Almost All Transverse-Free Plane Curves Are Trivially Transverse-Free
Alejandro Lopez, Bella Villarreal, Ren Watson, Jaedon Whyte
TL;DR
This paper analyzes the density of transverse-free plane curves over a finite field by building a density framework based on first-order local data and blocking-set combinatorics. It introduces simple sets encoding local transversality, proves their densities are computable and rational via Weil zeta functions, and leverages conditional independence to decompose global properties. The main result shows that the density of transverse-free curves is extremely small, with an explicit asymptotic mu(Ω) and refined terms for special q, and reveals that almost all such curves arise from trivially blocking configurations, i.e., singularities at all F_q-points on a line. The authors also develop bounds on minimal blocking sets and relate the asymptotics to blocking-set enumeration, concluding with a discussion of potential extensions to higher dimensions and the limits of current techniques.
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.