Most plane curves over finite fields are not blocking

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