Frobenius splitting of Hilbert schemes of points on surfaces

Abstract

Let X be a quasiprojective smooth surface defined over an algebraically closed field of positive characteristic. In this note we show that if X is Frobenius split then so is the Hilbert scheme Hilb $^n(X)$ of n points inX. In particular, we get the higher cohomology vanishing for ample line bundles on Hilb $^n(X)$ when X is projective and Frobenius split.