Almost All Transverse-Free Plane Curves Are Trivially Transverse-Free

TL;DR

This paper investigates the properties of transverse-free plane curves over finite fields, showing that almost all such curves are trivially transverse-free with singularities at every rational point of some line.

Contribution

It develops a combinatorial method to estimate the density of transverse-free curves and proves that nearly all contain singularities at all rational points of some line.

Findings

Nearly all transverse-free curves contain singularities at every $ ext{F}_q$-point of some line.

Develops a new combinatorial approach based on blocking sets for density estimation.

Provides upper bounds on the number of blocking sets of fixed size less than 2q.

Abstract

Call a curve $C \subset \mathbb{P}^2$ defined over $\mathbb{F}_q$ transverse-free if every line over $\mathbb{F}_q$ intersects $C$ at some closed point with multiplicity at least 2. In 2004, Poonen used a notion of density to treat Bertini Theorems over finite fields. In this paper we develop methods for density computation and apply them to estimate the density of the set of polynomials defining transverse-free curves. In order to do so, we use a combinatorial approach based on blocking sets of $\operatorname{PG}(2, q)$ and prove an upper bound on the number of such sets of fixed size $< 2q$. We thus obtain that nearly all transverse-free curves contain singularities at every $\mathbb{F}_q$-point of some line.