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

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 $\mathbb{F}_{q^4}$.

💡 Research Summary

This paper investigates classical geometric objects known as ovoids within the finite projective space PG(3, q), utilizing a specific algebraic model called the cyclic presentation. An ovoid in PG(3, q) is a set of q^2 + 1 points with the property that no line intersects it in more than two points. The two known families are elliptic quadrics (existing for all prime powers q) and Suzuki-Tits ovoids (existing only for q = 2^m with m ≥ 3 odd).

The cyclic presentation, explicitly described by Ball in