# Characterising elliptic solids of $Q(4,q)$, $q$ even

## TL;DR

This paper characterizes specific sets of solids in projective 4-space over even finite fields, showing they are either disjoint from a hyperoval or intersect a non-singular quadric in an elliptic quadric.

## Contribution

It provides a classification of elliptic solids in PG(4,q) for even q, identifying their structure relative to hyperovals and quadrics.

## Key findings

- Sets of solids are either disjoint from a hyperoval or meet a non-singular quadric in an elliptic quadric.
- Characterization applies to q even, q > 2.
- Provides a structural understanding of elliptic solids in projective geometry.

## Abstract

Let $E$ be a set of solids (hyperplanes) in $PG(4,q)$, $q$ even, $q>2$, such that every point of $PG(4,q)$ lies in either $0$, $\frac12q^3$ or $\frac12(q^3-q^2)$ solids of $E$, and every plane of $PG(4,q)$ lies in either $0$, $\frac12q$ or $q$ solids of $E$. This article shows that $E$ is either the set of solids that are disjoint from a hyperoval, or the set of solids that meet a non-singular quadric $Q(4,q)$ in an elliptic quadric.