On mixed projective curves

TL;DR

This paper investigates the topological properties of projective real algebraic varieties defined by mixed polar homogeneous polynomials, highlighting differences from classical hypersurfaces and showing topology isn't solely determined by degree.

Contribution

It introduces a foundational analysis of mixed projective curves, revealing their topological complexity beyond degree-based classification.

Findings

Topology varies independently of degree in mixed projective varieties

The behavior differs significantly from classical projective hypersurfaces

Basic properties of such varieties are characterized

Abstract

Let $f(\bfz,\bar\bfz)$ be a mixed polar homogeneous polynomial of $n$ variables $\bfz=(z_1,..., z_n)$. It defines a projective real algebraic variety $V:=\{[\bfz]\in \BC\BP^{n-1} | f(\bfz,\bar\bfz)=0 \}$ in the projective space $\BC\BP^{n-1}$. 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.