Hypergraph associahedra and compactifications of moduli spaces of points

TL;DR

This paper demonstrates that certain Hassett compactifications of moduli spaces of weighted stable rational curves are toric varieties associated with hypergraph associahedra, linking moduli space geometry with polyhedral combinatorics.

Contribution

It establishes that these moduli spaces are toric varieties with polytopes given by hypergraph associahedra, extending the understanding of their geometric and combinatorial structure.

Findings

Hassett compactifications are toric varieties with hypergraph associahedron polytopes.

The results connect moduli space compactifications with hypergraph combinatorics.

An explicit relationship between different moduli spaces via hypergraph inflation is shown.

Abstract

We prove that every Hassett compactification of the moduli space of weighted stable rational curves that admits both a reduction map from the Losev-Manin compactification and a reduction map to projective space is a toric variety, whose corresponding polytope is a hypergraph associahedron (also known as a nestohedron). In addition, we present an analogous result for the moduli space of labeled weighted points in affine space up to translation and scaling. These results are interconnected, and we make their relationship explicit through the concept of ``inflation" of a hypergraph associahedron.