$ one has to find the number of degree $d$ genus $g$ curves that pass through a certain generic configuration of points and at the same time are tangent to a certain generic configuration of lines. The total number of points and lines in these two configurations is $3d-1+g$ so that the answer is a finite integer number. In this paper we translate this classical problem to the corresponding enumerative problem of tropical geometry in the case when $g=0$. Namely, we show that the tropical problem is well-posed and establish a special case of the correspondence theorem that ensures that the corresponding tropical and classical numbers coincide. Then we use the floor diagram calculus to reduce the problem to pure combinatorics. As a consequence, we express genus 0 characteristic numbers of $\CC P^2$ in terms of open Hurwitz numbers.