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arxiv: 2604.07252 · v1 · submitted 2026-04-08 · 🧮 math.AG

A note on b-divisors and filtrations on a local ring

Lu Qi This is my paper

Pith reviewed 2026-05-10 17:19 UTC · model grok-4.3

classification 🧮 math.AG
keywords b-divisorsfiltrationsNoetherian local domainsbirational geometrylocal algebraalgebraic geometrydivisors
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The pith

A correspondence links filtrations on local rings to b-divisors.

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

The paper establishes a bijection between filtrations and b-divisors over a general class of Noetherian local domains. This identification lets algebraic properties of one object transfer directly to the other. The result is then applied in a global birational context to settle a conjecture about divisors. A reader would care because the link turns local ring-theoretic questions into geometric statements that may be easier to handle on models.

Core claim

We prove a correspondence between filtrations and b-divisors over a general class of Noetherian local domains. As an application in the global setting, this correspondence proves a recent conjecture.

What carries the argument

The bijection that associates to each filtration its corresponding b-divisor on the local ring.

If this is right

  • Algebraic invariants attached to filtrations can be computed or bounded using the geometry of b-divisors.
  • Questions about b-divisors on local rings reduce to questions about filtrations of ideals.
  • The local correspondence extends to yield global statements about divisors on varieties.

Where Pith is reading between the lines

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

  • The result may let researchers translate multiplicity computations into statements about exceptional divisors on blow-ups.
  • It could connect filtrations on Rees algebras to the geometry of the valuative tree.
  • Similar identifications might exist for other classes of rings such as excellent or henselian domains.

Load-bearing premise

The stated general class of Noetherian local domains admits a well-defined and bijective correspondence between the two objects.

What would settle it

An explicit Noetherian local domain together with a filtration that cannot be matched to any b-divisor would falsify the claimed correspondence.

read the original abstract

In this note, we prove a correspondence between filtrations and b-divisors over a general class of Noetherian local domains. As an application in the global setting, we prove a recent conjecture of Ro\'e-Urbinati.

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 / 1 minor

Summary. The manuscript proves a direct correspondence between filtrations on a general class of Noetherian local domains and b-divisors by constructing explicit inverse maps via valuation rings and associated graded pieces. As an application, it proves a recent conjecture of Roé-Urbinati in the global setting using standard localization and globalization arguments.

Significance. If the stated correspondence holds, the result supplies a concrete dictionary between local algebraic filtrations and geometric b-divisors that may streamline computations involving singularities and birational invariants. The resolution of the Roé-Urbinati conjecture constitutes a concrete advance in the global theory.

minor comments (1)
  1. The introduction would benefit from a single sentence recalling the precise technical conditions (e.g., excellence or existence of a resolution) imposed on the Noetherian local domains, even though they are defined later.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity; direct proof via explicit maps

full rationale

The paper establishes a correspondence between filtrations and b-divisors on a stated class of Noetherian local domains by constructing explicit inverse maps using valuation rings and associated graded pieces. This is a self-contained algebraic construction that does not rely on fitted parameters, self-citations as load-bearing premises, or any reduction of the claimed result to its own inputs by definition. The global application to the Roé-Urbinati conjecture proceeds by standard localization, again without circular steps. No enumerated circularity pattern is present.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No details on free parameters, axioms, or invented entities are supplied by the abstract; the ledger is therefore empty.

pith-pipeline@v0.9.0 · 5312 in / 980 out tokens · 31680 ms · 2026-05-10T17:19:15.032050+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

  1. [1]

    J.161(2012), no

    [BdFF12] S´ ebastien Boucksom, Tommaso de Fernex, and Charles Favre,The volume of an isolated singularity, Duke Math. J.161(2012), no. 8, 1455–1520.↑1, 2, 4, 6, 7, 8 [BFJ08] S´ ebastien Boucksom, Charles Favre, and Mattias Jonsson,Valuations and plurisubharmonic singulari- ties, Publ. Res. Inst. Math. Sci.44(2008), no. 2, 449–494.↑1 [BFJ09] ,Differentiabi...

    work page arXiv 2012