Most plane curves over finite fields are not blocking

Abstract

A plane curve C⊂P2 of degree d is called blocking if every Fq-line in the plane meets C at some Fq-point. We prove that the proportion of blocking curves among those of degree d is o(1) when d≥2q−1 and q→∞. We also show that the same conclusion holds for smooth curves under the somewhat weaker condition d≥3p and d,q→∞. Moreover, the two events in which a random plane curve is smooth and respectively blocking are shown to be asymptotically independent. Extending a classical result on the number of Fq-roots of random polynomials, we find that the limiting distribution of the number of Fq-points in the intersection of a random plane curve and a fixed Fq-line is Poisson with mean 1. We also present an explicit formula for the proportion of blocking curves involving statistics on the number of Fq-points contained in a union of k lines for k=1,2,…,q2+q+1.