In this paper we investigate an arithmetic analogue of the gonality of a smooth projective curve $C$ over a number field $k$: the minimal $e$ such there are infinitely many points $P \in C(\bar{k})$ with $[k(P):k] \leq e$. Developing techniques that make use of an auxiliary smooth surface containing the curve, we show that this invariant can take any value subject to constraints imposed by the gonality. Building on work of Debarre--Klassen, we show that this invariant is equal to the gonality for all sufficiently ample curves on a surface $S$ with trivial irregularity.

中文翻译：

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曲线上的低度点

在本文中，我们研究了数域 $k$ 上平滑投影曲线 $C$ 的方性的算术模拟：最小的 $e$ 这样有无限多个点 $P \in C(\bar{k}) $ 与 $[k(P):k] \leq e$。开发利用包含曲线的辅助光滑表面的技术，我们表明该不变量可以采用任何受共性强加约束的值。在 Debarre--Klassen 的工作的基础上，我们证明了这个不变量等于表面 $S$ 上所有足够充足的曲线的共性，具有微不足道的不规则性。