Toric graph associahedra and compactifications of \(M_{0,n}\) (Q5964811)

Given a graph \(G\), one can obtain a toric variety \(X(\mathcal{P}G)\) as a certain blowup of a projective space along subspaces determined by connected subgraphs of \(G\). The main result of this paper shows that the space \(X(\mathcal{P}G)\) is isomorphic to a Hassett weighted modular compactification of the moduli space \(M_{0,n}\) of \(n\)-pointed rational curves [\textit{B. Hassett}, Adv. Math. 173, No. 2, 316--352 (2003; Zbl 1072.14014)] precisely when \(G\) is an iterated cone over a discrete set. It generalizes the result of \textit{A. Losev} and \textit{Y. Manin} [Mich. Math. J. 48, 443--472 (2000; Zbl 1078.14536)] for the case when \(\mathcal{P}G\) is the permutohedron. The authors also provide a number of examples and figures to illustrate this beautiful correspondence.