Ovoids in the cyclic presentation of PG(3,q)

TL;DR

This paper explores the structure of ovoids in the cyclic presentation of PG(3,q), identifying elliptic quadrics and Suzuki-Tits ovoids through algebraic descriptions, enhancing understanding of finite projective geometries.

Contribution

It introduces a new algebraic characterization of ovoids, including elliptic quadrics and Suzuki-Tits ovoids, within the cyclic presentation of PG(3,q).

Findings

The set of $(q^2+1)$-th roots of unity forms an elliptic quadric.

Suzuki-Tits ovoids are characterized as zeros of specific polynomials.

Provides a new algebraic framework for understanding ovoids in cyclic projective spaces.

Abstract

We consider the cyclic presentation of $PG(3,q)$ whose points are in the finite field $\mathbb{F}_{q^4}$ and describe the known ovoids therein. We revisit the set $\mathcal{O}$, consisting of $(q^2+1)$-th roots of unity in $\mathbb{F}_{q^4}$, and prove that it forms an elliptic quadric within the cyclic presentation of $PG(3,q)$. Additionally, following the work of Glauberman on Suzuki groups, we offer a new description of Suzuki-Tits ovoids in the cyclic presentation of $PG(3,q)$, characterizing them as the zeroes of a polynomial over $\mathbb{F}_{q^4}$.