MODULI OF VECTOR-BUNDLES, FROBENIUS SPLITTING, AND INVARIANT-THEORY

Let X be an irreducible projective curve of genus g over an algebraica lly closed field of positive characteristic not equal 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 consequen ce 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 sp aces are F-split, and further, that they are Cohen-Macaulay. This invo lves showing that the ring of invariants of g copies of 2 x 2 matrices (under the adjoint action of SL(2)) is F-split and Cohen-Macaulay.