# Low degree points on curves

## TL;DR

This paper explores an arithmetic analogue of gonality for curves over number fields, establishing conditions under which the minimal degree of points with bounded field extension degree matches gonality, especially for ample curves on certain surfaces.

## Contribution

It introduces an arithmetic invariant related to gonality, develops new techniques using auxiliary surfaces, and proves the invariant equals gonality for ample curves on surfaces with trivial irregularity.

## Key findings

- The invariant can take any value constrained by gonality.
- For sufficiently ample curves on surfaces with trivial irregularity, the invariant equals gonality.
- Developed techniques using auxiliary surfaces to analyze points of bounded degree.

## 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.