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 $𝒮$ be a locally free sheaf on $C$ and let $c\in H^1(C,𝒮)$ be a cohomology class with corresponding torsor $T\to C$.

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

Proof, Write another proof