On mixed plane curves of degree 1

Let $f(\text{bfz},\overline{\text{bfz}})$ be a mixed strongly polar homogeneous polynomial of $3$ variables $\text{bfz}=(z_1,z_2,z_3)$. It defines a Riemann surface $V:=\{[\text{bfz}]\in {\text{BP}}^2\mathrm{\mid }f(\text{bfz},\overline{\text{bfz}})=0\}$ in the complex projective space ${\text{BP}}^2$. 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}.