Stable bundles as Frobenius morphism direct image

Abstract

Let X be a smooth projective curve of genus g⩾2 over an algebraically closed field k of characteristic p>0, and let F:X→X1 be the relative Frobenius morphism. We show that a vector bundle E on X1 is the direct image under F of some stable bundle on X if and only if the instability of F⁎E is equal to (p−1)(2g−2).