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Genus 0 characteristic numbers of the tropical projective plane

Part of: Projective and enumerative geometry

Published online by Cambridge University Press:  18 November 2013

Benoît Bertrand ,
Erwan Brugallé  and
Grigory Mikhalkin
Benoît Bertrand
Affiliation:
Institut Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse Cedex 9, France email benoit.bertrand@math.univ-toulouse.fr
Erwan Brugallé
Affiliation:
Université Pierre et Marie Curie, Paris 6, 4 place Jussieu, 75 005 Paris, France email brugalle@math.jussieu.fr
Grigory Mikhalkin
Affiliation:
Section de mathématiques Université de Genève, Villa Battelle, 7 route de Drize, 1227 Carouge, Suisse, Switzerland email grigory.mikhalkin@unige.ch
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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 $ \mathbb{C} {P}^{2} $ in terms of open Hurwitz numbers.

Keywords

tropical geometrycharacteristic numbersenumerative geometryprojective plane

MSC classification

Secondary: 14T05: Tropical geometry 14N10: Enumerative problems (combinatorial problems)
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 1 , January 2014 , pp. 46 - 104
Copyright
© The Author(s) 2013 

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