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}.
Categories: math.AG