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

Abstract

We continue the program started in \cite{M1} to understand the combinatorial commutative algebra of the projective coordinate rings of the moduli stack $\mathcal{M}_{C, \vec{p}}(SL_2(\C))$ of quasi-parabolic $SL_2(\C)$ 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 $\mathcal{M}_{C, \vec{p}}(SL_2(\C))$ follow from closure properties of this category with respect to fiber products.

23 pages, 14 figures, added new figures, stronger main results

References in corpus (3)

• Generalized theta linear series on moduli spaces of vector bundles on curves
• Presentations of Semigroup Algebras of Weighted Trees
• Sagbi Bases of Cox-Nagata Rings

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