On the proportion of transverse-free plane curves

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.

19 pages