A modular equality for $m$-ovoids of elliptic quadrics

TL;DR

This paper establishes a modular equality condition for the existence of m-ovoids in elliptic quadrics, significantly narrowing the possible values of m and providing new characterizations for specific cases.

Contribution

It introduces a modular equality criterion for m-ovoids in elliptic quadrics, improving bounds on m and characterizing certain small cases.

Findings

Modular equality restricts m-ovoids in elliptic quadrics.

New lower bound on m for existence of m-ovoids.

Characterization of m-ovoids in specific small cases.

Abstract

An $m$-ovoid of a finite polar space $\mathcal{P}$ is a set $\mathcal{O}$ of points such that every maximal subspace of $\mathcal{P}$ contains exactly $m$ points of $\mathcal{O}$. In the case when $\mathcal{P}$ is an elliptic quadric $\mathcal{Q}^-(2r+1, q)$ of rank $r$ in $\mathbb{F}_q^{2r+2}$, we prove that an $m$-ovoid exists only if $m$ satisfies a certain modular equality, which depends on $q$ and $r$. This condition rules out many of the possible values of $m$. Previously, only a lower bound on $m$ was known, which we slightly improve as a byproduct of our method. We also obtain a characterization of the $m$-ovoids of $\mathcal{Q}^{-}(7,q)$ for $q = 2$ and $(m, q) = (4, 3)$.