The dual complex of \(\bar{M}_{0,n}\) via phylogenetics

Abstract

The moduli space \(\bar{M}_{0,n}\) of stable rational n-pointed curves has divisorial boundary with simple normal crossings. In this brief note I observe that the dual complex is a flag complex; that is, a collection of irreducible boundary divisors has nonempty intersection if and only if the pairwise intersections are nonempty. Rather than proving this directly, I translate the statement to a setting in phylogenetics where it is widely used and multiple explicit proofs have been written. It appears this result is known by experts but lacks a detailed reference in the literature, except recently for \(n=7\).