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

TL;DR

This paper shows that the dual complex of the boundary divisors of the moduli space of stable rational curves is a flag complex, connecting algebraic geometry with phylogenetics and providing a clearer understanding of its combinatorial structure.

Contribution

The paper identifies the dual complex of ar{M}_{0,n} as a flag complex by translating the problem into phylogenetics, offering a new perspective and filling a gap in the literature.

Findings

The dual complex is a flag complex.

The result is known informally but lacked a detailed reference.

Connection established between algebraic geometry and 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 $n=7$.