On tight sets of hyperbolic quadrics

TL;DR

This paper establishes a modular equation condition for the parameter of tight sets in hyperbolic quadrics, significantly restricting possible parameters and generalizing previous results for specific cases.

Contribution

It introduces a new modular condition on the parameter of tight sets in hyperbolic quadrics, extending prior work from specific cases to general odd-rank quadrics.

Findings

At least half of the potential parameters are ruled out by the modular condition.

The condition generalizes previous results for $ ext{Q}^+(5,q)$.

Provides a new criterion for the existence of tight sets in hyperbolic quadrics.

Abstract

We prove that the parameter $x$ of a tight set $\mathcal{T}$ of a hyperbolic quadric $\mathsf{Q}^+(2n+1,q)$ of an odd rank $n+1$ satisfies ${x\choose 2}+w(w-x)\equiv 0\mod q+1$, where $w$ is the number of points of $\mathcal{T}$ in any generator of $\mathsf{Q}^+(2n+1,q)$. As this modular equation should have an integer solution in $w$ if such a $\mathcal{T}$ exists, this condition rules out roughly at least one half of all possible parameters $x$. It generalizes a previous result by the author and K. Metsch shown for tight sets of a hyperbolic quadric $\mathsf{Q}^+(5,q)$ (also known as Cameron-Liebler line classes in $\mathrm{PG}(3,q)$).