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 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 \(X(\mathcal {P}{G})\) 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 with the permutohedron.
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Acknowledgments
This work was completed as part of the 2014 Summer Undergraduate Mathematics Research at Yale (SUMRY) program, where the first author was a participant and the second and third authors were mentors. We are grateful to all involved in the SUMRY program for the vibrant research community that they helped create. It is a pleasure to thank Dagan Karp, who actively collaborated with the third when the ideas in the present text were at their early stages. We thank Satyan Devadoss for his encouragement, as well as permission to include Fig. 2 from [5]. Finally, we thank the referee for their careful reading and comments. The authors were supported by NSF Grant CAREER DMS-1149054 (PI: Sam Payne).
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da Rosa, R.F., Jensen, D. & Ranganathan, D. Toric graph associahedra and compactifications of \(M_{0,n}\) . J Algebr Comb 43, 139–151 (2016). https://doi.org/10.1007/s10801-015-0629-7
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DOI: https://doi.org/10.1007/s10801-015-0629-7