On rationality of the intersection points of a line with a plane quartic

TL;DR

This paper investigates conditions under which lines intersect smooth plane quartics over finite fields in rational points, establishing existence results for such lines and analyzing the probability of rational flexes, with special attention to field characteristics.

Contribution

It proves the existence of lines intersecting quartics with all rational points over finite fields for large q and explores rational flex probabilities, including special cases for characteristic 3.

Findings

Existence of a line with all rational intersection points for q ≥ 127.

Existence of a tangent line with all rational intersection points for certain field characteristics.

Analysis of rational flex probability, revealing unique behavior in characteristic 3.

Abstract

We study the rationality of the intersection points of certain lines and smooth plane quartics C defined over F_q. For q \geq 127, we prove the existence of a line such that the intersection points with C are all rational. Using another approach, we further prove the existence of a tangent line with the same property as soon as the characteristic of F_q is different from 2 and q \geq 66^2+1. Finally, we study the probability of the existence of a rational flex on C and exhibit a curious behavior when the characteristic of F_q is equal to 3.