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arXiv · 2410.04126

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

Kanat Abdukhalikov·Simeon Ball·Duy Ho·Tabriz Popatia
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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 $\mathbb{F}_{q^4}$.

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Kanat Abdukhalikov, Simeon Ball, Duy Ho, Tabriz Popatia. 2024-10-05. Ovoids in the cyclic presentation of PG(3,q). https://doi.org/10.1007/s10623-025-01695-9

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