Models of torsors and the fundamental group scheme

TL;DR

This paper investigates the extension of torsors from the generic fiber to models over a base scheme, focusing on the fundamental group scheme and torsor models in algebraic geometry.

Contribution

It provides partial solutions for modeling torsors over schemes, especially using Néron blow-ups, and interprets the fundamental group scheme via Tannaka duality.

Findings

Existence of torsor models on certain schemes with Néron blow-ups

Interpretation of the fundamental group scheme as a Tannaka Galois group

Partial answers for finite group scheme torsors

Abstract

Given a relative faithfully flat pointed scheme over the spectrum of a discrete valuation ring $X \to S$ this paper is motivated by the study of the natural morphism from the fundamental group scheme of the generic fiber $X_\eta $ to the generic fiber of the fundamental group scheme of $X$. Given a torsor $T \to X_\eta $ under an affine group scheme $G$ over the generic fiber of $X$, we address the question to find a model of this torsor over $X$, focusing in particular on the case where $G$ is finite. We obtain partial answers to this question, showing for instance that, when $X$ is integral and regular of relative dimension $1$, such a model exists on some model of $X_{\eta}$ obtained by performing a finite number of N\'eron blow-ups along a closed subset of the special fiber of $X$. In the first part we show that the relative fundamental group scheme of $X$ has an interpretation as the Tannaka Galois group of a tannakian category constructed starting from the universal torsor.