Cross-ratio degrees and triangulations

Abstract

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.