On tight sets of 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 $\left(\genfrac{}{}{0ex}{}{x}{2}\right)+w(w-x)\equiv 0\mathrm{mod}q+1$, 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 $\mathrm{PG}(3,q)$).