On $m$-ovoids of $Q^+(7,q)$ with $q$ odd

TL;DR

This paper constructs new $(q+1)$-ovoids of the hyperbolic quadric $Q^+(7,q)$ for odd prime powers by combining smaller ovoids of elliptic quadrics, and also finds specific $m$-ovoids in $Q^+(7,3)$, expanding understanding of these geometric structures.

Contribution

It introduces a novel method to construct $(q+1)$-ovoids of $Q^+(7,q)$ using glueing techniques from elliptic quadrics, and explicitly constructs certain $m$-ovoids in $Q^+(7,3)."

Findings

Constructed $(q+1)$-ovoids of $Q^+(7,q)$ not derived from 1-systems.

Developed a method to build $m$-ovoids for specific $m$ in $Q^+(7,3)."

Analyzed intersection properties of elliptic quadrics to facilitate ovoid construction.

Abstract

In this paper, we provide a construction of $(q+1)$-ovoids of the hyperbolic quadric $Q^+(7,q)$, $q$ an odd prime power, by glueing $(q+1)/2$-ovoids of the elliptic quadric $Q^-(5,q)$. This is possible by controlling some intersection properties of (putative) $m$-ovoids of elliptic quadrics. It yields eventually $(q+1)$-ovoids of $Q^+(7,q)$ not coming from a $1$-system. Secondly, we also construct $m$-ovoids for $m \in \{ 2,4,6,8,10\}$ in $Q^+(7,3)$. Therefore we first investigate how to construct spreads of $\pg(3,q)$ that have as many secants to an elliptic quadric as possible.