论文标题
将热带Kontsevich的公式概括为多个交叉比例
Generalizing tropical Kontsevich's formula to multiple cross-ratios
论文作者
论文摘要
Kontsevich的公式是递归，该递归计算$ \ Mathbb {p} _ {\ Mathbb {c}}}^2 $通过$ 3D-1 $一般位置点的理性程度$ D $曲线数量。肯特维奇（Kontsevich）通过考虑满足给定点条件以外的额外条件的曲线来证明这一点。这些至关重要的额外条件是两个线条条件和称为交叉比例的条件。 本文解决了一个问题，是否有肯特维奇将军的公式，该公式可容纳多个交叉比例。使用热带几何形状，我们获得了这样的递归公式。为此，我们使用对应定理ARXIV：1509.07453，该定理将所讨论的代数几何数字与热带的几何数字相关联。事实证明，我们获得的一般热带kontsevich的公式不仅能够计算我们要寻找的代数几何数字，还可以计算尚无通信定理的进一步的热带数字。 我们表明，我们的递归一般肯特维奇的公式意味着原始的kontsevich的公式，并且初始值是Kontsevich的fomula数字提供和纯粹的组合数字，即所谓的交叉比例多重性。
Kontsevich's formula is a recursion that calculates the number of rational degree $d$ curves in $\mathbb{P}_{\mathbb{C}}^2$ passing through $3d-1$ general positioned points. Kontsevich proved it by considering curves that satisfy extra conditions besides the given point conditions. These crucial extra conditions are two line conditions and a condition called cross-ratio. This paper addresses the question whether there is a general Kontsevich's formula which holds for more than one cross-ratio. Using tropical geometry, we obtain such a recursive formula. For that we use a correspondence theorem arXiv:1509.07453 that relates the algebro-geometric numbers in question to tropical ones. It turns out that the general tropical Kontsevich's formula we obtain is capable of not only computing the algebro-geometric numbers we are looking for, but also of computing further tropical numbers for which there is no correspondence theorem yet. We show that our recursive general Kontsevich's formula implies the original Kontsevich's formula and that the initial values are the numbers Kontsevich's fomula provides and purely combinatorial numbers, so-called cross-ratio multiplicities.
