On Mixed Plane Curves of Polar Degree 1

Abstract

Let $f(\mathbf{z, \bar{z}})$ be a mixed strongly polar homogeneous polynomial of 3 variables $z = (z1, z2, z3)$. It defines a Riemann surface $V := \{[\mathbf{z}] \in \mathbb{P}^2 | f(\mathbf{z, \bar{z}) =0\}$ in the complex projective space $\mathbb{P}^2$. We will show that for an arbitrary given $g \geq 0$, there exists a mixed polar homogenous 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 polar weighted homogenous polynomials of the twisted join type.