TORIC GRAPH ASSOCIAHEDRA AND COMPACTIFICATIONS OF M0,n

To any graph G one can associate a toric variety X(PG), 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(PG) is isomorphic to a Hassett compactification of 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 to the permutohedron.