Extention of Finite Solvable Torsors over a Curve

TL;DR

This paper proves that under certain conditions, finite solvable torsors over the generic fiber of a smooth surface can be extended over the entire surface after scalar extension and blow-up modifications.

Contribution

It establishes extension results for finite solvable torsors over a fibered surface in positive characteristic, generalizing previous cases.

Findings

Extension of torsors after scalar extension

Extension after blowing up the surface

Applicable to torsors with p-group or length-2 normal series

Abstract

Let $R$ be a discrete valuation ring with fraction field $K$ and with algebraically closed residue field of positive characteristic $p$. Let $X$ be a smooth fibered surface over $R$ with geometrically connected fibers endowed with a section $x\in X(R)$. Let $G$ be a finite solvable $K$-group scheme and assume that either $|G|=p^n$ or $G$ has a normal series of length 2. We prove that every quotient pointed $G$-torsor over the generic fiber $X_{\eta}$ of $X$ can be extended to a torsor over $X$ after eventually extending scalars and after eventually blowing up $X$ at a closed subscheme of its special fiber $X_s$.