26 research outputs found
On rationality of the intersection points of a line with a plane quartic
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.Comment: 17 pages. Theorem 2 now includes the characteristic 2 case;
Conjecture 1 from the previous version is proved wron