Frobenius splitting and invariant theory

Abstract

We consider varieties over an algebraically closed field k of characteristic p>0. Given a linear representation of a reductive group, we prove that the ring of invariants is F-regular provided the associated projective quotient is Frobenius-split, the twisting sheaves are Cohen-Macaulay (C-M), and a mild technical condition is met. As an example of how this can be used, we show that the ring of invariants (under the adjoint action of SL(3)) of g copies of M₃ is C-M. (Here M₃ denotes the vector space of 3 × 3 matrices over k and p > 3.) The method of proof involves an induction, and is potentially of wide applicability. As a corollary we obtain that the moduli space of rank 3 and degree 0 bundles on a smooth projective curve of genus g is C-M.