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arxiv: 2606.27116 · v1 · pith:CEPYDJODnew · submitted 2026-06-25 · 🧮 math.NT · math.AG

Adelic Line Bundles, Arithmetic Positivity and Diophantine Geometry

Xinyi Yuan This is my paper

Pith reviewed 2026-06-26 02:54 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords adelic line bundlesarithmetic positivityDiophantine geometryequidistribution theoremBogomolov conjecturequasi-projective varietiesarithmetic heights
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The pith

Adelic line bundles on quasi-projective varieties carry arithmetic positivity that yields an equidistribution theorem and the uniform Bogomolov conjecture.

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

The paper lays out the basic definitions and properties of adelic line bundles on quasi-projective varieties and introduces the notion of arithmetic positivity for these bundles. It then derives an equidistribution result from this positivity and applies the same framework to establish the uniform Bogomolov conjecture. A reader would follow the chain because the constructions translate height functions and positivity conditions into statements about the distribution of rational points, directly addressing longstanding questions in Diophantine geometry.

Core claim

The article establishes that arithmetic positivity of adelic line bundles on quasi-projective varieties implies both an equidistribution theorem for sequences of points with controlled heights and the uniform version of the Bogomolov conjecture, with the positivity condition serving as the bridge between the arithmetic data encoded in the bundles and the geometric conclusions about point distributions.

What carries the argument

Adelic line bundles on quasi-projective varieties equipped with arithmetic positivity, which encodes both finite and infinite place data to control heights and produce equidistribution and height lower bounds.

If this is right

  • Positivity on adelic line bundles forces equidistribution of small-height points with respect to a suitable measure on the variety.
  • The uniform Bogomolov conjecture holds for the varieties where the adelic positivity condition can be verified.
  • Height functions arising from these bundles give effective lower bounds that are uniform across families of varieties.
  • Applications extend to any Diophantine problem reducible to controlling arithmetic heights via line bundle data.

Where Pith is reading between the lines

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

  • The same positivity framework may apply to other conjectures in arithmetic geometry that rely on height inequalities.
  • Making the definitions explicit for quasi-projective rather than projective varieties widens the range of varieties where equidistribution can be tested directly.
  • Readers can now check positivity for concrete bundles without re-deriving the foundational comparison theorems.

Load-bearing premise

The presentation correctly summarizes the standard definitions and theorems on adelic line bundles and arithmetic positivity from earlier literature.

What would settle it

An explicit sequence of points on a quasi-projective variety whose heights satisfy the positivity condition yet fail to equidistribute or violate the uniform Bogomolov lower bound.

read the original abstract

This article is an expository account of the basics of adelic line bundles on quasi-projective varieties, arithmetic positivity of adelic line bundles, and applications of positivity to an equidistribution theorem and the uniform Bogomolov conjecture.

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

0 major / 0 minor

Summary. This article is an expository account of the basics of adelic line bundles on quasi-projective varieties, arithmetic positivity of adelic line bundles, and applications of positivity to an equidistribution theorem and the uniform Bogomolov conjecture.

Significance. If the exposition faithfully reproduces the cited results from the literature, the paper offers a consolidated overview that may help readers navigate the connections between adelic geometry, arithmetic positivity, equidistribution, and the uniform Bogomolov conjecture. Its value is primarily in organization and accessibility rather than novel theorems or derivations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We are pleased that the expository organization and accessibility of the material on adelic line bundles, arithmetic positivity, equidistribution, and the uniform Bogomolov conjecture are viewed as valuable contributions.

Circularity Check

0 steps flagged

Expository summary with no internal derivations or predictions

full rationale

The paper is explicitly an expository account of existing literature on adelic line bundles, arithmetic positivity, equidistribution, and the uniform Bogomolov conjecture, with no new claims, proofs, or derivations presented. No equations, predictions, or load-bearing steps appear that could reduce to self-definitions, fitted inputs, or self-citation chains. The content is self-contained as a faithful summary of external results, yielding no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No new free parameters, axioms, or invented entities are introduced because the work is purely expository and relies on standard background from algebraic and arithmetic geometry.

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Reference graph

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