Smooth mixed projective curves and a conjecture

Abstract

Let $f(\mathbf{z},\bar{\mathbf{z}})$ be a strongly mixed homogeneous polynomial of 3 variables $\mathbf{z}=({\mathbf{z}}_1,{\mathbf{z}}_2,{\mathbf{z}}_3)$ of polar degree $q$ with an isolated singularity at the origin. It defines a smooth Riemann surface $C$ in the complex projective space ${\mathbb{P}}^2$. The fundamental group of the complement ${\mathbb{P}}^2\setminus C$ is cyclic group of order $q$ if $f$ is homogeneous polynomial without $\bar{\mathbf{z}}$. We propose a conjecture that this may be even true for mixed homogeneous polynomials by giving several supporting examples.

arXiv comment15 figures

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