Genus 0 characteristic numbers of the tropical projective plane
Genus 0 characteristic numbers of the tropical projective plane
复制标题
热带投影平面的属 0 特征数
作者：
Benoît Bertrand;E. Brugallé;G. Mikhalkin
求复射影平面$ \mathbb{C} {P}^{2} $的特征数是Zeuthen在世纪前提出的计数几何的经典问题。对于一个给定的$d$和$g$一个必须找到的次数$d$ genus $g$曲线，通过一定的通用配置的点，并在同一时间是切线的某一通用配置的线。在这两种配置中，点和线的总数是3d- 1+ g，所以答案是一个有限整数。本文将这一经典问题转化为当g= 0时相应的热带几何计数问题。也就是说，我们证明了热带问题是适定的，并建立了一个特殊情况下的对应定理，确保相应的热带和经典号码相吻合。然后，我们使用地板图演算，以减少问题的纯组合。因此，我们用开Hurwitz数来表示$ \mathbb{C} {P}^{2} $的亏格0特征数.
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.