Isolated and parameterized points on curves

TL;DR

This paper explores isolated and parameterized points on algebraic curves, explaining their geometric significance, and demonstrates that only finitely many isolated points exist on any curve, with examples illustrating various behaviors.

Contribution

It introduces the concepts of $ ext{P}^1$- and AV-parameterized points within the arithmetic of curves and connects these to Faltings' finiteness result, providing new insights into the structure of points on curves.

Findings

Faltings' theorem implies finitely many isolated points on any curve.

Parameterizations of points reveal geometric reasons for low-degree point behaviors.

Examples illustrate diverse behaviors of degree d points on curves.

Abstract

We give a self-contained introduction to isolated points on curves and their counterpoint, parameterized points, that situates these concepts within the study of the arithmetic of curves. In particular, we show how natural geometric constructions of infinitely many degree d points on curves motivate the definitions of $\mathbb{P}^1$- and AV-parameterized points and explain how a result of Faltings implies that there are only finitely many isolated points on any curve. We use parameterized points to deduce properties of the density degree set and show that parameterized points of very low degree arise for a unique geometric reason. The paper includes several examples that illustrate the possible behaviors of degree d points.