Prym varieties associated with \(n\)-cyclic coverings of a hyperelliptic curve
type: Item
instance of: scholarly article
title: Prym varieties associated with \(n\)-cyclic coverings of a hyperelliptic curve
zbMATH Open document ID: 1049.14018
author: Angela Ortega
published in: Mathematische Zeitschrift
publication date: 2003-12-16T00:00:00Z
review text: Given a smooth, projective, hyperelliptic curve \(H\) of genus \(g\), for every \(\eta\) of order \(n\) in the Jacobian \(JH\), let \(f : C \rightarrow H\) be the \(n\)-cyclic covering associated to \(\eta\). The Prym variety \(P:= \text{Prym} (C/M)\) is given by the zero component of \(\operatorname{Ker} N_f\), \(N_f : JC \rightarrow JH\). \(P\) is an abelian subvariety of \(JC\); the main result in the paper shows that \(P\) is isomorphic to a product \(JC_0\times JC_1\) of jacobians of curves. The curves \(JC_0, JC_1\) can be found via a hyperelliptic structure \(C\rightarrow {\mathbb P}^1\) obtained from the one on \(H\); since \(\text{Gal} (C/{\mathbb P}^1) \cong D_n = \langle j, \sigma\rangle\), where \(j^2=\sigma^n=1\), there are involutions \(j_{\nu}=j\sigma ^\nu\), \(\nu = 0,\dots,n-1\) which define double ramified coverings \(f_\nu : C \rightarrow C_\nu := C/\langle j_\nu \rangle\). The jacobians \(JC_\nu\) are included in \(P\) and it turns out that \(P \cong JC_0\times JC_1\).
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14H45
zbMATH DE Number: 2018246
zbMATH Keywords: Jacobian
zbMATH Keywords: Prym variety
zbMATH Keywords: hyperelliptic curves
reviewed by: Alessandro Gimigliano
MaRDI profile type: Publication
full work available at URL: s00209-003-0522-2
OpenAlex ID: W1522945086
DOI: 10.1007/S00209-003-0522-2
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description: scientific article; zbMATH DE number 2018246