On unramified normal coverings of hyperelliptic curves
type: Item
instance of: scholarly article
title: On unramified normal coverings of hyperelliptic curves
zbMATH Open document ID: 1123.14019
author: Yolanda Fuertes
author: G. González-Diez
published in: Journal of Pure and Applied Algebra
publication date: 2007-01-09T00:00:00Z
review text: It is well known that if \(C\) is a compact Riemann surface, then the number of unramified coverings of \(C\) with group of covering transformations isomorphic to \({\mathbb Z}_{2} \times {\mathbb Z}_{2}\) is equal to \((2^{4g}-3\cdot 2^{2g}+2)/6\) (\({\mathbb Z}_{2}\) denotes the cyclic group of order \(2\)). In the paper under review, under the additional assumption that \(C\) is a hyperelliptic surface, the authors determine defining equations for each of these covers and give complete conditions for when any two such surfaces are isomorphic. This extends the results of \textit{R. Horiuchi} [J. Math. Kyoto Univ. 19, 497--523 (1979; Zbl 0427.30038)] where defining equations for those covers which are themselves hyperelliptic are determined. Suppose that \(C\) is a hyperelliptic surface of genus \(g\) with defining equation \[ y^2=\prod_{d=1}^{2g+2} (x-\mu_{d} ) \] and let \(X=\{ 1,2,\dots ,2g+2\}\). Suppose that \(\widetilde{C}\) is some unramified cover of \(C\) with group of covering transformations isomorphic to \({\mathbb Z}_{2} \times {\mathbb Z}_{2}\) generated by \(\alpha\) and \(\beta\) and let \(\pi_{1} \colon \widetilde{C} \rightarrow C_{1}=\widetilde{C} /\langle \alpha \rangle\) and \(\pi_{2} \colon \widetilde{C} \rightarrow C_{2}=\widetilde{C} /\langle \beta \rangle\) be the quotient maps and surfaces. Note that \(C_{1}\) and \(C_{2}\) are both unramified covers of \(C\) with group of covering transformations isomorphic to \({\mathbb Z}_{2}\). Let \(f_{1} \colon C_{1} \rightarrow C\) and \(f_{2} \colon C_{2} \rightarrow C\) denote the corresponding quotient maps. The proof follows from two important results: First, the authors show that \(\widetilde{C}\) is isomorphic to the fibre product \(C_{1} \times C_{2}\) defined by the surfaces \(C_{1}\), \(C_{2}\), and \(C\) and the maps between them, \(\pi_{1}\), \(\pi_{2}\), \(f_{1}\) and \(f_{2}\). Secondly, in previous work by the authors [in: The geometry of Riemann surfaces and abelian varieties. III Iber…
reviewed by: Aaron Wootton
Mathematics Subject Classification ID: 14H37
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 30F10
zbMATH DE Number: 5083181
zbMATH Keywords: hyperelliptic curves
zbMATH Keywords: automorphisms of curves
MaRDI profile type: Publication
full work available at URL: j.jpaa.2006.05.008
OpenAlex ID: W2053812046
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DOI: 10.1016/J.JPAA.2006.05.008
description: scientific article; zbMATH DE number 5083181