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Using combinatorial methods, we determine that a projective coordinate ring of the moduli of parabolic principal $\mathrm{SL}_2$-bundles on a marked projective curve is not Gorenstein when the genus and number of marked points are greater than $1$.

The Gorenstein Property for Projective Coordinate Rings of the Moduli of Parabolic $\mathrm{SL}_2$-Principal Bundles on a Smooth Curve