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

TL;DR

This paper investigates the algebraic and combinatorial structure of coordinate rings of moduli stacks of quasi-parabolic $SL_2( ext{C})$ bundles on curves, establishing bounds, Koszul properties, and categorical polytope frameworks.

Contribution

It introduces bounds on polynomial degrees for presentations, proves Koszul properties for certain line bundle squares, and develops a polytope category framework to analyze these algebraic structures.

Findings

Effective line bundle squares produce Koszul coordinate rings.

Polytope category with term-orders formalizes combinatorial properties.

Closure under fiber products explains many algebraic results.

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.