Projective variety/Locally free sheaf/First cohomology class/Corresponding torsor/Ext and projective bundle/description

For a vector bundle ${}{\mathcal {S}}$ and a cohomology class ${}c\in H^{1}(C,{\mathcal {S}})$ one can construct a geometric object: Because of ${}H^{1}(C,{\mathcal {S}})\cong \operatorname {Ext} ^{1}({\mathcal {O}}_{C},{\mathcal {S}})$, the class defines an extension

$0\longrightarrow {\mathcal {S}}\longrightarrow {{\mathcal {S}}'}\longrightarrow {\mathcal {O}}_{C}\longrightarrow 0$

and hence projective bundles

${\mathbb {P} }({\mathcal {S}}^{*})\subset {\mathbb {P} }({{\mathcal {S}}'}^{*})$

and the corresponding torsor

$T={\mathbb {P} }({{\mathcal {S}}'}^{*})-{\mathbb {P} }({\mathcal {S}}^{*}).$

This geometric object is the torsor or principal fiber bundle or affine-linear bundle given by ${}c$. The sheaf ${}{\mathcal {S}}$ acts on it by translations.