Integral models for moduli spaces of \(G\)-torsors
type: Item
instance of: scholarly article
title: Integral models for moduli spaces of \(G\)-torsors
zbMATH Open document ID: 1293.14004
publication date: 2012-11-02T00:00:00Z
review text: This article makes a generalization of an application of the Abramovich-Vistoli theory of twisted stable maps to higher dimensions. It is a general technique to use the Abromovich-Vistoli theory to construct compactifications of moduli spaces of curves with level structure. For the following, let \(g\) be an integer \(\geq 2\). Given \(C/S\) a smooth proper curve of genus \(g\) over a scheme \(S\). Let \(P\) be a finite set of primes which includes all residue characteristics of \(S\). For any section \(s:S\rightarrow C\) there exists a pro-object \(\pi_1(C/S,s)\) in the category of locally constant sheaves of finite groups on \(S\) whose fiber over a geometric \(\overline{t}\rightarrow S\) is equal to the maximal quotient of \(\pi_1(C_{\overline t},s_{\overline t})\) prime to \(P\). For a finite group \(G\) of order not divisible by any of the primes in \(P\), \(\mathcal H om^{\text{ext}}(\pi_1(C/S,s),G)\) denotes the sheaf of homomorphisms \(\pi_1(C/S,s)\rightarrow G\) modulo the action of \(\pi_1(C/S,s)\) given by conjugation. Then \(\mathcal H om^{\text{ext}}(\pi_1(C/S,s),G)\) is locally constant on \(S\) and is canonically independent of the section \(s\). It turns out that for any smooth curve \(C/S\) this sheaf can be canonically defined even when there does not exist a section. A Teichmüller structure of level \(G\) on \(C/S\) is defined as a section of \(\mathcal H om^{\text{ext}}(\pi_1(C/S),G)\) which étale locally on \(S\) can be represented by a surjective homomorphism \(\pi_1(C/S),s)\rightarrow G\) for a section \(s\). The stack \({}_G\mathcal M_g\) over \(\mathbb Z[1/|G|]\) is defined by the condition that to any \(\mathbb Z[1/|G|]\)-scheme \(S\) is associated the groupoid of pairs \((C/S,\sigma)\), where \(C\) is a smooth proper curve of genus \(g\) over \(S\) and \(\sigma\) is a Teichmüller structure of level \(G\) on \(C/S\). The connection with the theory of Abramovich-Vistoli is described in short form as follows: Let \({}_G\mathcal K_g^\circ\) be…
reviewed by: Arvid Siqveland
Mathematics Subject Classification ID: 14J15
Mathematics Subject Classification ID: 14D06
Mathematics Subject Classification ID: 14D20
zbMATH DE Number: 6101193
zbMATH Keywords: Abramovich-Vistoli theory
zbMATH Keywords: log scheme
zbMATH Keywords: Teichmüller structure
zbMATH Keywords: level structure
zbMATH Keywords: principal bundle
zbMATH Keywords: tame group scheme
zbMATH Keywords: G-torsor
zbMATH Keywords: twisted group torsor
author: Martin Olsson
MaRDI profile type: Publication
full work available at URL: aif.2728
OpenAlex ID: W2328769749
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DOI: 10.5802/AIF.2728
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published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 6101193