Torsor over projective curve/Strongly semistable/Affineness criterion/Fact

Let ${}C$ denote a smooth projective curve over an algebraically closed field ${}K$ and let ${}{\mathcal {S}}$ be a strongly semistable vector bundle over ${}C$ together with a cohomology class ${}c\in H^{1}(C,{\mathcal {S}})$ .

Then the torsor

{}T(c)

is an affine scheme if and only if

${}\operatorname {deg} {\left({\mathcal {S}}\right)}<0$ and ${}c\neq 0$ ( ${}F^{e}(c)\neq 0$

for all

{}e

in positive characteristic^[1]).

Write a proof

↑ Here one has to check only finitely many ${}e$ s and there exist good estimates how far one has to go. Also, in a relative situation, this is only an extra condition for finitely many prime numbers.

[1] Here one has to check only finitely many ${}e$ s and there exist good estimates how far one has to go. Also, in a relative situation, this is only an extra condition for finitely many prime numbers.

[1]