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arxiv: 2606.29044 · v1 · pith:3UCRXUOAnew · submitted 2026-06-27 · 🧮 math.AG

Dimension filtrations in birational localisation

David Kumallagov This is my paper

Pith reviewed 2026-06-30 08:09 UTC · model grok-4.3

classification 🧮 math.AG
keywords birational localizationdimension filtrationsmooth varietiesfully faithful functorbirational morphismsalgebraic geometrycategorical localization
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The pith

The birational localization of smooth varieties of dimension at most n is fully faithful only for n=0.

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

The paper examines the categories L_n formed by inverting all birational morphisms inside the category of smooth varieties of dimension at most n over a field. It shows that the natural map from each such L_n into the unrestricted birational localization is fully faithful precisely when n equals zero. For every n at least 1 the transition maps from L_n into any higher L_N produce infinitely many distinct images already on certain endomorphism sets. The argument rests on a sharp dimension threshold that decides exactly when a product with affine space becomes an isomorphism after localization. A reader would care because the result separates the case of points from all positive-dimensional cases in a categorical model of birational geometry.

Core claim

Let S_b be the class of birational morphisms between smooth varieties over a field F, and let L_n = S_b^{-1} d_≤n Sm(F). The natural functor L_n to S_b^{-1} Sm(F) is fully faithful exactly for n=0. For every n≥1 and every N≥n+1, the transition functor L_n to L_N has an infinite fibre on an endomorphism set. If dim X + r ≤ n then the morphism X × A^r to X is invertible in L_n precisely when dim X + r ≤ n-1. Proper and projective analogues hold as well.

What carries the argument

The category L_n obtained by localizing the subcategory of smooth varieties of dimension at most n by the class of all birational morphisms, together with the transition functors induced by inclusions of these subcategories before localization.

If this is right

  • The functor L_n to the full birational localization is fully faithful if and only if n equals zero.
  • For n at least 1 every transition L_n to L_N has an infinite fibre already on some endomorphism set.
  • Under the bound dim X + r ≤ n the morphism X × A^r to X is invertible in L_n exactly when the total dimension is at most n-1.
  • The same dimension-threshold statements hold after replacing smooth varieties by proper or projective ones.

Where Pith is reading between the lines

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

  • The threshold suggests that birational localization does not commute with naive dimension cut-offs once positive dimensions appear.
  • The result limits the extent to which one can approximate the full birational category by working only with bounded-dimensional pieces.
  • Similar dimension-dependent failures of full faithfulness may appear when other classes of morphisms are inverted instead of birational ones.
  • One could test the threshold by computing endomorphism rings in L_n for explicit low-dimensional varieties such as curves or surfaces.

Load-bearing premise

The categories L_n are constructed by inverting birational morphisms only inside the subcategory of smooth varieties of dimension at most n, with transition functors coming from the inclusions of these subcategories prior to localization.

What would settle it

Exhibit a concrete smooth variety X and integer r with dim X + r equal to some n ≥ 1 such that X × A^r to X becomes invertible in L_n even though dim X + r exceeds n-1, or produce an endomorphism set whose fibre under L_n to L_N is finite for some n ≥ 1 and N > n.

read the original abstract

Let \(S_b\) be the class of birational morphisms between smooth varieties over a field \(F\), and let \(L_n=S_b^{-1}d_{\leq n}\Sm(F)\). Kahn and Sujatha asked whether the natural functor \(L_n\to S_b^{-1}\Sm(F)\) is fully faithful. We prove that it is fully faithful exactly for \(n=0\). More strongly, for every \(n\geq1\) and every \(N\geq n+1\), the transition functor \(L_n\to L_N\) has an infinite fibre on an endomorphism set. The proof identifies a sharp dimension threshold: if \(\dim X+r\leq n\), then \(X\times\mathbb A^r\to X\) is invertible in \(L_n\) precisely when \(\dim X+r\leq n-1\). We also give proper and projective analogues.

