On a Conjecture of Cameron and Liebler

Abstract

Cameron–Liebler line classes arose from an attempt by Cameron and Liebler to classify those collineation groups ofPG(n,q) which have the same number of orbits on points as on lines. They satisfy several equivalent properties; among them, constant intersection with spreads. Cameron and Liebler conjectured that, apart from some ‘obvious’ examples, no sets of lines of this type exist inPG(3,q). This paper introduces a connection between Cameron–Liebler line classes inPG(3,q) and blocking sets inPG(2,q), and uses it to provide the strongest results to date concerning the non-existence of certain of these sets. In addition, a complete classification of Cameron–Liebler line classes inPG(3,3) is obtained, with the main result being that there is, essentially, a unique counterexample to Cameron and Liebler's conjecture in this space.