Characterization of non-singular hyperplanes of $H\left(s,q^2\right)$ in $\mathrm{P G}\left(s, q^2\right)$

TL;DR

This paper provides a combinatorial method to characterize hyperplanes associated with non-singular hermitian varieties in projective spaces, based on intersection properties with points and subspaces.

Contribution

It introduces a new combinatorial characterization of hermitian hyperplanes using intersection numbers, extending previous geometric approaches.

Findings

Established necessary and sufficient conditions for hyperplanes to be part of hermitian varieties.

Extended prior geometric characterizations with a purely combinatorial approach.

Provided a method applicable to higher-dimensional projective spaces.

Abstract

In this paper, we present a combinatorial characterization of the hyperplanes associated with non-singular hermitian varieties ${H}\left(s, q^2\right)$ in the projective space $\mathrm{PG}\left(s,q^2\right)$ where $s\geq3$ and $q>2$. By analyzing the intersection numbers of hyperplanes with points and co-dimension $2$ subspaces, we establish necessary and sufficient conditions for a hyperplane to be part of the hermitian variety. This approach extends previous characterizations of hermitian varieties based on intersection properties, providing a purely combinatorial method for identifying their hyperplanes.