On mixed plane curves of degree 1

TL;DR

This paper constructs a family of mixed polar homogeneous polynomials of degree 1 that define complex projective surfaces of any genus, introducing a new class of weighted homogeneous polynomials called polar weighted homogeneous polynomials of twisted join type.

Contribution

It introduces polar weighted homogeneous polynomials of twisted join type and demonstrates their use in constructing surfaces of arbitrary genus.

Findings

Existence of degree 1 mixed polar homogeneous polynomials for any genus g.

Introduction of a new polynomial class: polar weighted homogeneous polynomials of twisted join type.

Construction method for complex surfaces with prescribed genus.

Abstract

Let $f(\bfz,\bar\bfz)$ be a mixed strongly polar homogeneous polynomial of $3$ variables $\bfz=(z_1,z_2, z_3)$. It defines a Riemann surface $V:=\{[\bfz]\in \BP^{2}\,|\,f(\bfz,\bar\bfz)=0 \}$ in the complex projective space $\BP^{2}$. We will show that for an arbitrary given $g\ge 0$, there exists a mixed polar homogeneous polynomial with polar degree 1 which defines a projective surface of genus $g$. For the construction, we introduce a new type of weighted homogeneous polynomials which we call {\em polar weighted homogeneous polynomials of twisted join type}.