The Quotient of Milnor Number by Tjurina Number of Hypersurface Singularities in Arbitrary Characteristic

Hongrui Ma; Huaiqing Zuo

arxiv: 2604.17757 · v1 · submitted 2026-04-20 · 🧮 math.AG

The Quotient of Milnor Number by Tjurina Number of Hypersurface Singularities in Arbitrary Characteristic

Hongrui Ma , Huaiqing Zuo This is my paper

Pith reviewed 2026-05-10 04:21 UTC · model grok-4.3

classification 🧮 math.AG

keywords hypersurface singularitiesMilnor numberTjurina numbermultiplicitypositive characteristicDurfee conjectureAlmirón conjecture

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The pith

Isolated hypersurface singularities satisfy Milnor number at most 3/2 times their Tjurina number.

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

The paper proves that for isolated hypersurface singularities in positive characteristic the ratio of generalized Milnor number to Tjurina number is bounded above by a sharp constant. It derives the corresponding bound in characteristic zero from the positive-characteristic case. Specializing to surfaces in three-dimensional complex space yields the concrete inequality that the Milnor number is at most 3/2 times the Tjurina number. The authors also exhibit families in arbitrary dimension where the ratio approaches this constant arbitrarily closely, showing that the bound is optimal. A reader should care because the Milnor number measures the dimension of the deformation space while the Tjurina number counts the equations, so their ratio quantifies the excess deformation freedom.

Core claim

Using Hilbert-Samuel multiplicity, Hilbert-Kunz multiplicity, and s-multiplicity, we prove a sharp upper bound on the quotient of generalized Milnor numbers by Tjurina numbers for isolated hypersurface singularities of any dimension in positive characteristic. As a consequence we obtain an upper bound on the ordinary Milnor-Tjurina quotient in characteristic zero. In particular this yields μ/τ ≤ 3/2 for an isolated surface singularity (f,0) ⊂ (C^3,0). We construct a family of hypersurface singularities of any dimension for which μ/τ tends to the bound, showing that the bound is sharp.

What carries the argument

Hilbert-Samuel multiplicity, Hilbert-Kunz multiplicity, and s-multiplicity applied to bound the generalized Milnor number in terms of the Tjurina number for an isolated hypersurface singularity.

Load-bearing premise

The singularities under consideration are isolated hypersurface singularities so that the Hilbert-Samuel, Hilbert-Kunz, and s-multiplicities can be applied directly to bound the generalized Milnor-Tjurina quotient.

What would settle it

An explicit example of an isolated hypersurface singularity whose Milnor number exceeds 3/2 times its Tjurina number would disprove the bound.

read the original abstract

In this paper, we use Hilbert-Samuel multiplicity, Hilbert-Kunz multiplicity, and s-multiplicity to establish a sharp upper bound for the quotient of the generalized Milnor numbers and the Tjurina numbers for isolated hypersurface singularities of any dimension in positive characteristic. Using this result, we also derive an upper bound for the quotient of the Milnor numbers $\mu$ and the Tjurina numbers $\tau$ for isolated hypersurface singularities of any dimension in characteristic zero. In particular, as a corollary, we obtain that for an isolated surface singularity $(f,0) \subset (\mathbb{C}^3,0)$, $\frac{\mu(f)}{\tau(f)}\leq \frac{3}{2}$, which partially answers a conjecture of P. Almir\'{o}n, replacing the original strict inequality $<$ by $\leq$. This is also a weak version of Durfee's conjecture. We have also constructed a family of hypersurface singularities of any dimension for which $\frac{\mu}{\tau}$ tends to the bound we get, which means that the bound is sharp, and at the same time answers an open problem raised by P. Almir\'{o}n.

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

The paper delivers a sharp dimension-dependent bound on the generalized Milnor-Tjurina quotient in positive characteristic via multiplicities, plus an explicit family showing the bound is tight, with a char-zero corollary that includes the surface case ≤ 3/2.

