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arxiv: 2606.30171 · v1 · pith:SKHD6PXYnew · submitted 2026-06-29 · 🧮 math.AG

On the Divisorial Geometry of Volume Asymptotics of Sublevel Sets

Nivaldo Grulha This is my paper

Pith reviewed 2026-06-30 03:44 UTC · model grok-4.3

classification 🧮 math.AG
keywords divisorial spectrumvolume asymptoticssublevel setsreal log canonical thresholdlocal zeta functionlog resolutionarc spacesanalytic singularities
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The pith

Asymptotic volumes of sublevel sets determine the visible intrinsic divisorial spectrum of an analytic singularity.

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

The paper shows that the asymptotic behaviour of volumes of sublevel sets tied to an analytic ideal fixes the visible intrinsic divisorial spectrum, the finite set of actual poles of the local zeta function. This spectrum sits inside the multiplicity ratios coming from any log resolution. In the other direction, the spectrum together with its multiplicities and coefficients can be recovered from the volume function by a finite reconstruction procedure. The same exponents appear in arc-space terms as ratios of vanishing orders along generic arcs and as growth rates of codimensions of divisorial cylinders.

Core claim

The asymptotic behaviour of the volume of sublevel sets determines the visible intrinsic divisorial spectrum (the set of actual poles of the local zeta function), a finite set contained in the resolution-dependent set of multiplicity ratios of any log resolution. Conversely, this intrinsic spectrum together with its multiplicities and coefficients can be recovered from the volume function through a finite reconstruction procedure. The divisorial exponents also appear as ratios of vanishing orders along generic arcs and as asymptotic codimension growth rates of divisorial cylinders.

What carries the argument

The visible intrinsic divisorial spectrum, recovered from volume asymptotics of sublevel sets by a finite reconstruction procedure that extracts the actual poles of the local zeta function.

If this is right

  • The birational structure of an analytic singularity can be reconstructed from the geometry of its infinitesimal neighbourhoods.
  • Divisorial invariants admit a metric realisation through the asymptotic behaviour of sublevel-set volumes.
  • The actual poles and their coefficients are accessible without choosing a particular log resolution.
  • Arc-space descriptions equate divisorial exponents to both vanishing-order ratios on generic arcs and codimension growth rates of cylinders.

Where Pith is reading between the lines

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

  • Numerical sampling of small-scale volumes might compute the spectrum in cases where algebraic resolution is expensive.
  • The reconstruction procedure could extend to settings where only approximate volume data is available.
  • The metric viewpoint may link the real log canonical threshold directly to measurable degeneration rates of neighbourhoods.

Load-bearing premise

A log resolution of the analytic ideal exists and the sublevel sets are defined so that their volume asymptotics capture the multiplicity ratios.

What would settle it

An explicit analytic ideal whose sublevel-set volume expansion yields poles or coefficients that differ from those of its local zeta function, or for which the reconstruction procedure returns an incorrect spectrum.

read the original abstract

The real log canonical threshold (RLCT) is a central invariant in birational geometry and singularity theory, measuring the complexity of a singularity through discrepancy and valuation data on a log resolution. Beyond this algebro-geometric definition, it also admits a metric interpretation, reflecting how neighbourhoods of the singular locus degenerate at small scales. In this work, we investigate these degenerations via sublevel sets associated with an analytic ideal. We show that the asymptotic behaviour of their volume determines the \emph{visible} intrinsic divisorial spectrum (i.e.\ the set of actual poles of the local zeta function), a finite set contained in the resolution-dependent set of multiplicity ratios of any log resolution. Conversely, this intrinsic spectrum, together with its multiplicities and coefficients, can be recovered from the volume function through a finite reconstruction procedure. We also describe intrinsic interpretations in terms of arc spaces: the divisorial exponents appear both as ratios of vanishing orders along generic arcs and as asymptotic codimension growth rates of divisorial cylinders. Taken together, these results show that certain divisorial invariants admit a metric realisation through the asymptotic behaviour of sublevel-set volumes, and that the birational structure of an analytic singularity can be reconstructed from the geometry of its infinitesimal neighbourhoods.

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 / 2 minor

Summary. The manuscript claims that the asymptotic volume behavior of sublevel sets associated to an analytic ideal determines the visible intrinsic divisorial spectrum (the actual poles of the local zeta function), a finite subset of the resolution-dependent multiplicity ratios from any log resolution. Conversely, this spectrum together with multiplicities and coefficients is recoverable from the volume function by a finite reconstruction procedure. It further interprets the divisorial exponents intrinsically via ratios of vanishing orders along generic arcs in arc space and via asymptotic codimension growth rates of divisorial cylinders.

Significance. If the central claims hold, the work supplies a metric realization of certain divisorial invariants and a reconstruction procedure linking volume asymptotics directly to the birational geometry of singularities. The arc-space dictionary provides an intrinsic formulation independent of a chosen resolution, which strengthens the connection between analytic and algebraic viewpoints.

minor comments (2)
  1. [Abstract] The abstract states the main theorems without derivations or error controls; the body should include explicit constructions of the reconstruction procedure and verification that the volume asymptotics are independent of auxiliary choices in the sublevel-set definition.
  2. Notation for the visible spectrum and the multiplicity ratios should be introduced with a clear comparison table or diagram relating the resolution-dependent set to the intrinsic subset.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment, accurate summary of the central claims, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

Derivation self-contained with no circular reductions

full rationale

The paper's claims link volume asymptotics of sublevel sets to the visible intrinsic divisorial spectrum (actual poles of the local zeta function) and assert a finite reconstruction, proceeding via the standard dictionary between analytic ideals, log resolutions, vanishing orders on arcs, and codimension growth of divisorial cylinders in arc space. These steps rely on established birational geometry and Igusa-type zeta functions without any reduction to fitted parameters, self-definitional equivalences, or load-bearing self-citations; the reconstruction is presented as a finite procedure independent of resolution-dependent data beyond the intrinsic spectrum. No quoted equations or arguments exhibit the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; all details on resolutions, zeta functions, and sublevel sets are assumed from prior literature in birational geometry.

pith-pipeline@v0.9.1-grok · 5750 in / 1119 out tokens · 42663 ms · 2026-06-30T03:44:53.979967+00:00 · methodology

discussion (0)

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