标题： 低次数点上的光滑平面曲线 显示英文标题

标题： Points of Low Degree on Smooth Plane Curves

摘要： 本文的目的是将法尔廷斯最近证明的S.朗猜想应用于光滑平面曲线。 设$C$是由次数为$d$的方程定义的具有整数系数的光滑平面曲线。 我们证明对于$d\ge 7$，曲线$C$只有有限多个定义域在$Q$上次数为$\le d-2$的点，并且对于$d\ge 8$，所有但有限多个定义域在$Q$上次数为$\le d-1$的$C$的点都是通过$C$的有理点的有理直线的交点。 显示英文摘要

摘要： The purpose of this note is to provide some applications of Faltings' recent proof of S. Lang's conjecture to smooth plane curves. Let $C$ be a smooth plane curve defined by an equation of degree $d$ with integral coefficients. We show that for $d\ge 7$, the curve $C$ has only finitely many points whose field of definition has degree $\le d-2$ over $Q$, and that for $d\ge 8$, all but finitely many points of $C$ whose field of definition has degree $\le d-1$ over $Q$ arise as points of intersection of rational lines through rational points of $C$.