Projective curve/Vector bundle/Cohomology class/Finite annihilation and torsor/Fact

Let ${}C$ denote a smooth projective curve over an algebraically closed 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 the following conditions are equivalent.

1. There exists a finite morphism

   $\varphi \colon C'\longrightarrow C$

   from a smooth projective curve ${}C'$ such that ${}\varphi ^{*}(c)=0$.

2. There exists a projective curve ${}Z\subseteq T$.

Proof, Write another proof