arxiv: 2604.27484 · v3 · submitted 2026-04-30 · 🧮 math.AG

Recognition: 2 theorem links

· Lean Theorem

On the normalized local volume of a non-closed point

Donghyeon Kim

Authors on Pith no claims yet

Pith reviewed 2026-05-15 06:46 UTC · model grok-4.3

classification 🧮 math.AG

keywords normalized local volumenon-closed pointclosed pointalgebraic geometryschemelocal invariantmultiplicity

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The pith

The normalized local volume of a non-closed point equals an expression in the normalized local volumes of closed points.

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

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.

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.

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.

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.

Where Pith is reading between the lines

These are 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.

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.

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.

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.

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.

Desk Editor's Note private letter to a colleague

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

The main thing to know about this paper is that it claims a simple reduction: the normalized local volume of a non-closed point can be expressed directly in terms of the normalized local volumes of closed points. This is presented as a short technical note in algebraic geometry. What the paper does well is to highlight this compatibility in a clean way. It takes an existing invariant and shows how it behaves under the distinction between closed and non-closed points, which could make some theoretical arguments or computations more straightforward for those working in the area. The focus is narrow, which keeps the note brief and on target. The soft spots are mainly in the verification department. With only the abstract available, there are no proof steps to examine, no explicit formulas to check for consistency, and no way to see how the reduction is derived. This makes it impossible to assess if the result is genuinely new or if it follows immediately from standard properties in the literature. The soundness score is low for this reason, and the novelty is uncertain without seeing the citations. If the full manuscript has a solid proof, that would change things, but based on what's here, the central claim rests on unexamined details. This paper is for a small group of specialists in algebraic geometry who deal with local volume invariants, likely in the context of singularities or resolution of singularities. A reader who is already comfortable with the definitions of normalized local volume would be the one to get value from the reduction formula. I recommend that it goes to peer review. The claim is precise enough that a referee could evaluate the proof and suggest improvements or point out overlaps with prior work. Even if revisions are needed, it's worth the time for a technical note like this.

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)

[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.
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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 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.

pith-pipeline@v0.9.0 · 5294 in / 983 out tokens · 45229 ms · 2026-05-15T06:46:24.226766+00:00 · methodology

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discussion (0)

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

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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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