交叉比度和三角测量

IF 0.8 3区数学 Q2 MATHEMATICS Bulletin of the London Mathematical Society Pub Date : 2024-09-17 DOI:10.1112/blms.13148

Rob Silversmith

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摘要

交叉比度问题计算 P 1 $\mathbb {P}^1$ 上 n 个 $n$ 点的配置，其中有 n - 3 个 $n-3$ 规定的交叉比。交叉比度问题出现在组合学和几何的许多角落，但它们的结构一般还不太清楚。有趣的是，研究该问题的各种特例可以得到既多样又丰富的组合结构。在本文中，我们证明了一类以 n $n$ -gon 的三角形为索引的交叉比率度的简单封闭公式；这些度与 M 0 , n $M_{0,n}$ 的实部几何以及正几何相关联。

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Cross-ratio degrees and triangulations

The cross-ratio degree problem counts configurations of $n$ points on $P^{1}$ with $n - 3$ prescribed cross-ratios. Cross-ratio degrees arise in many corners of combinatorics and geometry, but their structure is not well-understood in general. Interestingly, examining various special cases of the problem can yield combinatorial structures that are both diverse and rich. In this paper, we prove a simple closed formula for a class of cross-ratio degrees indexed by triangulations of an $n$ -gon; these degrees are connected to the geometry of the real locus of $M_{0, n}$ , and to positive geometry.