Finite field/Smooth projective curve/Vector bundle/Affineness and nonexistence of curves/Fact

Let ${}C$ denote a smooth projective curve over the algebraic closure of a finite field ${}K$, let ${}{\mathcal {S}}$ be a locally free sheaf on ${}C$ and let ${}c\in H^{1}(C,{\mathcal {S}})$ be a cohomology class with corresponding torsor ${}T\rightarrow C$.

Then ${}T$ is affine if and only if it does not contain any projective curve.

Proof, Write another proof