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

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.