Rational curve with Galois point and extendable Galois automorphism
type: Item
instance of: scholarly article
title: Rational curve with Galois point and extendable Galois automorphism
zbMATH Open document ID: 1167.14017
author: Hisao Yoshihara
published in: Journal of Algebra
publication date: 2009-04-29T00:00:00Z
review text: Let \(C\) be an irreducible plane projective curve of degree \(d \geq 3\) over an algebraically closed field \(k\) of characteristic zero, and \(k(C)\) its function field. Let \(P\) be a point of projective plane \(\mathbb{P}^2\) and \(\pi_P : \mathbb{P}^2 \dashrightarrow \mathbb{P}^1\) be the projection with center \(P\). The restriction \(\pi\) to \(C\) gives a dominant rational map \(\tilde{\pi} : C \dashrightarrow \mathbb{P}^1\) which induces a finite field extension \(k(C)/k(\mathbb{P}^1)\). \(P\) is called a Galois point for \(C\) if \(k(C)/k(\mathbb{P}^1)\) is a Galois extension. Let \(G\) be the corresponding Galois group. The author studies when a birational transformation of \(C\) induced by an element of \(G\) can be extended to a projective or birational transformation of the plane. Previously [J. Algebra 239, No. 1, 340--355 (2001; Zbl 1064.14023)], the author showed that if \(C\) is smooth and \(d\geq 4\), then \(G\) is cyclic and an element of \(G\) can be extended to a projective transformation of \(\mathbb{P}^2\). It is not so when \(C\) has a singular point. It is shown in the paper under review that in general case an element of \(G\) can not be extended to a birational transformation of \(\mathbb{P}^2\). This is done by presenting several examples. In addition, the defining equation of a rational curve with a Galois point are presented, and a special birational transformation is introduced in order to simplify the defining equation.
reviewed by: Vasyl I. Andriychuk
Mathematics Subject Classification ID: 14H37
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 14E05
zbMATH DE Number: 5549370
zbMATH Keywords: plane curve
zbMATH Keywords: Galois point
zbMATH Keywords: Galois group
zbMATH Keywords: birational transformation
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.11.035
OpenAlex ID: W2076644342
cites work: Galois Lines for Normal Elliptic Space Curves
cites work: Q3952229
cites work: Birational geometry of plane curves
cites work: Q4263461
cites work: On dihedral Galois coverings arising from Lissajous's curves
cites work: Field theory for function fields of plane quartic curves
cites work: Function field theory of plane curves by dual curves.
DOI: 10.1016/J.JALGEBRA.2008.11.035
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description: scientific article; zbMATH DE number 5549370