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

Abstract

To any graph G, one can associate a toric variety $$X(\mathcal {P}G)$$X(PG), obtained as a blowup of projective space along coordinate subspaces corresponding to connected subgraphs of G. The polytopes of these toric varieties are the graph associahedra, a class of polytopes that includes the permutohedron, associahedron, and stellahedron. We show that the space $$X(\mathcal {P}{G})$$X(PG) is isomorphic to a Hassett compactification of $$M_{0,n}$$M0,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 with the permutohedron.