On rationality of the intersection points of a line with a plane quartic
Number Theory
2010-07-07 v2 Algebraic Geometry
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.
Keywords
Cite
@article{arxiv.1006.0873,
title = {On rationality of the intersection points of a line with a plane quartic},
author = {Roger Oyono and Christophe Ritzenthaler},
journal= {arXiv preprint arXiv:1006.0873},
year = {2010}
}
Comments
17 pages. Theorem 2 now includes the characteristic 2 case; Conjecture 1 from the previous version is proved wrong