Mathematics – Commutative Algebra
Scientific paper
2011-05-10
Mathematics
Commutative Algebra
18 pages, 8 figures
Scientific paper
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.
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