Projektive Kurve/Vektorbündel/Kohomologieklasse/Endliche Annulierung und Kurven im Torsor/Fakt/en

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

1. There exists a finite morphism

   $\phi :C^{\prime }⟶C$

   from a smooth projective curve $C^{\prime }$ such that  ${\phi }^{\ast }(c)=0$. 

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