A new family of tight sets in $\mathcal{Q}^{+}(5,q)$

TL;DR

This paper introduces a new infinite family of tight sets in hyperbolic quadrics, which correspond to Cameron--Liebler line classes in projective space, providing counterexamples to a longstanding conjecture and confirming a conjecture about affine plane sets.

Contribution

It describes a novel infinite family of tight sets in hyperbolic quadrics and links them to Cameron--Liebler line classes, challenging previous conjectures and confirming a new one.

Findings

Existence of a new infinite family of tight sets in $ ext{Q}^+(5,q)$ for specific q.

Correspondence between these sets and Cameron--Liebler line classes with parameter $(q^2-1)/2$.

Counterexamples to Cameron-Liebler conjecture and proof of Rodgers' conjecture.

Abstract

In this paper, we describe a new infinite family of $\frac{q^{2}-1}{2}$-tight sets in the hyperbolic quadrics $\mathcal{Q}^{+}(5,q)$, for $q \equiv 5 \mbox{ or } 9 \bmod{12}$. Under the Klein correspondence, these correspond to Cameron--Liebler line classes of ${\rm PG}(3,q)$ having parameter $\frac{q^{2}-1}{2}$. This is the second known infinite family of nontrivial Cameron--Liebler line classes, the first family having been described by Bruen and Drudge with parameter $\frac{q^{2}+1}{2}$ in ${\rm PG}(3,q)$ for all odd $q$. The study of Cameron--Liebler line classes is closely related to the study of symmetric tactical decompositions of ${\rm PG}(3,q)$ (those having the same number of point classes as line classes). We show that our new examples occur as line classes in such a tactical decomposition when $q \equiv 9 \bmod 12$ (so $q = 3^{2e}$ for some positive integer $e$), providing an infinite family of counterexamples to a conjecture made by Cameron and Liebler in 1982; the nature of these decompositions allows us to also prove the existence of a set of type $\left(\frac{1}{2}(3^{2e}-3^{e}), \frac{1}{2}(3^{2e}+3^{e}) \right)$ in the affine plane ${\rm AG}(2,3^{2e})$ for all positive integers $e$. This proves a conjecture made by Rodgers in his PhD thesis.