Probabilities of incidence between lines and a plane curve over finite fields

TL;DR

This paper investigates the probabilities of a random line intersecting a finite field curve in a specific number of points, analyzing their limits over field extensions using Chebotarev density theorem, and extends to intersections with random curves via Veronese maps.

Contribution

It applies a variant of Chebotarev density theorem to compute intersection probabilities and introduces its use in incidence geometry over finite fields.

Findings

Limits of intersection probabilities exist under certain conditions.

Chebotarev density theorem can be used to compute these limits.

Probabilities for intersections with random curves are also derived.

Abstract

We study the probability for a random line to intersect a given plane curve, defined over a finite field, in a given number of points defined over the same field. In particular, we focus on the limits of these probabilities under successive finite field extensions. Supposing absolute irreducibility for the curve, we show how a variant of Chebotarev density theorem for function fields can be used to prove the existence of these limits, and to compute them under a mildly stronger condition, known as simple tangency. Partial results have already appeared in the literature, and we propose this work as an introduction to the use of Chebotarev theorem in the context of incidence geometry. Finally, Veronese maps allow us to compute similar probabilities of intersection between a given curve and random curves of given degree.