Genus 0 characteristic numbers of tropical projective plane

Details Genus 0 characteristic numbers of tropical projective plane Genus 0 characteristic numbers of tropical projective plane

2011-05-10

arxiv.org/abs/1105.2004v2

Mathematics

Algebraic Geometry

50 pages, 21 figures

Scientific paper

Finding the so-called characteristic numbers of the complex projective plane $CP^2$ 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 in the same time are tangent to 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 compute genus 0 characteristic numbers of $CP^2$ in terms of open Hurwitz numbers.

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