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Journal of Algebraic Combinatorics

, Volume 43, Issue 1, pp 139–151 | Cite as

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

  • Rodrigo Ferreira da Rosa
  • David Jensen
  • Dhruv Ranganathan
Article

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.

Keywords

Graph associahedra Permutohedron Hassett space  Moduli space of curves Toric variety 

Notes

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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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Rodrigo Ferreira da Rosa
    • 1
  • David Jensen
    • 2
  • Dhruv Ranganathan
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA
  2. 2.Department of MathematicsUniversity of KentuckyLexingtonUSA

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