Smooth mixed projective curves and a conjecture

Abstract

Let $f(\bf z,\bar{\bf z})$ be a strongly mixed homogeneous polynomial of 3 variables $\bf z=(z_1,z_2,z_3)$ of polar degree $q$ with an isolated singularity at the origin. It defines a smooth Riemann surface $C$ in the complex projective space $\mathbb P^2$. The fundamental group of the complement $\mathbb P^2\setminus C$ is cyclic group of order $q$ if $f$ is homogeneous polynomial without $\bar{\bf z}$. We propose a conjecture that this may be even true for mixed homogeneous polynomials by giving several supporting examples.