直线与平面四次曲线交点的合理性
On rationality of the intersection points of a line with a plane quartic
摘要：
本文研究了定义在F_q上的光滑平面四次线C与某些直线交点的合理性。对于q \geq 127，我们证明了存在一条直线使得与C的交点都是有理的。利用另一种方法，进一步证明了当F_q的特征不等于2且q \geq 66 ^2 +1时，存在具有相同性质的切线.最后，我们研究了C上存在有理伸缩的概率，并在特征数F_q = 3时表现出一种奇怪的行为。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
