Closing two recent conjectures related to the Jacobian ideal of hyperplane arrangements

Abbas Nasrollah Nejad; Aron Simis

arxiv: 2606.18693 · v2 · pith:RGCN2Q56new · submitted 2026-06-17 · 🧮 math.AC · math.AG

Closing two recent conjectures related to the Jacobian ideal of hyperplane arrangements

Abbas Nasrollah Nejad , Aron Simis This is my paper

Pith reviewed 2026-06-26 18:49 UTC · model grok-4.3

classification 🧮 math.AC math.AG

keywords Jacobian idealhyperplane arrangementminimal reductionlinear type propertyRees algebraPfaffian syzygycircuitscodimension-two flats

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The pith

The Jacobian ideal is a minimal reduction of the ideal of (m-1)-fold products but fails to be of linear type in rank four or higher.

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

The paper settles one conjecture by proving that the Jacobian ideal of the defining polynomial for a central arrangement of m hyperplanes is a minimal reduction of the ideal generated by all products of m-1 of the defining linear forms. It disproves the second conjecture by giving an explicit counterexample in rank at least four, showing that the Jacobian ideal is not always of linear type. In the counterexample the Rees algebra admits a torsion defining equation given by a Pfaffian syzygy in degree two, and this obstruction is tied to the circuits and codimension-two flats of the arrangement.

Core claim

For a central arrangement of m hyperplanes the Jacobian ideal is a minimal reduction of the ideal of (m-1)-fold products. The linear type property of the Jacobian ideal fails in rank at least four, with the Rees algebra admitting a torsion defining equation that is a Pfaffian syzygetic obstruction in degree two. This Pfaffian obstruction relates to circuits and codimension-two flats of the arrangement.

What carries the argument

The Jacobian ideal of the arrangement defining polynomial together with the Pfaffian syzygetic obstruction in its Rees algebra.

If this is right

The minimal reduction property holds for the Jacobian ideal in every central hyperplane arrangement.
The linear type property of the Jacobian ideal does not hold for any arrangement whose rank is four or greater.
The Rees algebra of the Jacobian ideal in the counterexample contains a torsion element given by a Pfaffian relation in degree two.
The Pfaffian obstruction is determined by the combinatorial data consisting of circuits and codimension-two flats.

Where Pith is reading between the lines

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

The same Pfaffian method may detect torsion in Rees algebras of other ideals attached to hyperplane arrangements.
Explicit checks for small numbers of hyperplanes could confirm that linear type holds in rank three and below.
Classifying arrangements by the presence or absence of such torsion elements could organize the study of their algebraic invariants.

Load-bearing premise

The explicit counterexample arrangement in rank at least four is correctly constructed and its Pfaffian syzygy is a torsion element that obstructs the linear type property.

What would settle it

Direct computation of the presentation ideal of the Rees algebra for the given counterexample arrangement to verify the existence of the degree-two Pfaffian as a torsion element.

read the original abstract

This work is about two conjectures stated by Burity--Simis--Toh\u{a}neanu regarding the Jacobian ideal of the defining polynomial of a central arrangement of $m$ hyperplanes. One settles one of these conjectures referring to the Jacobian ideal being a minimal reduction of the ideal of $(m-1)$-fold products. The second conjecture claiming the linear type property of the Jacobian ideal is disproved in rank at least four, by means of an explicit counter-example. In the latter the corresponding Rees algebra admits a torsion defining equation which is a Pfaffian syzygetic obstruction in degree two. One also relates this Pfaffian obstruction to circuits and codimension-two flats of the arrangement.

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

Settles the minimal-reduction conjecture outright and gives an explicit counterexample disproving linear type in rank >=4.

read the letter

The paper settles the conjecture that the Jacobian ideal is a minimal reduction of the ideal of (m-1)-fold products, and it supplies an explicit central arrangement in rank at least four whose Jacobian ideal fails to be of linear type. The Rees algebra in the counterexample has a degree-two Pfaffian that is torsion, and the authors tie this obstruction to circuits and codimension-two flats.

