Coordinate rings for the moduli stack of SL2(C) quasi-parabolic principal bundles on a curve and toric fiber products

We continue the program started in Manon (2010) [M1] to understand the combinatorial commutative algebra of the projective coordinate rings of the moduli stack MC,p→(SL2(C)) of quasi-parabolic SL2(C) principal bundles on a generic marked projective curve. We find general bounds on the degrees of polynomials needed to present these algebras by studying their toric degenerations. In particular, we show that the square of any effective line bundle on this moduli stack yields a Koszul projective coordinate ring. This leads us to formalize the properties of the polytopes used in proving our results by constructing a category of polytopes with term orders. We show that many of results on the projective coordinate rings of MC,p→(SL2(C)) follow from closure properties of this category with respect to fiber products.