Flex curves and their applications
type: Item
instance of: scholarly article
title: Flex curves and their applications
zbMATH Open document ID: 0952.14020
DOI: 10.1023/A:1005004123844
author: M. Oka
published in: Geometriae Dedicata
publication date: 2001-01-14T00:00:00Z
review text: This article is concerned with the topology of the complement of an algebraic curve in the complex projective plane. In previous work, the author produced an example of two irreducible complex plane projective curves \(C_1\) and \(C_2\), both of degree 12 and having 27 ordinary cusps, such that the fundamental groups \(\pi_1 (\mathbb{P}^2- C_1)\) and \(\pi_1 (\mathbb{P}^2- C_2)\) are (nonabelian and) not isomorphic. In the present article, he produces a further example of an irreducible plane curve \(C_3\), such that \(\pi_1 (\mathbb{P}^2- C_3)\) is abelian. Moreover, the Alexander polynomial \(\Delta_{C_3} (t)\) of \(C_3\) is not equal to that of \(C_1\). A consequence is the fact that the space parametrizing plane projective curves of degree 12 having 27 cusps has at least three connected components, something that was unknown before. The author's technique consists in using certain cyclic transformations \(\varphi: \mathbb{C}^2\to \mathbb{C}^2\). A curve \(C\) in the target plane, when pulled back to the domain via \(\varphi\), will acquire more singularities. So, starting from a suitable relatively simple curve, it is possible to construct new ones, with more interesting features. Care must be taken to see what happens ``at infinity'', i.e. when one considers the extension of \(\varphi\) to \(\mathbb{P}^2\), as well as to control the fundamental groups under consideration and the pertinent Alexander polynomials. In previous work the author considered transformations as above ramified along a line. In the present paper he introduces, given a plane curve \(C\) on a nonsingular point \(P\in C\), such that the tangent line to \(C\) at \(P\) is not vertical, the notion of flex curve of degree \(l\) at \(P\). For a fixed positive integer \(l\), this is the (unique) curve \({\mathcal F}^{(l)} (P)\) of degree \(\leq l\) whose multiplicity of intersection with \(C\) at \(P\) is \(>l\). Then the author considers certain mappings \(\varphi\) as above, ramified along \({\ma…
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14F35
Mathematics Subject Classification ID: 14F45
Mathematics Subject Classification ID: 32S50
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 14H20
zbMATH DE Number: 1295084
zbMATH Keywords: flex curve
zbMATH Keywords: flex covering
zbMATH Keywords: topology of the complement of an algebraic curve
zbMATH Keywords: fundamental groups
zbMATH Keywords: Alexander polynomial
zbMATH Keywords: parametrizing plane projective curves
reviewed by: Augusto Nobile
MaRDI profile type: Publication
full work available at URL: a:1005004123844
OpenAlex ID: W183878102
description: scientific article; zbMATH DE number 1295084