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 / 1 minor

Summary. The paper defines L_n = S_b^{-1} d_≤n Sm(F), the localization at birational morphisms S_b of the category of smooth F-varieties of dimension ≤n. It proves that the natural functor L_n → S_b^{-1} Sm(F) is fully faithful if and only if n=0. More strongly, for every n≥1 and N≥n+1 the transition L_n → L_N has infinite fibre on an endomorphism set. The key technical result is a sharp dimension threshold: if dim X + r ≤ n then X × A^r → X is invertible in L_n precisely when dim X + r ≤ n-1. Analogous statements are proved for the proper and projective variants.

Significance. If the central claims hold, the work resolves the question of Kahn and Sujatha on full faithfulness of these functors and introduces a dimension filtration on birational localizations. The precise threshold for invertibility of affine-space morphisms supplies a concrete structural result that distinguishes the filtered categories. The proper and projective analogues extend the scope. No machine-checked proofs or parameter-free derivations are present, but the direct categorical construction is the main strength.

major comments (1)
  1. [proof of the dimension threshold (likely §3 or §4)] The non-invertibility statement when dim X + r = n (which underpins both the threshold and the infinite-fibre claim for n≥1) is established by exhibiting a detecting functor Φ : d_≤n Sm(F) → T that inverts every morphism in S_b yet fails to invert the critical X × A^r → X. The manuscript must contain an explicit verification that Φ sends the entire class S_b to isomorphisms (rather than a proper subclass); any gap in this verification would invalidate the threshold and therefore the non-full-faithfulness result for n≥1.
minor comments (1)
  1. [Introduction] The abstract states the threshold cleanly, but the introduction would benefit from a one-paragraph roadmap indicating where the detecting functor is constructed and where its verification that it inverts all of S_b appears.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for greater explicitness in the verification of the detecting functor. We address the single major comment below.

read point-by-point responses

  1. Referee: [proof of the dimension threshold (likely §3 or §4)] The non-invertibility statement when dim X + r = n (which underpins both the threshold and the infinite-fibre claim for n≥1) is established by exhibiting a detecting functor Φ : d_≤n Sm(F) → T that inverts every morphism in S_b yet fails to invert the critical X × A^r → X. The manuscript must contain an explicit verification that Φ sends the entire class S_b to isomorphisms (rather than a proper subclass); any gap in this verification would invalidate the threshold and therefore the non-full-faithfulness result for n≥1.

    Authors: We agree that the verification must be fully explicit. In the manuscript the functor Φ is constructed in §3 as the composition of the function-field functor with the localization at birational maps; the argument that every morphism in S_b becomes an isomorphism under Φ is given by noting that birational morphisms induce isomorphisms of function fields and that Φ is defined on all of d_≤n Sm(F). Nevertheless, to remove any possible ambiguity we will insert a dedicated lemma (new Lemma 3.x) that states and proves Φ(S_b) consists entirely of isomorphisms, with a direct reference back to the definition of Φ. This is a presentational clarification only and does not alter any statements or proofs. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is self-contained from explicit definitions

full rationale

The paper defines L_n explicitly as the localization S_b^{-1} d_≤n Sm(F) and states a direct proof of the dimension threshold for invertibility of X×A^r → X after localization, together with the resulting non-full-faithfulness for n≥1. No step reduces a claimed result to a fitted parameter, a self-citation chain, or a self-definitional equivalence; the threshold statement is presented as a consequence of the localization construction rather than an input. The derivation therefore stands on the category-theoretic setup without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper works entirely within the existing framework of category localization applied to smooth varieties and birational morphisms; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • domain assumption The category of smooth varieties over a field F admits a well-defined localization with respect to the class of birational morphisms.
    This is the foundational setup used to define L_n and the functors under study.

pith-pipeline@v0.9.1-grok · 5662 in / 1345 out tokens · 64327 ms · 2026-06-30T08:09:00.371968+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

6 extracted references · 3 canonical work pages

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