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arxiv: 2604.02975 · v1 · submitted 2026-04-03 · 🧮 math.NT · math.AG

Points of low degree on curves over function fields

Si\`ena van Schaick This is my paper

Pith reviewed 2026-05-13 18:37 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords algebraic curvespoints of low degreefunction fieldsspecializationFaltings theorem
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The pith

The classification of curves with infinitely many points of degree at most 5 carries over from number fields to function fields of characteristic zero.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that smooth projective curves over function fields of characteristic zero that admit infinitely many points of degree d at most 5 must be geometrically of the same types already classified over number fields. Prior results by Faltings, Harris-Silverman, Abramovich-Harris, and Kadets-Vogt give this classification for d from 1 to 5 over number fields; the new work shows the list remains complete when the base field is instead a function field. The proof proceeds by a specialization argument that maps the function-field curve to a number-field curve while preserving the infinitude of low-degree points. A reader cares because this shows the geometric constraints on low-degree points are insensitive to whether the constant field is finite or infinite, at least in characteristic zero.

Core claim

Smooth projective curves over function fields of characteristic zero that possess infinitely many points of degree at most 5 are geometrically classified by the same list of types that holds over number fields, obtained by reducing via specialization to the known number-field case.

What carries the argument

A specialization argument that reduces a curve over a function field to a curve over a number field while preserving the infinitude of points of degree d less than or equal to 5.

Load-bearing premise

The specialization from the function-field curve to a number-field curve keeps infinitely many distinct low-degree points and introduces no characteristic-dependent obstructions.

What would settle it

A smooth projective curve over a function field of characteristic zero that has infinitely many points of degree at most 5 but is not geometrically equivalent to any of the curve types already classified over number fields.

read the original abstract

We show that the geometric classification of smooth projective curves admitting infinitely many points of degree $d\leq 5$ extends from number fields to function fields of characteristic 0. Over number fields, this classification was established by Faltings for $d=1$, Harris--Silverman for $d=2$, Abramovich--Harris for $d=3,4$ and Kadets--Vogt for $d=4,5$. Our approach uses a specialization argument to reduce the problem over function fields to the number field case.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper claims that the geometric classification of smooth projective curves admitting infinitely many points of degree d≤5, previously established over number fields by Faltings (d=1), Harris-Silverman (d=2), Abramovich-Harris (d=3,4), and Kadets-Vogt (d=4,5), extends to function fields of characteristic 0. The proof proceeds by a specialization argument that reduces the function-field case directly to the known number-field theorems.

Significance. If the reduction is valid, the result unifies the classification of curves with infinitely many low-degree points across all characteristic-zero fields, showing that the geometric obstructions identified over number fields persist over function fields. This is a natural and useful extension of existing theorems, though its impact is primarily consolidative rather than introducing new geometric phenomena.

major comments (1)
  1. [Specialization argument] Specialization section: the argument that an infinite set S of points of degree ≤5 over the function field K specializes to an infinite set of points of degree ≤5 over the number field k requires an explicit density or Hilbert-irreducibility statement guaranteeing that the residue-field degrees remain ≤5 for infinitely many places of K. Without such control (e.g., on ramification in the residue extensions or smoothness of the model), the infinitude may drop after specialization, undermining the reduction to the Kadets-Vogt/Abramovich-Harris theorems.
minor comments (2)
  1. [Introduction] Clarify the precise statement of the base field K (function field of a curve over an algebraically closed field of char 0) and the model used for specialization.
  2. [Specialization argument] Add a reference to the precise form of Hilbert irreducibility or density theorem invoked to preserve degrees.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need for greater explicitness in the specialization argument. We address the comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: Specialization section: the argument that an infinite set S of points of degree ≤5 over the function field K specializes to an infinite set of points of degree ≤5 over the number field k requires an explicit density or Hilbert-irreducibility statement guaranteeing that the residue-field degrees remain ≤5 for infinitely many places of K. Without such control (e.g., on ramification in the residue extensions or smoothness of the model), the infinitude may drop after specialization, undermining the reduction to the Kadets-Vogt/Abramovich-Harris theorems.

    Authors: We agree that an explicit control on the specialization is necessary to preserve the degree bound. In the revised version we will insert a new lemma (placed immediately before the main specialization argument) that invokes a function-field version of Hilbert irreducibility (as stated, for instance, in the work of Fried–Jarden or in the arithmetic-geometric setting of Caporaso–Harris–Mazur). The lemma will show that, after passing to a finite extension of the base if needed and working with a smooth proper model, there exists a dense set of places of good reduction such that the residue-field extensions of the specialized points have degree at most 5. This guarantees that the image of the infinite set S remains infinite after specialization, allowing the direct appeal to the known number-field theorems. The revision adds only the missing density statement and does not change the geometric conclusions. revision: yes

Circularity Check

0 steps flagged

No circularity: specialization reduces to independent external theorems

full rationale

The paper's derivation consists of a single reduction step: a specialization argument that maps a smooth projective curve over a function field of characteristic zero, together with an infinite set of points of degree at most 5, to a curve over a number field while preserving the infinitude and degree bound. The target classification is then supplied entirely by prior results of Faltings, Harris-Silverman, Abramovich-Harris, and Kadets-Vogt, none of which are authored by the present writer or depend on the current manuscript. No equation or definition in the paper is shown to be equivalent to its own input by construction, no fitted parameter is relabeled as a prediction, and no uniqueness theorem or ansatz is imported via self-citation. The argument is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the validity of the cited number-field classifications and on the correctness of a specialization map in characteristic zero; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The geometric classification over number fields holds as stated by Faltings, Harris-Silverman, Abramovich-Harris, and Kadets-Vogt.
    The paper reduces the function-field statement to these prior theorems.
  • domain assumption Specialization preserves the infinitude of points of degree ≤5 in characteristic zero.
    Invoked to transfer the infinitude property from the specialized number-field curve back to the original function-field curve.

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