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arXiv · 1005.1449

On mixed plane curves of degree 1

Mutsuo Oka
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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}.

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Mutsuo Oka. 2010-05-10. On mixed plane curves of degree 1. https://arxiv.org/abs/1005.1449

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