On the fundamental group of the complement of certain affine hypersurfaces
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 \(q\).
scientific article; zbMATH DE number 1215551