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

TL;DR

The paper advances the understanding of ovoids of the hyperbolic quadric $Q^+(7,q)$ by restricting to low-degree parametrizations $(f_1,f_2,f_3)$ with $\deg f_i\le 3$ and translating the problem into the study of the algebraic hypersurface $\mathcal{S}_{f_1,f_2,f_3}$ in $\mathrm{PG}(6,q)$. It leverages Lang-Weil-type bounds to obtain nonexistence results and then analyzes the factorization patterns of $\mathcal{S}_{f_1,f_2,f_3}$ under Frobenius to classify degree-2 and degree-3 ovoids; in particular, the degree-2 classification shows that, for large $q$, the Kantor ovoid with $q=2^h$ is the only possibility. For degree-3, the hypersurface must split into four hyperplanes or two quadrics, and the authors derive explicit forms and parametric families for $f_i$ in several arithmetic regimes, including complete results in some cases and open problems in others (notably uniqueness questions and $p=3$). Overall, the work highlights how algebraic hypersurfaces and Frobenius symmetries constrain low-degree ovoids and connects Kantor-type ovoids to degree-3 realizations within a unifying framework of $\mathcal{S}_{f_1,f_2,f_3}$.

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.

Theorems & Definitions (9)

• Definition 1.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 4.1
• Theorem 5.1
• Theorem 5.2
• Theorem 5.3
• Theorem 5.5