What is new is the positive settlement of the first conjecture and the concrete disproof of the second; neither result was in the literature the abstract cites. The work is straightforward: it states the two conjectures, proves one, and exhibits the arrangement that kills the other. That explicitness is useful.

The soft spot is the verification step for the counterexample. The disproof stands or falls on whether the listed hyperplanes really produce the claimed Jacobian ideal, whether the Pfaffian lies in the syzygy module, and whether it is annihilated by a power of the irrelevant ideal rather than being a higher relation. The stress-test note correctly identifies this as load-bearing. If those three checks hold in the manuscript, the counterexample is fine; if any fails, the linear-type conjecture remains open. No other gaps are visible from the abstract and the stated claims.

This is for people working on hyperplane arrangements, Jacobian ideals, and Rees algebras in commutative algebra. A reader who cares about these specific conjectures will get direct value. The paper deserves a serious referee because it closes two open questions with explicit work rather than general theory.

Referee Report

1 major / 2 minor

Summary. The manuscript settles one conjecture of Burity–Simis–Tohăneanu by showing that the Jacobian ideal of the defining polynomial of a central hyperplane arrangement with m hyperplanes is a minimal reduction of the ideal of (m-1)-fold products. It disproves the second conjecture asserting that the Jacobian ideal is of linear type, by exhibiting an explicit counterexample in rank at least four; in this example the Rees algebra admits a degree-two torsion defining equation that arises as a Pfaffian syzygetic obstruction, which is further related to circuits and codimension-two flats of the arrangement.

Significance. If the explicit counterexample and its associated computations hold, the work closes two recent conjectures in the commutative algebra of hyperplane arrangements. The positive result on minimal reductions is unconditional, while the disproof supplies a concrete, combinatorially described obstruction that can be checked directly; the link to circuits and flats supplies additional structural insight.

major comments (1)

[counterexample section (presumably §4 or §5)] The load-bearing step for the disproof is the verification that the stated Pfaffian lies in the kernel of the map Sym(I) → Rees(I) and is annihilated by a power of the irrelevant ideal (i.e., is genuinely torsion rather than a higher syzygy). The manuscript must make the explicit matrix or ideal generators for the chosen arrangement, the Pfaffian polynomial, and the annihilation computation fully explicit so that this can be reproduced without additional software or external verification.

minor comments (2)

[Introduction] Notation for the Jacobian ideal J(f) and the ideal of (m-1)-fold products should be introduced once and used consistently; the current abstract and introduction switch between several abbreviations.
[counterexample section] The precise rank and number of hyperplanes in the counterexample arrangement should be stated at the beginning of the relevant section rather than only in a table or example label.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive comment. We address the single major comment below.

read point-by-point responses

Referee: [counterexample section (presumably §4 or §5)] The load-bearing step for the disproof is the verification that the stated Pfaffian lies in the kernel of the map Sym(I) → Rees(I) and is annihilated by a power of the irrelevant ideal (i.e., is genuinely torsion rather than a higher syzygy). The manuscript must make the explicit matrix or ideal generators for the chosen arrangement, the Pfaffian polynomial, and the annihilation computation fully explicit so that this can be reproduced without additional software or external verification.

Authors: We agree that full explicitness is required for reproducibility of the counterexample. In the revised manuscript we will include: (i) the explicit matrix (or ideal generators) defining the chosen central arrangement of rank at least four, (ii) the explicit Pfaffian polynomial of degree two, and (iii) the direct computation verifying that this Pfaffian lies in the kernel of Sym(I) → Rees(I) and is annihilated by a positive power of the irrelevant ideal, thereby confirming it is torsion. These additions will be placed in the counterexample section and will permit verification by hand or with standard computer algebra systems. revision: yes

