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

TL;DR

This paper investigates the Cox ring of the moduli space of stable pointed rational curves using permutohedral spaces, revealing polynomial subrings and relations that aid in understanding its algebraic structure.

Contribution

It introduces polynomial subrings within the Cox ring of ,n and shows relations are generated by Plfccer relations for degree 6, advancing the algebraic understanding of these moduli spaces.

Findings

Identifies  choose 2 polynomial subrings of the Cox ring.

Provides a combinatorial approach to the Riemann-Roch problem for ,n.

Shows relations in Cox(,6) are generated by Plfccer relations.

Abstract

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