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Contracting low degree points on curves

Maarten Derickx

TL;DR

This work addresses how low-degree points on curves over number fields distribute and whether they arise as pullbacks along nontrivial morphism factorizations. It proves that, under a genus bound $g(X) > \deg(f)g(Y) + (2d-1)(\deg f-1)$, all but finitely many degree-$d$ points on $X$ come from pulling back degree-reduced points along a nontrivial factorisation $X\to Z\to Y$, with $\deg f_1>1$. The results yield finiteness statements for points with $K(f(x))=K(x)$, a finite-sources framework, and generalizations of Kadets–Vogt and Kawhaja–Siksek, framed via a geometric pullback mechanism. Key tools include Faltings’ theorem on parameterised points and the Castelnuovo–Severi inequality, which together provide a unified explanation for how small-degree points relate to pullback structures. The corollaries extend to a Single Source perspective, shedding light on density-degree sets and enabling new paths for studying infinitude of primitive points across degrees.

Abstract

The main result of this article is that all but finitely many points of small enough degree on a curve can be written as a pullback of a smaller degree point. The main theorem has several corollaries that yield improvements on results of Kadets and Vogt, Khawaja and Siksek, and Vojta under a slightly stronger assumption on the degree of the points.

Contracting low degree points on curves

TL;DR

This work addresses how low-degree points on curves over number fields distribute and whether they arise as pullbacks along nontrivial morphism factorizations. It proves that, under a genus bound , all but finitely many degree- points on come from pulling back degree-reduced points along a nontrivial factorisation , with . The results yield finiteness statements for points with , a finite-sources framework, and generalizations of Kadets–Vogt and Kawhaja–Siksek, framed via a geometric pullback mechanism. Key tools include Faltings’ theorem on parameterised points and the Castelnuovo–Severi inequality, which together provide a unified explanation for how small-degree points relate to pullback structures. The corollaries extend to a Single Source perspective, shedding light on density-degree sets and enabling new paths for studying infinitude of primitive points across degrees.

Abstract

The main result of this article is that all but finitely many points of small enough degree on a curve can be written as a pullback of a smaller degree point. The main theorem has several corollaries that yield improvements on results of Kadets and Vogt, Khawaja and Siksek, and Vojta under a slightly stronger assumption on the degree of the points.

Paper Structure

This paper contains 9 sections, 13 theorems, 13 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Comparison with existing results.
  4. Comparing Corollary 1.2
  5. Comparing Corollary 1.3
  6. Comparing Corollary 1.4
  7. Background
  8. Proof of Theorem 1.1
  9. Proof of Corollary 1.4

Key Result

Theorem 1.1

Let $X$ and $Y$ be nicenice here means smooth, projective, and geometrically integral curves over a number field $K$, and $f: X \to Y$ be a dominant morphism. Let $d$ be an integer such that Then there is a finite set $T_d$ of closed degree $d$ points such that for all closed points $x \in X \setminus T_d$ of degree $d$ there exists a decomposition $X \stackrel{f_1}{\to} Z \stackrel{f_2}{\to} Y$

Figures (1)

  • Figure 4.1:

Theorems & Definitions (20)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • proof
  • Corollary 1.4: Single Source Theorem
  • Definition 1.5
  • Theorem 2.1: SongTucker
  • Theorem 2.2: KadetsVogt
  • Theorem 2.3: Single Source Theorem KhawajaSiksekSingleSource
  • Definition 3.1: $\mathbb{P}^1$-parameterised point
  • ...and 10 more