Jan De Beule: A new infinite family of tight sets of the hyperbolic quadric Q^+(5,q)
Abstract:
Cameron-Liebler line classes occur in the study of collineation groups of
PG(3,q) with equally many orbits on the points and the lines. A
Cameron-Liebler line class with parameter x is, shortly spoken, a set of
lines of PG(3,q) meeting every spread in exactly x lines. This is equivalent
with a so-called $x$-tight set of the Klein quadric. It has been conjectured
that no "non-trivial" Cameron-Liebler line classes exist. This conjecture
has been disproven (long time ago already). The first example of an infinite
family is due to Bruen and Drudge, who constructed geometrically a
Cameron-Liebler line class of parameter (q^2+1)/2, for odd q. Between the
time of their construction and now, more non-trivial (but sporadic?)
examples are found, and also many parameters have been excluded, especially
by Metsch and Gavrilyuk, in recent results. Morgan Rodgers found in his PhD
thesis many interesting examples of Cameron-Liebler line classes with
parameter (q^2-1)/2 for different odd values of q. Together with Morgan,
Jeroen Demeyer (Ghent University) and Klaus Metsch, we could turn some of
these examples into an infinite family. This result will be presented in the
talk.