Models of torsors over curves

Abstract: Let $R$ be a complete discrete valuation ring with fraction field $K$ and with algebraically closed residue field. Let $X$ be a faithfully flat $R$-scheme of finite type of relative dimension 1 and $G$ be any affine $K$-group scheme of finite type. We prove that every $G$-torsor $Y$ over the generic fibre $X_{\eta}$ of $X$ can be extended to a torsor over ${X'}$ under the action of an affine and flat $K$-group scheme of finite type $G'$ where $X'$ is obtained by $X$ after a finite number of N\'eron blowing ups. Moreover if $G$ is finite and \'etale (resp. admits a finite and flat model) we find $X'$ such that $G'$ is finite and \'etale (resp. finite and flat) after, if necessary, extending scalars. We provide examples explaining the new techniques.