Most plane curves over finite fields are not blocking

Abstract

A plane curve $C\subset P^2$ of degree d is called blocking if every $F_q$-line in the plane meets C at some $F_q$-point. We prove that the proportion of blocking curves among those of degree d is $o\left(1\right)$ when $d\ge 2q-1$ and $q\to \infty$. We also show that the same conclusion holds for smooth curves under the somewhat weaker condition $d\ge 3p$ and $d,q\to \infty$. 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 $F_q$-roots of random polynomials, we find that the limiting distribution of the number of $F_q$-points in the intersection of a random plane curve and a fixed $F_q$-line is Poisson with mean 1. We also present an explicit formula for the proportion of blocking curves involving statistics on the number of $F_q$-points contained in a union of k lines for $k=1,2,\dots ,q^2+q+1$.