On Hermitian varieties in $\mathrm{PG}(6,q^2)$

TL;DR

This paper characterizes the non-singular Hermitian variety in PG(6, q^2) by analyzing its point count and intersection properties with solids among certain hypersurfaces.

Contribution

It provides a characterization of the Hermitian variety in PG(6, q^2) based on point counts and intersection properties, distinguishing it from other hypersurfaces.

Findings

Hermitian variety characterized by point count and intersection with solids

Unique properties among degree q+1 hypersurfaces in PG(6, q^2)

Conditions exclude singular or other hypersurfaces

Abstract

In this paper we characterize the non-singular Hermitian variety ${\mathcal H}(6,q^2)$ of $\mathrm{PG}(6, q^2)$, $q\neq2$ among the irreducible hypersurfaces of degree $q+1$ in $\mathrm{PG}(6, q^2)$ not containing solids by the number of its points and the existence of a solid $S$ meeting it in $q^4+q^2+1$ points.