Low degree points on curves

Geoffrey Smith; Isabel Vogt

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arXiv · 1906.02328

Low degree points on curves

Geoffrey Smith·Isabel Vogt

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Abstract

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.

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Geoffrey Smith, Isabel Vogt. 2019-06-05. Low degree points on curves. https://arxiv.org/abs/1906.02328

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