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arxiv: 1906.09593 · v1 · pith:4QLPTT4Jnew · submitted 2019-06-23 · 🧮 math.AG · math.AC

Characterizing the increase of the residual order under blowup in positive characteristic

Herwig Hauser , Stefan Perlega This is my paper

Pith reviewed 2026-05-25 17:56 UTC · model grok-4.3

classification 🧮 math.AG math.AC
keywords residual orderblowuppositive characteristicresolution of singularitiescoefficient idealmultiplicityhypersurfacealgebraic geometry
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The pith

In positive characteristic the residual order may increase under blowup even when multiplicity stays constant.

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

The paper examines the residual order, an invariant used after multiplicity when resolving singularities by successive blowups. In characteristic zero this quantity stays non-increasing under blowups of permissible centers, but the authors show that the version defined by taking the maximum value over all smooth local hypersurfaces can grow in positive characteristic. They give a detailed description of the geometric situations that produce this growth. The analysis is presented as a step toward constructing an adjusted invariant that remains stable under the same operations in every characteristic.

Core claim

The residual order, defined as the maximum over all smooth local hypersurfaces of the order of the coefficient ideal minus exceptional multiplicities, may increase under blowup of a permissible center while the local multiplicity remains constant. The article analyzes in detail the circumstances under which this increase occurs in positive characteristic.

What carries the argument

The residual order defined as the maximum value of the coefficient ideal order taken over all smooth local hypersurfaces.

If this is right

  • The increase occurs while the local multiplicity remains constant.
  • The definition used in characteristic zero does not transfer directly to positive characteristic.
  • The characterization identifies the precise conditions under which growth happens.
  • The analysis supplies information needed to construct a modified invariant that does not increase.

Where Pith is reading between the lines

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

  • One might need to select a distinguished hypersurface rather than the global maximum to restore non-increasing behavior.
  • The same phenomenon could appear in other invariants that rely on coefficient ideals during resolution.
  • Low-dimensional explicit computations of blowups could be used to test the listed circumstances.

Load-bearing premise

Taking the maximum over all smooth local hypersurfaces correctly identifies the relevant coefficient ideal behavior that controls the resolution process.

What would settle it

An explicit example of a permissible blowup in positive characteristic where the residual order increases but the geometric circumstances fall outside the cases described in the characterization.

read the original abstract

In characteristic zero, the residual order constitutes, after the local multiplicity, the second key invariant for the resolution of singularities. It is defined as the order of the coefficient ideal in a local hypersurface of maximal contact, minus the exceptional multiplicities. It does not increase under blowup in permissible centers as long as the local multiplicity remains constant. In positive characteristic, however, the residual order (defined now as the maximum over all smooth local hypersurfaces) may increase under blowup. In the article we analyze in detail the circumstances when this happens. This may help to develop a modification of the residual order which does work in positive characteristic.

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 / 3 minor

Summary. The paper examines the residual order in positive characteristic, defined as the maximum order of the coefficient ideal taken over all smooth local hypersurfaces. It asserts that, unlike the characteristic-zero case, this quantity can increase under permissible blowups even when the local multiplicity remains constant, and supplies a detailed case-by-case analysis of the geometric circumstances under which the increase occurs. The work is framed as a step toward constructing a modified invariant that behaves well in positive characteristic.

Significance. If the case analysis is accurate, the result isolates a concrete obstruction that prevents direct transfer of the residual-order invariant from characteristic zero to positive characteristic. This is a modest but useful clarification for the resolution-of-singularities literature; the explicit enumeration of increase scenarios may guide the search for a replacement invariant. No machine-checked proofs or parameter-free derivations are present, but the observational characterization itself is the paper’s main contribution.

minor comments (3)
  1. The abstract and introduction repeatedly contrast the new definition with the characteristic-zero notion of “order of the coefficient ideal in a local hypersurface of maximal contact.” A short paragraph in §1 or §2 explicitly recalling the precise characteristic-zero definition (including the subtraction of exceptional multiplicities) would help readers who are not specialists.
  2. Notation for the residual order (presumably denoted r or ord_res) and for the exceptional divisor multiplicities should be introduced once, with a clear statement of dependence on the choice of hypersurface, before the case analysis begins.
  3. The manuscript would benefit from a single summary table or diagram that lists, for each enumerated case, the ambient dimension, the multiplicity, the center, and whether the residual order increases.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review and recommendation of minor revision. The referee summary accurately describes the manuscript's content and contribution.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper offers a case-by-case observational characterization of when the residual order (explicitly defined as the maximum over all smooth local hypersurfaces) can increase under permissible blowup while multiplicity is constant. No derivation, prediction, or uniqueness claim is present that reduces by construction to fitted inputs, self-citations, or prior ansatzes from the same authors. The central claim is an examination of the consequences of the given definition rather than a self-referential computation, making the work self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated or identifiable from the provided text.

pith-pipeline@v0.9.0 · 5629 in / 937 out tokens · 26769 ms · 2026-05-25T17:56:33.021601+00:00 · methodology

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Reference graph

Works this paper leans on

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