Toric graph associahedra and compactifications of $M_{0,n}$

TL;DR

This paper explores the relationship between graph associahedra and moduli space compactifications, showing that certain toric varieties derived from graphs are isomorphic to Hassett compactifications of $M_{0,n}$, generalizing known cases.

Contribution

It characterizes when the toric variety from a graph associahedron is isomorphic to a Hassett compactification of $M_{0,n}$, extending previous results to iterated cones over discrete sets.

Findings

Toric varieties from graph associahedra correspond to specific Hassett compactifications.

The isomorphism holds when the graph is an iterated cone over a discrete set.

Generalizes the relation between the permutohedron and the Losev--Manin space.

Abstract

To any graph $G$ one can associate a toric variety $X(\mathcal{P}G)$, obtained as a blowup of projective space along coordinate subspaces corresponding to connected subgraphs of $G$. The polytope of this toric variety is the graph associahedron of $G$, a class of polytopes that includes the permutohedron, associahedron, and stellahedron. We show that the space $X(\mathcal{P}{G})$ is isomorphic to a Hassett compactification of $M_{0,n}$ precisely when $G$ is an iterated cone over a discrete set. This may be viewed as a generalization of the well-known fact that the Losev--Manin moduli space is isomorphic to the toric variety associated to the permutohedron.