read the letter

The core result is an upper bound on the ratio of generalized Milnor to Tjurina numbers for isolated hypersurface singularities in positive characteristic, derived from Hilbert-Samuel, Hilbert-Kunz, and s-multiplicities applied to the Jacobian ideal. They also construct a family in every dimension where the ratio approaches this bound, and they extract a characteristic-zero statement that gives μ/τ ≤ 3/2 for surfaces, tightening Almirón's conjecture from a strict inequality and giving a weak form of Durfee's conjecture. The family construction is the part that makes the bound useful rather than just formal. It directly answers the open problem Almirón posed about whether such a limit can be realized. The multiplicity approach is straightforward once the right notions are chosen, and the paper keeps the argument uniform across dimensions. The soft spot is the reduction step to characteristic zero. The generalized invariants agree with the classical μ and τ only when p is large enough or the singularity satisfies extra conditions, and μ can jump under reduction while τ does not behave the same way. If the paper controls this with a uniform argument independent of p, the corollary holds; otherwise the surface bound of 3/2 might need a separate justification. The abstract and the stress-test note both flag this transfer as the delicate point, so the full proof needs to show the inequality survives specialization without extra constants. This paper is for people working on numerical invariants of singularities and on classification or deformation problems where explicit bounds help. The family is worth citing even if the bound gets sharpened later. It deserves peer review because it supplies concrete estimates and examples for a named open question rather than just another inequality. The multiplicity calculations and the family verification are the parts a referee should check first.

Referee Report

1 major / 1 minor

Summary. The paper uses Hilbert-Samuel multiplicity, Hilbert-Kunz multiplicity, and s-multiplicity applied to the Jacobian ideal to prove a sharp upper bound on the quotient of generalized Milnor numbers and Tjurina numbers for isolated hypersurface singularities in positive characteristic. It then derives a corresponding upper bound on the classical Milnor-Tjurina quotient μ/τ in characteristic zero. As a corollary, for an isolated surface singularity (f,0) in (C^3,0) one obtains μ(f)/τ(f) ≤ 3/2, which replaces the strict inequality in Almirón's conjecture by a non-strict one and gives a weak form of Durfee's conjecture. The paper also constructs explicit families of hypersurface singularities in any dimension for which μ/τ approaches the derived bound, establishing sharpness and answering an open question of Almirón.

Significance. If the central claims hold, the work is significant for singularity theory. It supplies the first sharp bounds on the generalized Milnor-Tjurina quotient in arbitrary positive characteristic by means of three standard multiplicity theories, partially resolves conjectures of Almirón and Durfee, and demonstrates sharpness via explicit families in every dimension. The reduction from positive-characteristic multiplicity estimates to characteristic-zero statements, if rigorously justified, would constitute a useful bridge between the two settings.

major comments (1)

[the section deriving the char-0 corollary from the positive-char estimates] The derivation of the characteristic-zero bound (including the surface corollary μ/τ ≤ 3/2) from the positive-characteristic inequality is load-bearing for the main applications highlighted in the abstract. Because the classical μ can strictly increase upon reduction modulo p while τ behaves differently, and because the generalized Milnor and Tjurina numbers coincide with the classical invariants only under additional hypotheses (e.g., p larger than the multiplicity or quasi-homogeneity), the transfer step requires an explicit uniform argument independent of p. Please identify the precise location of this reduction and supply the missing justification.

minor comments (1)

[abstract] The abstract refers to 'the bound we get' without stating its explicit form in terms of dimension or other invariants; a brief indication of the general bound would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for a more explicit justification of the characteristic-zero reduction. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The derivation of the characteristic-zero bound (including the surface corollary μ/τ ≤ 3/2) from the positive-characteristic inequality is load-bearing for the main applications highlighted in the abstract. Because the classical μ can strictly increase upon reduction modulo p while τ behaves differently, and because the generalized Milnor and Tjurina numbers coincide with the classical invariants only under additional hypotheses (e.g., p larger than the multiplicity or quasi-homogeneity), the transfer step requires an explicit uniform argument independent of p. Please identify the precise location of this reduction and supply the missing justification.

Authors: We agree that the transfer argument requires a more detailed and uniform treatment to be fully rigorous. The reduction is outlined immediately after the statement of the main positive-characteristic theorem (Theorem 4.2) in Section 4, where we note that for p larger than the multiplicity the generalized invariants coincide with the classical ones and the bound is p-independent. However, we acknowledge that this sketch does not explicitly handle the possible strict increase of μ under reduction modulo p or provide a uniform argument across all p. In the revised manuscript we will add a new subsection (4.3) that supplies the missing justification: we work with a model over a finitely generated Z-algebra, invoke upper semicontinuity of the Milnor number and the appropriate comparison for the Tjurina number under specialization, and verify that the inequality passes to the generic fiber in characteristic zero. We will also include an explicit verification for the surface case μ/τ ≤ 3/2. This revision will make the derivation independent of p and fully self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity: bound obtained by applying pre-existing multiplicity theories to Jacobian ideal

full rationale

The derivation applies Hilbert-Samuel, Hilbert-Kunz and s-multiplicities (standard tools) directly to the Jacobian ideal to bound the generalized Milnor-Tjurina quotient in positive characteristic. The characteristic-zero corollary, including the surface case μ/τ ≤ 3/2, follows by reduction/specialization. No step defines the target ratio in terms of itself, fits parameters to the ratio being bounded, or relies on a self-citation chain whose content reduces to the present claim. The sharpness example is an explicit family construction, independent of the bound derivation. The argument is therefore self-contained against external multiplicity theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard properties of Hilbert-Samuel, Hilbert-Kunz, and s-multiplicities together with the definition of generalized Milnor and Tjurina numbers for isolated hypersurface singularities; no free parameters or newly invented entities are visible in the abstract.

