Genus 0 characteristic numbers of the tropical projective plane
Benoit Bertrand, Erwan Brugalle, Grigory Mikhalkin

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 is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given and one has to find the number of degree genus 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 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 . 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…
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