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arxiv: 2606.20343 · v1 · pith:GJGHM35Gnew · submitted 2026-06-18 · 🧮 math.AG · math.AC

Plane curve singularities and Fitting ideals

Alexandru Dimca , Gabriel Sticlaru This is my paper

Pith reviewed 2026-06-26 15:19 UTC · model grok-4.3

classification 🧮 math.AG math.AC
keywords plane curve singularitiesFitting idealsTjurina idealMilnor numberTjurina numberalgebraic geometry
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The pith

Fitting ideals of the Tjurina ideal for non-quasi-homogeneous plane curve singularities exhibit special properties when the Milnor number exceeds the Tjurina number by at most two.

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

The paper examines the Fitting ideals linked to the Tjurina ideal in plane curve singularities that are not quasi-homogeneous. It finds that when the difference between the Milnor number and the Tjurina number is two or less, these ideals display special properties. A sympathetic reader would care because this relates the numerical invariants of the singularity to algebraic structures of its ideals, potentially aiding classification or computation of such singularities.

Core claim

When the Milnor number minus the Tjurina number is at most 2 for a non-quasi-homogeneous plane curve singularity, the Fitting ideals associated to the Tjurina ideal have special properties.

What carries the argument

The Fitting ideals associated to the Tjurina ideal, which encode information about the relations and syzygies in the module structure of the Tjurina ideal.

If this is right

  • The properties allow for better understanding of the structure when the numerical difference is small.
  • It distinguishes these cases from others where the difference is larger.
  • Provides a way to relate the numerical invariants to ideal-theoretic properties.

Where Pith is reading between the lines

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

  • This could lead to algorithms for computing invariants in low difference cases.
  • Connections to deformation theory of singularities.
  • Perhaps extends to higher dimensional cases or other types of singularities.

Load-bearing premise

The singularities under consideration are non-quasi-homogeneous plane curve singularities.

What would settle it

Finding a non-quasi-homogeneous plane curve singularity with Milnor-Tjurina difference at most 2 where the Fitting ideals do not exhibit the claimed special properties would falsify the claim.

read the original abstract

In this note we investigate the Fitting ideals associated to the Tjurina ideal of a non quasi-homogeneous plane curve singularity. Special properties occur when the difference between Milnor number and Tjurina number is at most 2.

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 / 0 minor

Summary. The manuscript is a short note investigating the Fitting ideals associated to the Tjurina ideal of a non-quasi-homogeneous plane curve singularity. It states that special properties occur when the difference between the Milnor number μ and the Tjurina number τ is at most 2.

Significance. If the special properties are made explicit with supporting calculations or examples, the note could add a modest observation to the literature on Fitting ideals and Tjurina algebras for plane curves. The restriction to the non-quasi-homogeneous case and the μ − τ ≤ 2 threshold is a clearly delimited scope; the contribution would lie in any concrete description of the Fitting ideals under that numerical condition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our short note. The referee's summary accurately captures the scope and focus of the work. We agree that the delimited condition μ − τ ≤ 2 for non-quasi-homogeneous singularities is the natural setting in which to examine the Fitting ideals of the Tjurina ideal, and the manuscript supplies the explicit description of those ideals under this numerical hypothesis.

Circularity Check

0 steps flagged

No circularity: descriptive investigation within explicit domain

full rationale

The paper is a short note investigating Fitting ideals of the Tjurina ideal for non-quasi-homogeneous plane curve singularities and observing that special properties hold when μ − τ ≤ 2. No derivation, prediction, or first-principles result is advanced that reduces by construction to fitted inputs, self-definitions, or self-citation chains. The non-quasi-homogeneous condition is stated explicitly as the setting rather than smuggled in, and the central statement is scoped and observational rather than a load-bearing theorem derived from prior self-citations. The work is self-contained against external benchmarks with no equations or reductions that collapse to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no details on free parameters, axioms, or invented entities are provided.

pith-pipeline@v0.9.1-grok · 5543 in / 1098 out tokens · 25377 ms · 2026-06-26T15:19:15.414935+00:00 · methodology

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