Models of torsors over curves

TL;DR

This paper demonstrates how to extend torsors over curves from the generic fiber to models over modified schemes using Néron blow-ups, with special cases for finite étale and flat group schemes.

Contribution

It introduces new techniques for extending torsors over curves through Néron blow-ups, including cases with finite étale and flat group schemes.

Findings

Extension of torsors over curves via Néron blow-ups.

Existence of models with finite étale or flat group schemes.

Examples illustrating the new methods.

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.