Curves with only triple ramification.
type: Item
instance of: scholarly article
title: Curves with only triple ramification.
zbMATH Open document ID: 1057.14035
author: Stefan Schröer
publication date: 2004-08-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0206091
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116097
review text: Let \(C\) be a smooth proper curve of genus \(g\) over an algebraically closed field \(k\) of characteristic \(p \geq 0\). In the paper under review, the author poses the following question: Does there exist a generically étale finite map \(C \to {\mathbb P}^{1}\) whose ramification points all have index \(e=3\)? The main result of this paper is the following: Suppose \(p \neq 3\) and assume \(g \geq 2\). Let \(S\subset M_{g}\) be the closure of the set of all closed points such that the corresponding curve \(C\) admits a generically étale map \(C \to {\mathbb P}^{1}_{k}\) with only triple ramification. Then we have \(\dim(S) \geq 2g-3\). Furthermore, the author extends a result on elliptic complex curves of \textit{M. Fried, E. Klassen} and \textit{Y. Kopeliovich} [Proc. Am. Math. Soc., 129, 111--119 (2001; Zbl 0978.30026)] to all characteristics \(p \neq 3\), proving that for all but finitely many \(j\)-invariants \(j \in k\), the corresponding elliptic curve \(E\) admits a finite generically étale map \(C \to {\mathbb P}^{1}_{k}\) with only triple ramification. For the proof, the author uses deformation theory and the moduli space of stable curves.
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 14H51
zbMATH DE Number: 2093470
zbMATH Keywords: smooth curves
zbMATH Keywords: ramification points
zbMATH Keywords: families
zbMATH Keywords: moduli space of curves
reviewed by: Paolo A. Oliverio
MaRDI profile type: Publication
OpenAlex ID: W1626612316
arXiv ID: math/0206091
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DOI: 10.5802/AIF.2004
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published in: Annales de l’institut Fourier
description: scientific article