Resolution of nonsingularities of families of curves
type: Item
instance of: scholarly article
title: Resolution of nonsingularities of families of curves
zbMATH Open document ID: 1078.14037
author: Akio Tamagawa
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2005-07-01T00:00:00Z
review text: Let \(S\) be a 1-dimensional, noetherian, normal, integral, separated scheme and denote by \(K\) the function field of \(S\). A pair \((S', K')\) is a finite extension of \((S,K)\) if \(K'\) is a finite extension of \(K\) and \(S'\) is the integral closure of \(S\) in \(K\) (hence \(K'\) coincides with the function field of \(S'\)). In the present paper the author studies the following problem: for a given closed point \(x\) of a special fiber of a generically smooth family \(X \rightarrow S\) of stable curves is there a covering \(Y \rightarrow X\) that is generically étale, where \(Y\) is also a family of stable curves, such that the image in \(X\) of the non-smooth locus of \(Y\) contains \(x\)? Among many results, the author proves the following. Assume that \(S\) is strictly of mixed characteristic, and that the residue field of each closed point of \(S\) is algebraic over \(\mathbb{F}_p\), for some \(p > 0\). Let \((X^*, D)\) be a generically smooth, stable marked curve over \(S\). Then for each closed point \(x \in X^*\) there exists a finite extension \((S', K')\) of \((S,K)\) and a generically tame stable covering \((Y^*_{S'}, E_{S'})\) of \((X^*_{S'}, D_{S'})\) over \(S'\) such that the image in \(X^*\) of the vertical components in the fibers of \(Y^*_{S'} \rightarrow X^*_{S'}\) contains \(x\) (in particular the image of the non-smooth locus of \(Y^*_{S'}\) in \(X^*\) contains \(x\)).
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 14D06
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14H05
zbMATH DE Number: 2185483
zbMATH Keywords: coverings of stable curves
reviewed by: Cicero Carvalho
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1145475448
OpenAlex ID: W2026204140
cites work: Stable maps and Hurwitz schemes in mixed characteristics
cites work: Smoothness, semi-stability and alterations
cites work: Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic. -- Erratum
cites work: The tame fundamental groups of a formal neighbourhood of a divisor with normal crossings on a scheme
cites work: Q4539743
cites work: Arithmetic Moduli of Elliptic Curves. (AM-108)
cites work: The projectivity of the moduli space of stable curves, II: The stacks $M_{g,n}$
cites work: Q4264427
DOI: 10.2977/PRIMS/1145475448
description: scientific article; zbMATH DE number 2185483