On semistable vector bundles over curves (Pre-published version)

Abstract

Let X be a geometrically irreducible smooth projective curve de ned over a eld k, and let E be a vector bundle on X. Then E is semistable if and only if there is a vector bundle F on X such that Hi(X; F E) = 0 for i = 0; 1. We give an explicit bound for the rank of F. The proof uses a result of Popa for the case that k is algebraically closed.