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arxiv: 2506.11624 · v2 · submitted 2025-06-13 · 🧮 math.NT · math.AG

A geometric determinant method and geometric dimension growth

Tijs Buggenhout , Yotam I. Hendel , Floris Vermeulen This is my paper

Pith reviewed 2026-05-19 09:43 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords dimension growthdeterminant methodrational points over function fieldsgeometric Manin conjectureBombieri-Pila theoremspaces of rational curvesprojective varieties
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The pith

For irreducible projective varieties over C(t) of degree at least 2, the space of points of degree less than b has dimension at most b times dim X.

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

The paper proves a geometric form of the dimension growth conjecture for irreducible projective varieties X defined over the function field C(t). It shows that the auxiliary variety X(b), which parametrizes the C(t)-rational points of bounded degree on X, satisfies dim X(b) ≤ b dim X for all b ≥ 1. When the degree d of X is at least 6, the number of irreducible components of X(b) that attain this maximal dimension is bounded by a polynomial in d that is independent of b. The result specializes to uniform bounds on the dimension of spaces of rational curves of degree b when X is defined over C. The proof rests on a new geometric adaptation of Heath-Brown's determinant method that works over function fields and controls both dimension and component counts.

Core claim

For an irreducible projective variety X defined over C(t) of degree d ≥ 2, dim X(b) ≤ b dim X for every b ≥ 1; moreover, when d ≥ 6 the number of irreducible components of X(b) of dimension b dim X is bounded by a polynomial in d independent of b.

What carries the argument

Geometric version of Heath-Brown's p-adic determinant method adapted to varieties over C(t), which produces both the dimension bound and the uniform component bound for d ≥ 6.

If this is right

  • Uniform upper bounds on the dimension of the space of degree-b rational curves on any projective variety defined over C.
  • An analogue of the Bombieri-Pila theorem that counts points of bounded degree on affine curves over C(t).
  • A corresponding bound for points on projective curves over C(t).
  • More uniform estimates than those coming from geometric Manin's conjecture, applicable in greater generality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same method may produce dimension-growth statements over other function fields of characteristic zero.
  • Component bounds independent of b could be useful for studying moduli spaces of maps from curves to X.
  • The determinant technique might adapt to give height bounds in arithmetic settings over number fields.

Load-bearing premise

The adapted determinant method applies to varieties over C(t) and produces the stated component bound once the degree reaches 6.

What would settle it

An explicit irreducible projective variety X of degree 6 over C(t) such that for arbitrarily large b the number of maximal-dimensional irreducible components of X(b) exceeds every fixed polynomial in the degree d.

read the original abstract

We study a geometric version of the dimension growth conjecture. While it is closely related in spirit to themes arising in geometric Manin's conjecture, it applies in greater generality and provides more uniform bounds. For an irreducible projective variety $X$ defined over $\mathbb{C}(t)$, the set $X(b)$ of $\mathbb{C}(t)$-rational points on $X$ of degree less than $b$ has a natural structure of an algebraic variety over $\mathbb{C}$. We study the dimension and irreducibility of $X(b)$ when $X$ has degree $d \ge 2$, and obtain a geometric analogue of the classical dimension growth conjecture, namely that $\dim X(b) \le b\dim X $ for every $b \ge 1$. In particular, when $X$ is defined over $\mathbb{C}$, this provides uniform bounds on the dimension of the space of degree $b$ rational curves on $X$. We also develop a geometric version of Heath-Brown's $p$-adic determinant method for varieties defined over $\mathbb{C}(t)$. This allows us to show that as soon as $d \ge 6$, the number of irreducible components of $X(b)$ of dimension $b\dim X$ is bounded by a polynomial in $d$ which is independent of $b$. As a further application, we obtain an analogue of the Bombieri--Pila theorem for affine curves, as well as a corresponding result for projective curves.

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 / 0 minor

Summary. The paper develops a geometric analogue of Heath-Brown's p-adic determinant method for varieties over the function field ℂ(t) and applies it to prove a geometric version of the dimension growth conjecture. For an irreducible projective variety X of degree d ≥ 2 over ℂ(t), it shows that the variety X(b) of ℂ(t)-points of degree less than b satisfies dim X(b) ≤ b ⋅ dim X for all b ≥ 1. When d ≥ 6, the number of irreducible components of X(b) attaining this maximal dimension is bounded by a polynomial in d that is independent of b. Further applications include uniform bounds on the space of degree-b rational curves on varieties defined over ℂ and geometric analogues of the Bombieri–Pila theorem for affine and projective curves.

Significance. If the uniformity of the determinant estimates holds as claimed, the results give b-independent component bounds that are stronger and more general than those typically obtained from geometric Manin's conjecture in this setting. The development of the geometric determinant method itself is a substantial technical contribution with potential for further applications in arithmetic geometry over function fields, and the paper ships explicit polynomial bounds together with the method.

major comments (1)
  1. [§4] §4 (application of the geometric determinant method): the proof that the component bound remains polynomial in d and independent of b requires explicit control on the orders of vanishing and the degrees of the local completions at places whose degree grows with b. The text should clarify whether the Wronskian-type determinant estimates in the geometric setting absorb this growth without introducing a b-dependent factor (cf. the uniformity statement in the abstract).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, their positive assessment of the geometric determinant method and its applications, and their recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [§4] §4 (application of the geometric determinant method): the proof that the component bound remains polynomial in d and independent of b requires explicit control on the orders of vanishing and the degrees of the local completions at places whose degree grows with b. The text should clarify whether the Wronskian-type determinant estimates in the geometric setting absorb this growth without introducing a b-dependent factor (cf. the uniformity statement in the abstract).

    Authors: We thank the referee for this observation. The Wronskian-type determinant estimates developed in Section 3 are formulated over the function field ℂ(t) and rely on a choice of basis of global sections whose orders of vanishing are controlled uniformly by the degree d and the dimension of X, independent of b. When applying these estimates in Section 4 to places whose degree grows with b, the local completions are normalized by the place degree in the valuation; this normalization cancels any potential growth, so that the resulting upper bound on the number of maximal-dimensional components remains a polynomial in d with no b-dependent factor. The uniformity is already implicit in the passage from the determinant inequality to the component count in the proof of Theorem 4.5, but we agree that an explicit sentence or short paragraph making this cancellation visible would improve readability. We will add such a clarification in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; geometric method provides independent bounds

full rationale

The paper develops a new geometric adaptation of Heath-Brown's determinant method for varieties over C(t) and applies it to obtain the stated dimension bound dim X(b) ≤ b dim X together with a d-polynomial bound on maximal-dimensional components when d ≥ 6. No equations or steps in the abstract reduce the claimed predictions to fitted parameters, self-definitions, or load-bearing self-citations; the derivation chain is presented as self-contained once the geometric determinant estimates are established. This is the normal case of an independent analytic tool yielding uniform bounds.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The results rest on standard facts from algebraic geometry (projective varieties, dimension theory, irreducibility) and the construction of a new geometric determinant method; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)
  • standard math Standard properties of dimension and irreducibility for projective varieties over algebraically closed fields of characteristic zero.
    Invoked implicitly when defining X(b) and stating dimension bounds.

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