We continue the program started in \cite{M1} to understand the commutative
algebra of the projective coordinate rings of line bundles on the moduli
$\mathcal{M}_{C, \vec{p}}(SL_2(\C))$ of quasi-parabolic principal bundles on a
marked projective curve. We prove a general theorem about presentations of
these rings, which implies that for generic marked curves $(C, \vec{p})$ the
square of any effective line bundle has projective coordinate ring generated in
degree 1 with a presenting ideal generated in degree 3. When the genus of the
curve $C$ is less than or equal to 2, we find that the square of any such line
bundle gives a Koszul projective coordinate ring. Both theorems are obtained by
studying toric degenerations of the 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, and studying its closure
properties under fiber products.