On the Non-Existence of Certain Cameron-Liebler Line Classesin PG(3, q)

Abstract

Our main result is a non-existence theorem for certain families of lines in three dimensional projective space PG(3, q) over a finite field GF(q). Specifically, a Cameron-Liebler line class in PG(3, q) is a set of lines which intersects every spread of PG(3, q) in the same number x of lines (this number is called its parameter). These sets arose in connection with an attempt by Cameron and Liebler to determine the subgroups of PGL(n+1, q) which have the same number of orbits on points (of PG(n, q)) as on lines; they satisfy several equivalent properties. Here we prove that for 2 < x ≤ √q, no Cameron-Liebler line class of parameter x exists in PG(3, q). A relevant general question on incidence matrices is described.