Smooth mixed projective curves and a conjecture

Journal of Singularities
volume 18 (2018), 329-349

Received: 24 November 2017. Accepted: 9 May 2018.

DOI: 10.5427/jsing.2018.18q

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Abstract:

Let f(z,\bar z) be a strongly mixed homogeneous polynomial of three variables z=(z_1, z_2, z_3) of polar degree q with an isolated singularity at the origin. It defines a smooth Riemann surface C in the complex projective space P^2. The fundamental group of the complement π_1(P^2-C) is a cyclic group of order q if f is a homogeneous polynomial without \bar z. We propose a conjecture that this may be even true for mixed homogeneous polynomials by giving several supporting examples.

Keywords:

Mixed homogeneous, Milnor fiber

Mathematical Subject Classification:

(2000) 14J17, 14N99

Author(s) information:

Mutsuo Oka
Department of Mathematics
Tokyo University of Science
Kagurazaka 1-3
Shinjuku-ku, Tokyo 162-8601
email: oka@rs.kagu.tus.ac.jp