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We continue the program started in \cite{M1} to understand the combinatorialcommutative algebra of the projective coordinate rings of the moduli stack$\mathcal{M}_{C, \vec{p}}(SL_2(\C))$ of quasi-parabolic $SL_2(\C)$ principalbundles on a generic marked projective curve. We find general bounds on thedegrees of polynomials needed to present these algebras by studying their toricdegenerations. In particular, we show that the square of any effective linebundle on this moduli stack yields a Koszul projective coordinate ring. Thisleads us to formalize the properties of the polytopes used in proving ourresults by constructing a category of polytopes with term-orders. We show thatmany of results on the projective coordinate rings of $\mathcal{M}_{C,\vec{p}}(SL_2(\C))$ follow from closure properties of this category withrespect to fiber products.

Coordinate rings for the moduli of $SL_2(\C)$ quasi-parabolic principal bundles on a curve and toric fiber products