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We study the Cox ring of the moduli space of stable pointed rational curves, , via the closely related permutohedral (or Losev-Manin) spaces . Our main result establishes  polynomial subrings of , thus giving collections of boundary variables that intersect the ideal of relations of  trivially. As applications, we give a combinatorial way to partially solve the Riemann-Roch problem for , and we show that all relations in degrees of  arising from certain pull-backs from projective spaces are generated by the Plücker relations.

Permutohedral spaces and the Cox ring of the moduli space of stable pointed rational curves