Topological types of p-hyperelliptic real algebraic curves
type: Item
instance of: scholarly article
title: Topological types of p-hyperelliptic real algebraic curves
zbMATH Open document ID: 0592.14029
DOI: 10.1007/BF01161975
published in: Mathematische Zeitschrift
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173834
review text: Given a natural number p, a projective irreducible smooth algebraic curve V defined over \({\mathbb{R}}\) is called p-hyperelliptic if there exists a birational isomorphism \(\phi\) of V, of order 2, such that V/\(\phi\) has genus p. This work is concerned with the existence of such curves according to their genus g and the number k of connected components of V(\({\mathbb{R}})\). We prove that Harnack's condition \(1\leq k\leq g\) is sufficient if \(V\setminus V({\mathbb{R}})\) is connected. In case \(V\setminus V({\mathbb{R}})\) non-connected, the following conditions \(1\leq k\leq g+1\) \((g+k\equiv 1 (2)),\) and either \(k=g+1-2q\) for some q, \(0\leq q\leq p\), or \(\epsilon \leq k\leq 2p+2\) with \(\epsilon =1\) for even p, \(\epsilon =2\) for odd p, are necessary and sufficient for the existence of the curve.
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14F45
zbMATH DE Number: 3950716
zbMATH Keywords: topological types of p-hyperelliptic real algebraic curves
zbMATH Keywords: existence of p-hyperelliptic real algebraic curves
zbMATH Keywords: genus
author: Emilio Bujalance
author: José Javier Etayo
author: Jose Manuel Gamboa
MaRDI profile type: Publication
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OpenAlex ID: W2029710586
description: scientific article; zbMATH DE number 3950716