Yoshihara, Hisao Rational curve with Galois point and extendable Galois automorphism. (English) Zbl 1167.14017 J. Algebra 321, No. 5, 1463-1472 (2009). 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. Reviewer: Vasyl I. Andriychuk (Lviv) Cited in 1 ReviewCited in 11 Documents MSC: 14H37 Automorphisms of curves 14H05 Algebraic functions and function fields in algebraic geometry 12F10 Separable extensions, Galois theory 14E05 Rational and birational maps Keywords:plane curve; Galois point; Galois group; birational transformation Citations:Zbl 1064.14023 PDFBibTeX XMLCite \textit{H. Yoshihara}, J. Algebra 321, No. 5, 1463--1472 (2009; Zbl 1167.14017) Full Text: DOI References: [1] Duyaguit, C.; Yoshihara, H., Galois lines for normal elliptic space curves, Algebra Colloq., 12, 205-212 (2005) · Zbl 1076.14036 [2] Iitaka, S., Algebraic Geometry, An Introduction to Birational Geometry of Algebraic Varieties, Grad. Texts in Math., vol. 76 (1982), Springer-Verlag: Springer-Verlag New York, Heidelberg, Berlin · Zbl 0491.14006 [3] Iitaka, S., Birational geometry of plane curves, Tokyo J. Math., 22, 289-321 (1999) · Zbl 0982.14006 [4] Malle, G.; Matzat, B. H., Inverse Galois Theory, Springer Monogr. Math. (1999), Springer-Verlag: Springer-Verlag New York, Heidelberg, Berlin · Zbl 0940.12001 [5] K. Miura, On dihedral Galois coverings arising from Lissajous’ curves, preprint; K. Miura, On dihedral Galois coverings arising from Lissajous’ curves, preprint · Zbl 1163.14308 [6] Miura, K.; Yoshihara, H., Field theory for function fields of plane quartic curves, J. Algebra, 226, 283-294 (2000) · Zbl 0983.11067 [7] Yoshihara, H., Function field theory of plane curves by dual curves, J. Algebra, 239, 340-355 (2001) · Zbl 1064.14023 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.