Projective plane curves and the automorphism groups of their complements
type: Item
instance of: scholarly article
title: Projective plane curves and the automorphism groups of their complements
zbMATH Open document ID: 0535.14017
author: Hisao Yoshihara
published in: Journal of the Mathematical Society of Japan
publication date: 1985-01-01T00:00:00Z
review text: Let \(C\) be an irreducible algebraic curve of degree \(d\) on \({\mathbb{P}}^ 2={\mathbb{P}}^ 2({\mathbb{C}})\) and let \({\mathcal G}\) be the automorphism group of the open algebraic surface \({\mathbb{P}}^ 2-C\). In case \(d=1\), it is known that \({\mathcal G}\) is generated by linear transformations and de Jonquières transformations of \({\mathbb{P}}^ 2-C\). In case \(d=2\), the generators of a similar kind have been found by \textit{V. I. Danilov} and \textit{M. Kh. Gizatullin} [Math. USSR, Izv. 11, 51-98 (1977; translation from Izv. Akad. Nauk SSSR, Ser. Mat. 41, 54-103 (1977; Zbl 0357.14003)]. Here we study the structure of \({\mathcal G}\) and at the same time the property of C in the case when \(d\geq 3\). Let \({\mathcal L}\) be the linear part of \({\mathcal G}\), i.e., \(\{T\in Aut({\mathbb{P}}^ 2)| T(C)=C\}\) and let \(\Delta\) [resp. \(\Delta_ e]\) be the curve defined by \(XYZ=X^ 3+Y^ 3\) [resp. \(Y^ eZ^{d-e}=X^ d\), where \((e,d)=1\) and 1\(\leq e\leq d-2]\). Let N be the number of elements of \(f^{-1}(S)\), where \(f:\tilde C\to C\) is the normalization of C and S is the set of singular points of C. In case \(N=1\), we put \(R=d^ 2-(\sum^{p}_{i=1}e_ i\!^ 2)-e_ p+1,\) where \(e_ 1,...,e_ p\) are the multiplicities of all infinitely near singular points. Then the result is as follows: The order of \({\mathcal G}\) is finite if and only if C satisfies any one of the following three conditions. (1) \(C\) is not rational. (2) \(N\geq 2\) and \(C\) is projectively equivalent to neither \(\Delta\) nor \(\Delta_ e\). (3) \(N=1\) and \(R\leq - 1\). On the contrary, for the remaining curves with \(d\geq 3\), \({\mathcal G}\) has the following properties. (4) If \(C\) is projectively equivalent to \(\Delta_ e\), where \(e\geq 2\), then \({\mathcal G}={\mathcal L}\cong G_ m,\) which is the multiplicative group. (5) If \(C\) is projectively equivalent to \(\Delta\), then the order of \({\mathcal G}\) is countably infinite and \({\mathcal L}\) is the dihedral g…
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 3849402
zbMATH Keywords: open algebraic surface
zbMATH Keywords: degree
zbMATH Keywords: finite order
zbMATH Keywords: automorphism group of complement of plane algebraic curve
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03710087
OpenAlex ID: W2027914500
DOI: 10.2969/JMSJ/03710087
description: scientific article; zbMATH DE number 3849402