On mixed projective curves

Abstract

Let $f(\mathbf{z},\bar{\mathbf{z}})$ be a strongly polar homogeneous polynomial of $n$ variables $\mathbf{z} =(z_1,\dots, z_n)$. This polynomial defines a projective real algebraic variety $V = \{[\mathbf{z}] \in \mathbf{CP}^{n-1}\,|\,f(\mathbf{z},\bar{\mathbf{z}})=0 \}$ in the projective space $\mathbf{CP}^{n-1}$. The behavior is different from that of the projective hypersurface. The topology is not uniquely determined by the degree of the variety even if $V$ is non-singular. We study a basic property of such a variety.