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On mixed plane curves of degree 1 (1005.1449v1)
Published 10 May 2010 in math.AG
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}.