Isolated and parameterized points on curves

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.

💡 Research Summary

The paper provides a self‑contained introduction to the concepts of isolated points and parameterized points on algebraic curves, positioning these notions within the broader study of arithmetic geometry. After recalling basic definitions—closed points, degree, degree set D(C/F), index ind(C/F), density degree set δ(C/F), and potential density degree set ℘(C/F)—the authors illustrate the need for a finer classification beyond genus by examining hyperelliptic curves of genus g ≥ 2. Such curves have infinitely many degree‑2 points, all of which arise from a rational map to ℙ¹; this motivates the definition of ℙ¹‑parameterized points (a closed point x of degree d is ℙ¹‑parameterized if there exists a morphism π:C→ℙ¹ of degree d with π(x)∈ℙ¹(k)). Points that are not of this form are called ℙ¹‑isolated.

The authors then generalize to AV‑parameterized points, where the target is an abelian variety A (of any positive dimension). A point is AV‑parameterized if there is a morphism φ:C→A of degree d with φ(x)∈A(k). This captures families of points coming from higher‑dimensional parameter spaces.

Key technical tools are the Hilbert scheme HilbᵈX and the symmetric product SymᵈX. A degree‑d closed point on X corresponds to a rational point on HilbᵈX (and, via the Hilbert–Chow morphism, to a rational point on SymᵈX). When a morphism π:X→Y of degree d exists, the fibers π⁻¹(y) provide length‑d subschemes, giving a natural embedding Y↪HilbᵈX. The density of rational points on Y translates into Zariski‑density of degree‑d points on X, provided the fibers are integral.

The paper leverages the Mordell–Lang conjecture (proved by Faltings and Vojta) to show that an infinite set of degree‑d points must arise from a parameterization by either ℙ¹ or an abelian variety. Conversely, Faltings’ finiteness theorem for rational points on curves of genus ≥ 2, together with Riemann–Roch, implies that isolated points are always finite in number (Theorem 4.4.1).

A substantial part of the work studies the relationship between parameterized points and the density degree set. The authors prove that the minimal element of δ(C/k) coincides with the gonality γ(C) or a multiple thereof, and that the existence of a ℙ¹‑parameterized point of degree d forces d to belong to δ(C/k). They also show that low‑degree parameterized points (degree much smaller than the genus) can only occur when there is a unique low‑degree morphism from C to another curve; this is formalized in Theorem 6.0.1.

Section 4 constructs explicit infinite families of degree‑d points, motivating the definitions of ℙ¹‑ and AV‑parameterized points, and introduces the complementary notions of ℙ¹‑isolated and AV‑isolated points, linking them to the Ueno locus of a subvariety of an abelian variety. Section 5 derives several new properties of density degrees (Propositions 5.2.1 and 5.5.1) and connects them to classical invariants such as gonality. Section 6 focuses on very low degree parameterized points, proving that they must arise from a single low‑degree map, and discusses examples illustrating the full range of possible behaviors.

The paper concludes with a list of open problems, including uniform bounds for the number of non‑Ueno isolated points (as suggested by recent work of Gao, Ge, and Kühne), finer classifications of AV‑parameterized points, and the structure of potential density degree sets under field extensions.

Overall, the work offers a coherent framework that unifies geometric constructions of infinite families of points with deep arithmetic results, providing new insight into how the geometry of a curve controls the distribution of its algebraic points of arbitrary degree.