Structural properties of nested set complexes

TL;DR

This paper investigates the structural and topological features of nested set complexes of matroids, demonstrating their decomposability and applying these results to important moduli space complexes, revealing new combinatorial properties.

Contribution

It proves that nested set complexes are vertex decomposable and have convex ear decompositions, unifying and extending existing theorems, and applies these findings to the Deligne--Mumford--Knudsen moduli space.

Findings

Nested set complexes are vertex decomposable.

They admit convex ear decompositions.

The h-vector of these complexes is strongly flawless and top-heavy.

Abstract

We study structural and topological properties of nested set complexes of matroids with arbitrary building sets, proving that these complexes are vertex decomposable and admit convex ear decompositions. These results unify and generalize several recent and classical theorems on Bergman complexes and augmented Bergman complexes of matroids. As a first application, we show that the $h$-vector of a nested set complex is strongly flawless and, in particular, top-heavy. We then specialize to the boundary complex of the Deligne--Mumford--Knudsen moduli space $\overline{\mathcal{M}}_{0, n}$ of rational stable marked curves, which coincides with the complex of trees, establishing new structural decomposition theorems and deriving combinatorial formulas for its face enumeration polynomials.