Smooth mixed projective curves and a conjecture

TL;DR

This paper investigates the topology of smooth Riemann surfaces defined by strongly mixed homogeneous polynomials in three variables, proposing a conjecture that their complement's fundamental group is cyclic, extending known results from purely homogeneous polynomials.

Contribution

It introduces a conjecture that the fundamental group of the complement of such mixed projective curves is cyclic, supported by multiple examples, thus extending classical results.

Findings

The fundamental group is cyclic for certain mixed homogeneous polynomials.

Supporting examples suggest the conjecture may hold generally.

Extension of classical results from homogeneous to mixed polynomials.

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.