Ovoids of $Q^+(7,q)$ of low-degree

TL;DR

This paper classifies low-degree ovoids of the hyperbolic quadric $Q^+(7,q)$ in projective space, focusing on those parametrized by polynomials of degree at most 3, using algebraic hypersurface analysis.

Contribution

It provides a classification of low-degree ovoids of $Q^+(7,q)$, a problem not fully addressed before, by analyzing associated algebraic hypersurfaces.

Findings

Classification of ovoids with polynomial degree ≤ 3

Identification of algebraic hypersurfaces linked to ovoids

New constraints on ovoid structures in $Q^+(7,q)$

Abstract

Ovoids of the hyperbolic quadric $Q^+(7,q)$ of $\mathrm{PG}(7,q)$ have been extensively studied over the past 40 years, partly due to their connections with other combinatorial objects. It is well known that the points of an ovoid of $Q^+(7,q)$ can be parametrized by three polynomials $f_1(X,Y,Z)$, $f_2(X,Y,Z)$, $f_3(X,Y,Z)$. In this paper, we classify ovoids of $Q^+(7,q)$ of low degree, specifically under the assumption that $f_1(X,Y,Z)$, $f_2(X,Y,Z)$, $f_3(X,Y,Z)$ have degree at most 3. Our approach relies on the analysis of an algebraic hypersurface associated with the ovoid.