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

Beweis

If (1) holds, then the pull-back  ${\phi }^{\ast }(T)=T{\times }_C{C}^{\prime }$  is trivial (as a torsor), as it equals the torsor given by  ${\phi }^{\ast }(c)=0$.  Hence ${\phi }^{\ast }(T)$ is isomorphic to a vector bundle and contains in particular a copy of $C^{\prime }$. The image $Z$ of this copy is a projective curve inside $T$.

If (2) holds, then let $C^{\prime }$ be the normalization of $Z$. Since $Z$ dominates $C$, the resulting morphism

$\phi :C^{\prime }⟶C$

is finite. Since this morphism factors through $T$ and since $T$ annihilates the cohomology class by which it is defined, it follows that

 ${\phi }^{\ast }(c)=0$. 

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