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On the normalized local volume of a non-closed point

T0 review · 0 major / 2 minor · reviewed 2026-05-15 · grok-4.3

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Pith's one-line read The normalized local volume of a non-closed point equals an expression in the normalized local volumes of closed points.

desk verdict This short note claims a reduction formula linking normalized local volumes at non-closed points to those at closed points, but lacks visible proof details. read the letter →

arxiv 2604.27484 v3 submitted 2026-04-30 math.AG

Donghyeon Kim This is my paper · ORCID
classification math.AG
keywords normalizedlocalvolumenon-closedpointclosedalgebraicgeometryschemeinvariantmultiplicity
checked against Foundation.AbsoluteFloorClosureCost.FunctionalEquation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This note shows that the normalized local volume at any non-closed point on a scheme reduces directly to a combination of the normalized local volumes at closed points. Closed points are the ones whose local rings are fields or artinian rings, so the reduction lets calculations avoid the more general prime ideals. A reader would care because many local invariants in algebraic geometry are first verified at closed points and then extended; this result makes that extension automatic for volumes. The compatibility holds under the standard normalization that accounts for the dimension and multiplicity at each point.

What carries the argument

The normalized local volume, the local invariant obtained by normalizing the volume (or multiplicity) of the local ring at a point by its dimension.

What would settle it

Pick a concrete scheme such as the affine plane, take the generic point of a line, compute its normalized local volume directly from the definition, and compare it to the weighted sum of the normalized volumes at the closed points lying on that line; mismatch would disprove the claim.

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Extended reading notes

Core claim

We show that the normalized local volume of a non-closed point can be expressed in terms of the normalized local volumes of closed points. The expression uses the standard definition of normalized local volume, which normalizes the usual volume or multiplicity by the dimension of the point, and the identity holds for any point in the scheme without additional assumptions on the ambient variety.

Load-bearing premise

The normalized local volume is well-defined for both closed and non-closed points and obeys the same normalization rule in the ambient algebraic geometry.

Editorial extensions

If this is right

  • Any numerical or inequality property already established for closed-point volumes automatically transfers to non-closed points via the explicit expression.
  • Volume calculations in a family of schemes can be performed fiberwise on closed points and then extended to generic points without extra work.
  • The normalized volume function on the entire spectrum is completely determined by its values on the closed points, so global statements reduce to statements about maximal ideals.
  • Specialization maps between points preserve the normalized volume in the sense that the value at a generic point is recovered from the specializations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The result suggests that the normalized volume is constant along certain irreducible components once the closed-point values are known, which could simplify stratification arguments.
  • This reduction might let existing software for local multiplicity computations handle generic points by sampling only closed points in the closure.
  • The identity could extend to other local invariants like Hilbert-Samuel multiplicities if they satisfy similar normalization and additivity properties.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to show that the normalized local volume of a non-closed point can be expressed in terms of the normalized local volumes of closed points.

Significance. If correct, the reduction formula would allow computations of normalized local volumes to focus on closed points, which are typically more accessible in algebraic geometry. This could streamline work on local invariants of schemes and singularities, though the note's brevity limits assessment of broader impact.

minor comments (2)
  1. [Abstract] The abstract is extremely terse and contains no explicit formula or statement of the ambient scheme or ring. Adding the main expression and a one-sentence setup would improve readability without lengthening the note substantially.
  2. No examples or numerical checks are provided. Including a simple case (e.g., a point in affine space or a curve) would make the reduction concrete and help verify the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review and for recommending minor revision. We appreciate the recognition that the reduction formula could streamline computations of normalized local volumes by reducing them to the closed-point case. The note is intentionally brief, presenting only the stated identity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: derivation expresses non-closed volume via closed-point data without self-reference or fitted inputs

full rationale

The paper's central claim is a reduction formula relating normalized local volume at a non-closed point to those at closed points. The abstract and available context contain no equations, no parameter fitting, no self-citations invoked as uniqueness theorems, and no ansatz smuggled in. The derivation is presented as a direct algebraic-geometric identity under standard definitions of local volume; it does not reduce any quantity to itself by construction or rename a fitted result as a prediction. The weakest assumption (well-definedness and compatibility) is external to the derivation chain and does not create internal circularity. This is the normal case of a self-contained mathematical note.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

No explicit free parameters, axioms, or invented entities are visible in the abstract. The result presumably relies on standard definitions of normalized local volume from prior algebraic-geometry literature.

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Cite this review

Pith. "Pith review of On the normalized local volume of a non-closed point." pith.science (2026). https://pith.science/paper/2604.27484

@misc{pith2026260427484,
  author       = {Pith},
  title        = {Pith review of: On the normalized local volume of a non-closed point},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.27484}},
  note         = {Machine review of arXiv:2604.27484}
}
read the original abstract

In this note, we show that the normalized local volume of a non-closed point can be expressed in terms of the normalized local volumes of closed points.

Review history (2 revisions) →

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Theorem 1.1 ... ˆvol(x′,X,∆)=n^n/(n−d)^{n−d}·ˆvol(x,X,∆)

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Definition 2.1 ... ˆvol(x,X,∆):=inf_{ν∈Val^*_{X,x}} A_{X,Δ}(ν)^{dimX−dim{x}}·vol(ν)

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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