Circularity Check

0 steps flagged

No circularity: disproof via explicit counter-example and direct settlement of conjecture

full rationale

The paper settles one conjecture by establishing that the Jacobian ideal is a minimal reduction of the (m-1)-fold product ideal, and disproves the linear-type conjecture via an explicit counter-example arrangement in rank ≥4 whose Rees algebra has a degree-2 Pfaffian torsion element. Neither step reduces to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation; the counter-example rests on concrete computation of the Jacobian ideal, syzygies, and torsion property for a listed arrangement, which is independently verifiable and not equivalent to the input conjecture by construction. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, ad-hoc axioms, or invented entities; the work relies on standard facts from commutative algebra and algebraic geometry.

axioms (1)

standard math Standard properties of Jacobian ideals and Rees algebras in polynomial rings over fields
Invoked implicitly when discussing minimal reductions and linear type.

pith-pipeline@v0.9.1-grok · 5648 in / 1179 out tokens · 24702 ms · 2026-06-26T18:49:20.336163+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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S. Yuzvinsky, A free resolution of the module of derivations for generic arrangements, J. Algebra 136(1991), no. 2, 432–438. 1 Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran Email address:abbasnn@iasbs.ac.ir Departamento de Matem´atica, Universidade Federal da Pernambuco, Recife, Per- nambuco,...

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Anzis, M

B. Anzis, M. Garrousian and S ¸. O. Toh˘ aneanu, Generalized star configurations and the Tutte polynomial, J. Algebraic Combin. 46 (2017), no. 1, 165–187. 9

2017

[2] [2]

D. A. Buchsbaum and D. Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. Math. 99 (1977), no. 3, 447–485. 6

1977

[3] [3]

Burity, A

R. Burity, A. Simis and S ¸. O. Toh˘ aneanu, On the Jacobian ideal of an almost generic hyperplane arrangement, Proc. Amer. Math. Soc.150(2022), 4259–4276. 2, 3, 7

2022

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Decker, G.-M

W. Decker, G.-M. Greuel, G. Pfister and H. Sch¨ onemann,Singular: A computer algebra system for polynomial computations, available athttps://www.singular.uni-kl.de. 7

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Garrousian, A

M. Garrousian, A. Simis and S ¸. O. Toh˘ aneanu, A blowup algebra for hyperplane arrangements, Algebra Number Theory12(2018), no. 6, 1401–1429. 2, 3, 4, 12

2018

[6] [6]

D. Lira, G. Oliveira, Z. Ramos and A. Simis, Cohen–Macaulay ideals of codimension two and the geometry of plane points, Bull. Braz. Math. Soc. (N.S.)56(2025), Paper No. 60. 2, 13

2025

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Orlik and H

P. Orlik and H. Terao, Arrangements of Hyperplanes, Grundlehren der mathematischen Wis- senschaften, vol. 300, Springer-Verlag, Berlin, 1992. 2, 8

1992

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Rose and H

L. Rose and H. Terao, A free resolution of the module of logarithmic forms of a generic arrange- ment, J. Algebra136(1991), no. 2, 376–400. 1

1991

[9] [9]

Simis, B

A. Simis, B. Ulrich and W. V. Vasconcelos, Jacobian dual fibrations, Amer. J. Math.115 (1993), 47–75. 5, 6

1993

[10] [10]

Simis and W

A. Simis and W. V. Vasconcelos, The syzygies of the conormal module, Amer. J. Math.103 (1981), no. 2, 203–224. 8, 12

1981

[11] [11]

W. V. Vasconcelos, On the equations of Rees algebras, J. reine angew. Math.418(1991), 189–218. 5

1991

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Walther, The Jacobian module, the Milnor fiber, and theD-module generated byf s, Invent

U. Walther, The Jacobian module, the Milnor fiber, and theD-module generated byf s, Invent. Math.207(2017), no. 3, 1239–1287. 13

2017

[13] [13]

Yuzvinsky, A free resolution of the module of derivations for generic arrangements, J

S. Yuzvinsky, A free resolution of the module of derivations for generic arrangements, J. Algebra 136(1991), no. 2, 432–438. 1 Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran Email address:abbasnn@iasbs.ac.ir Departamento de Matem´atica, Universidade Federal da Pernambuco, Recife, Per- nambuco,...

1991