Finite field/Smooth projective curve/Vector bundle/Strongly semistable degree nonnegative/Finite annihilation/Fact/Proof

Proof

If the degree of ${}{\mathcal {S}}$ is positive, then a Frobenius pull-back ${}F^{e*}({\mathcal {S}})$ has arbitrary large degree and is still semistable. By Serre duality we get that ${}H^{1}(X,F^{e*}({\mathcal {S}}))=0$. So in this case we can annihilate the class by an iteration of the Frobenius alone.

So suppose that the degree is ${}0$. Then there exists by fact a finite morphism which trivializes the bundle. So we may assume that ${}{\mathcal {S}}\cong {\mathcal {O}}_{X}^{r}$. Then the cohomology class has several components ${}c_{i}\in H^{1}(X,{\mathcal {O}}_{X})$ and it is enough to annihilate them separately by finite morphisms. But this is possible by the parameter theorem of K. Smith (or directly using Frobenius and Artin-Schreier extensions).

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