Generalizing quadratic $\mathbb{R}$-Algebraic sets in $\mathbb{CP}^{n}$

TL;DR

This paper characterizes special quadratic algebraic sets in complex projective space based on their intersection properties with complex lines, advancing understanding of complex geometric structures.

Contribution

It introduces a new characterization of quadratic $ $-algebraic sets in $p^n$ through their line intersection behavior, connecting algebraic and topological properties.

Findings

Characterization of complex ellipsoids via line intersections.

Topological analysis of subsets with specific line intersection properties.

Insights into quadratic $ $-algebraic subsets in $p^n$.

Abstract

Motivated by our study of the complex Banach conjecture, we characterize a complex ellipsoids $\mathcal E$ as compact subsets of $\mathbb C^n$, with the property that every complex line intersect $\mathcal E$ either in a single point or in the complex affine image of the unit disk. This characterization leads to the main interest of this paper. We study the topological behavior of compact subsets of $\mathbb{CP}^n$ with the property that any complex line that intersects them does either at a single point, at the boundary of a complex disk, or along the entire line. In particular, we are interested in quadratic $\R$-algebraic subsets of $\mathbb{CP}^n$.