Double coverings between smooth plane curves
type: Item
instance of: scholarly article
title: Double coverings between smooth plane curves
zbMATH Open document ID: 1145.14027
author: Takeshi Harui
author: Jiryo Komeda
author: Akira Ohbuchi
published in: Kodai Mathematical Journal
publication date: 2008-08-14T00:00:00Z
review text: In this paper the authors classify the pairs of smooth plane curves \((C, C')\) such that there exists a covering \(\pi: C \rightarrow C'\) of degree 2. The main result is as follows. Theorem. Let \(C\) and \(C'\) be two smooth plane curves of degree \(d\) and \(d'\), respectively. Then there exists no double covering from \(C\) to \(C'\), except for the following cases: (i) \(C'\) is rational \((d'\leq 2\)) and \(C\) is rational or elliptic (\(d \leq 3\)); (ii) \(C\) and \(C'\) are elliptic (\(d = d'= 3\)); (iii) \(C'\) is elliptic and \(C\) is a bielliptic plane quartic (\(d'= 3\), \(d = 4\)). In particular, no smooth plane curve can be a double covering of a smooth plane curve of degree greater than 3.
Mathematics Subject Classification ID: 14H51
Mathematics Subject Classification ID: 14H45
zbMATH DE Number: 5312060
zbMATH Keywords: double covering of curves
zbMATH Keywords: plane curves
zbMATH Keywords: elliptic curves
reviewed by: Cicero Carvalho
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1214442797
OpenAlex ID: W2055061160
DOI: 10.2996/KMJ/1214442797
description: scientific article; zbMATH DE number 5312060