On the proportion of transverse-free plane curves

TL;DR

This paper investigates the likelihood that a random smooth plane curve over a finite field is tangent to all lines over that field, using advanced combinatorial and algebraic geometry techniques.

Contribution

It provides a partial answer to Favre's question by establishing that such curves are very likely to have a transverse line over the finite field.

Findings

High probability of existence of a transverse _q-line for random smooth plane curves

Application of Poonen's Bertini theorem and Schrijver's theorem in algebraic geometry context

Quantitative asymptotic results over finite fields

Abstract

We study the asymptotic proportion of smooth plane curves over a finite field $\mathbb{F}_q$ which are tangent to every line defined over $\mathbb{F}_q$. This partially answers a question raised by Charles Favre. Our techniques include applications of Poonen's Bertini theorem and Schrijver's theorem on perfect matchings in regular bipartite graphs. Our main theorem implies that a random smooth plane curve over $\mathbb{F}_q$ admits a transverse $\mathbb{F}_q$-line with very high probability.