# Characterising hyperbolic hyperplanes of a non-singular quadric in $PG(4,q)$

## TL;DR

This paper characterizes a specific set of hyperplanes in projective 4-space over a finite field, showing they correspond to hyperplanes intersecting a non-singular quadric in a hyperbolic quadric, based on combinatorial properties.

## Contribution

It provides a geometric characterization of hyperplanes related to a non-singular quadric in PG(4,q) using combinatorial intersection properties.

## Key findings

- The set of hyperplanes meeting a non-singular quadric in a hyperbolic quadric is uniquely characterized by specific point and plane intersection counts.
- The characterization applies to even q and relies on combinatorial conditions of hyperplanes and planes in PG(4,q).
- This result links geometric configurations with combinatorial intersection properties in finite projective spaces.

## Abstract

Let $H$ be a non-empty set of hyperplanes in $PG(4,q)$, $q$ even, such that every point of $PG(4,q)$ lies in either $0$, $\frac12q^3$ or $\frac12(q^3+q^2)$ hyperplanes of $ H$, and every plane of $PG(4,q)$ lies in $0$ or at least $\frac12q$ hyperplanes of $H$. Then $H$ is the set of all hyperplanes which meet a given non-singular quadric $Q(4,q)$ in a hyperbolic quadric.