Abstract
Let $X$ be an irreducible projective curve of genus $g$ over an algebraically closed field of positive characteristic $\ne 2,3$. In Part I, we prove, adapting the degeneration arguments of [N-TR], that moduli spaces of rank-two (parabolic) bundles on $X$ are Frobenius split [M-R] for generic smooth $X$. (A similar result holds for $X$ nodal. A consequence is the Verlinde formula in positive characteristic.) In Part II, we give a direct proof of the fact that the local rings of the moduli spaces are $\mathrm{F}$-split, and further, that they are Cohen-Macaulay. This involves showing that the ring of invariants of $g$ copies of $2\times 2$ matrices (under the adjoint action of $\mathrm{SL}(2)$) is a $\mathrm{F}$-split and Cohen-Macaulay.