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

We study the Cox ring of the moduli space of stable pointed rational curves, ${overline{M}_{0,n}}$ , via the closely related permutohedral (or Losev-Manin) spaces ${overline{L}_{n-2}}$ . Our main result establishes $left(begin{array}{ll} n 2 end{array}right)$ polynomial subrings of ${{rm Cox}(overline{M}_{0,n})}$ , thus giving collections of boundary variables that intersect the ideal of relations of ${{rm Cox}(overline{M}_{0,n})}$ trivially. As applications, we give a combinatorial way to partially solve the Riemann-Roch problem for ${overline{M}_{0,n}}$ , and we show that all relations in degrees of ${{rm Cox}(overline{M}_{0,6})}$ arising from certain pull-backs from projective spaces are generated by the Plücker relations.