On the fundamental group of the complement of certain affine hypersurfaces
type: Item
instance of: scholarly article
title: On the fundamental group of the complement of certain affine hypersurfaces
zbMATH Open document ID: 0937.14012
author: Keita Tono
published in: Manuscripta Mathematica
publication date: 2000-02-03T00:00:00Z
review text: Let \(f \in \mathbb C[x_{1},\dots, x_{m}]\) and \(g \in \mathbb C[x_{m+1}, \dots, x_{m+n}]\) be two weighted homogeneous polynomials that are neither constant nor power of some other polynomial; let \(p,q\) be two positive integers. In this short paper, the author shows that, under these hypotheses, the fundamental group of the complement of the affine hypersurface \(\{f^{p}+g^{q}=0\}\) in \(\mathbb C^{m+n}\) does not depend from \(f\) and \(g\), but only from \(p\) and \(q\). In particular it equals the well-known fundamental group of the complement of the affine plane curve \(\{x^{p}+y^{q}=0\}\). The author expresses explicitly the fundamental group of the (affine) zero locus of a weighted homogeneous polynomial \(h(x_{1}, \dots, x_{l})\) as semidirect product of the fundamental group of the affine hypersurface \(\{h=1\}\) and \(\mathbb Z\), using the \(\mathbb C^{*}\) action on the polynomial induced by the weights. Then he shows that in his hypotheses the locus \(\{f^{p}+g^{q}=1\}\) is homotopy equivalent to the join of \(p\) copies of \(\{f=1\}\) and q copies of \(\{g=1\}\) whose fundamental group depends only from \(p\) and \(q\), and finally he checks that also the automorphism defining the semidirect product depends only from \(p\) and \(q\).
Mathematics Subject Classification ID: 14F35
Mathematics Subject Classification ID: 32J25
Mathematics Subject Classification ID: 14H30
zbMATH DE Number: 1215551
zbMATH Keywords: weighted homogeneous polynomials
zbMATH Keywords: fundamental group
zbMATH Keywords: automorphism
reviewed by: Roberto Pignatelli
MaRDI profile type: Publication
full work available at URL: s002290050086
OpenAlex ID: W2355722272
DOI: 10.1007/S002290050086
description: scientific article; zbMATH DE number 1215551