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arXiv · 2502.02219

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

Daniele Bartoli·Nicola Durante·Giovanni Giuseppe Grimaldi·Marco Timpanella
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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.

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Daniele Bartoli, Nicola Durante, Giovanni Giuseppe Grimaldi, Marco Timpanella. 2025-02-04. Ovoids of $Q^+(7,q)$ of low-degree. https://arxiv.org/abs/2502.02219

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