Liftings of coverings of weakly ramified curves
type: Item
instance of: scholarly article
title: Liftings of coverings of weakly ramified curves
zbMATH Open document ID: 1108.14024
author: Gunther Cornelissen
author: Ariane Mézard
published in: Mathematische Zeitschrift
publication date: 2006-11-29T00:00:00Z
review text: Let \(X\) be a smooth projective curve over a perfect field \(k\) of characteristic \(p>0\) with a finite group \(G \subset \text{Aut}_k(X)\). Let \(R(X,G)\) be the versal equivariant deformation ring of \((X,G)\). The authors prove the following result: Theorem. Let us assume that the action of \(G\) is weakly ramified (i.e. for all \(P \in X\) with uniformizer \(x\), \(G_{P,2}:=\{\sigma \in G, \text{ord}_x(\sigma x-x)>2\}\) is trivial). Then one of the following properties is satisfied : 1) \(R(X,G)\) is cancelled by \(p\). 2) the non trivial ramification groups of \(G\) whose order is divisible by \(p\) are either a) \(\mathbb{Z}/p\mathbb{Z}\) ; b) the dihedral group \(\mathbb{Z}/p\mathbb{Z} \rtimes \mathbb{Z}/2\mathbb{Z} \) if \(p>2\) or \((\mathbb{Z}/2\mathbb{Z})^2\) if \(p=2\) ; c) \(A_4=(\mathbb{Z}/2\mathbb{Z})^2 \rtimes \mathbb{Z}/3\mathbb{Z}\) for \(p=2\). In any of these cases \(R(X,G)\) is a relative complete intersection over the ring of Witt vectors \(W(k)\); in particular \(R(X,G)\) is flat over \(W(k)\). Thus the characteristic of \(R(X,G)\) is either \(0\) or \(p\). As the authors point out, if the curve \(X\) is ordinary then the action of \(G\) is weakly ramified [\textit{S. Nakajima}, Trans. Am. Math. Soc. 303, 595--607 (1987; Zbl 0644.14010)].
reviewed by: Christophe Ritzenthaler
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 11G20
zbMATH DE Number: 5076560
zbMATH Keywords: versal equivariant deformation ring
zbMATH Keywords: characteristic \(p\)
MaRDI profile type: Publication
full work available at URL: s00209-006-0944-8
OpenAlex ID: W1971126764
cites work: Obstructions locales au relèvement de revêtements galoisiens de courbes lisses
cites work: Formal deformations of wildly ramified coverings of algebraic curves
cites work: Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic
cites work: Zur Entartung schwach verzweigter Gruppenoperationen auf Kurven
cites work: Q4210743
cites work: Q4440901
cites work: p-Ranks and Automorphism Groups of Algebraic Curves
cites work: On the deformation of Artin-Schreier to Kummer
cites work: Functors of Artin Rings
DOI: 10.1007/S00209-006-0944-8
Recommended article: Zur Entartung schwach verzweigter Gruppenoperationen auf Kurven
Recommended article: Equivariant splitting of the Hodge-de Rham exact sequence
Recommended article: Une formule de Riemann-Roch équivariante pour les courbes
Recommended article: Déformations de courbes avec action de groupe
Recommended article: On the tangent space of the deformation functor of curves with automorphisms
Recommended article: Level 𝑚 stratifications of versal deformations of 𝑝-divisible groups
Recommended article: Formal deformation of curves with group scheme action
Recommended article: Which weakly ramified group actions admit a universal formal deformation?
Recommended article: Equivariant Riemann-Roch theorems for curves over perfect fields
Recommended article: Universal deformation rings need not be complete intersections
description: scientific article; zbMATH DE number 5076560