Most plane curves over finite fields are not blocking

TL;DR

This paper shows that most high-degree plane curves over finite fields are not blocking, with the proportion tending to zero as the degree and field size grow, and explores related probabilistic properties of these curves.

Contribution

It establishes asymptotic results on the rarity of blocking curves among high-degree plane curves over finite fields and analyzes their intersection point distributions.

Findings

Proportion of blocking curves tends to zero for large degree and field size.

The number of intersection points with a fixed line follows a Poisson distribution.

Blocking and smoothness events are asymptotically independent.

Abstract

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 $d\geq 2q-1$ and $q \to \infty$. We also show that the same conclusion holds for smooth curves under the somewhat weaker condition $d\geq 3p$ 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 $\mathbb{F}_q$-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 $k=1, 2, \ldots, q^2+q+1$.