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On the fundamental group of the complement of linear torus curves of maximal contact. (English) Zbl 1202.14028

A plane projective curve \(C\) of degree \(d\) is called a curve of torus type \((p,q)\) if there is a defining polynomial \(F\) of \(C\) which can be written as \(F = F_{d/q}^q - F_{d/p}^p\), where \(p\), \(q\) are positive integers dividing \(d\) and \(F_{d/q}, F_{d/p}\) are homogeneous polynomials in \(X, Y, Z\) of degree \(d/q\) and \(d/p\) respectively. Furthermore \(C\) is torus type of maximal contact if \(\{F_{d/q}=0\}\) is smooth at the intersection locus \(\xi_0 =\{F_{d/q} = F_{d/p}=0\}\).
In the present paper, the author considers the special class of torus curves whose defining polynomial in the affine coordinates takes the form \(f(x,y)=f_p(x,y)^q - l(x,y)^{pq}\), where \(l(x,y)\) is a linear form. He uses the so-called van Kampen-Zariski method (see for details [M. Oka, Franco-Japanese singularities. Proceedings of the 2nd Franco-Japanese singularity conference, CIRM, Marseille-Luminy, France, September 9–13, 2002. Paris: Société Mathématique de France. Séminaires et Congrés 10, 209–232 (2005; Zbl 1093.14037)]) and computes the fundamental group of the complement of linear torus curves of maximal contact and shows that it is isomorphic to that of generic linear torus curves.
As an application, he gives two new Zariski triples.

MSC:

14H30 Coverings of curves, fundamental group
14H20 Singularities of curves, local rings
14H45 Special algebraic curves and curves of low genus

Citations:

Zbl 1093.14037
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