Asymptotic Probability of Incidence Relations Over Finite Fields

Abstract

Given four generic lines in FP3, we ask, "How many lines meet the four?" The answer depends on the field. When F = C, the answer is two. When F = R, the answer is either zero or two. If we work over a finite field there are only finitely many projective lines. We compute the probability four lines are met by two. The main result is that as q approaches infinity, this probability approaches 1/2. Asymptotically, the other half of the time zero lines will meet the four.