Models of torsors over curves

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 $X^{\mathrm{\prime }}$ under the action of an affine and flat $K$-group scheme of finite type $G^{\mathrm{\prime }}$ where $X^{\mathrm{\prime }}$ 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 $X^{\mathrm{\prime }}$ such that $G^{\mathrm{\prime }}$ is finite and 'etale (resp. finite and flat) after, if necessary, extending scalars. We provide examples explaining the new techniques.