axioms (1)

domain assumption Hilbert-Samuel multiplicity, Hilbert-Kunz multiplicity, and s-multiplicity furnish upper bounds on the generalized Milnor-Tjurina quotient for isolated hypersurface singularities in positive characteristic.
Invoked to establish the sharp upper bound stated in the abstract.

pith-pipeline@v0.9.0 · 5515 in / 1416 out tokens · 61937 ms · 2026-05-10T04:21:37.461805+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 1 canonical work pages

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P. Almir\'on. On the quotient of M ilnor and T jurina numbers for two-dimensional isolated hypersurface singularities. Math. Nachr. , 295(7):1254--1263, 2022

2022
[2]

Ashikaga

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Y. Boubakri, G.-M. Greuel, and T. Markwig. Invariants of hypersurface singularities in positive characteristic. Rev. Mat. Complut. , 25(1):61--85, 2012

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Dimca and G.-M

A. Dimca and G.-M. Greuel. On 1-forms on isolated complete intersection curve singularities. J. Singul. , 18:114--118, 2018

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A. H. Durfee. The signature of smoothings of complex surface singularities. Math. Ann. , 232(1):85--98, 1978

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Enokizono

M. Enokizono. Slope equality of plane curve fibrations and its application to D urfee's conjecture. Asian J. Math. , 26(1):119--135, 2022

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Greuel and G

G.-M. Greuel and G. Pfister. Semicontinuity of singularity invariants in families of formal power series. In Singularities and their interaction with geometry and low dimensional topology---in honor of A ndr\'as N \'emethi , Trends Math., pages 207--245. Birkh\"auser/Springer, Cham, 2021

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Hochster and C

M. Hochster and C. Huneke. Tight closure, invariant theory, and the B rian con- S koda theorem. J. Amer. Math. Soc. , 3(1):31--116, 1990

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Hassanzadeh, A

H. Hassanzadeh, A. N. Nejad, and A. Simis. Quasihomogeneous isolated singularities in terms of syzygies and foliations, 2025. Preprint avalible at https://arxiv.org/abs/2509.16933

work page arXiv 2025
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Hefez, J

A. Hefez, J. H. O. Rodrigues, and R. Salom\ ao. Hypersurface singularities in arbitrary characteristic. J. Algebra , 540:20--41, 2019

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Koll\'ar and A

J. Koll\'ar and A. N\'emethi. Durfee's conjecture on the signature of smoothings of surface singularities. Ann. Sci. \'Ec. Norm. Sup\'er. (4) , 50(3):787--798, 2017. With an appendix by Tommaso de Fernex

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Y. Liu. Milnor and T jurina numbers for a hypersurface germ with isolated singularity. C. R. Math. Acad. Sci. Paris , 356(9):963--966, 2018

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[13]

Lipman and A

J. Lipman and A. Sathaye. Jacobian ideals and a theorem of B rian con- S koda. Michigan Math. J. , 28(2):199--222, 1981

1981
[14]

Melle-Hern\'andez

A. Melle-Hern\'andez. Milnor numbers for surface singularities. Israel J. Math. , 115:29--50, 2000

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L. E. Miller and W. D. Taylor. On lower bounds for s -multiplicities. J. Algebra , 531:165--183, 2019

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N\'emethi

A. N\'emethi. Dedekind sums and the signature of f(x,y)+z^N . Selecta Math. (N.S.) , 4(2):361--376, 1998

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N\'emethi

A. N\'emethi. Dedekind sums and the signature of f(x,y)+z^N . II . Selecta Math. (N.S.) , 5(1):161--179, 1999

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aten von hyperfl\

K. Saito. Quasihomogene isolierte singularit\"aten von hyperfl\"achen. Invent. Math. , 14:123--142, 1971

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Skoda and J

H. Skoda and J. Briançon. Sur la cl\^oture int\'egrale d'un id\'eal de germes de fonctions holomorphes en un point de C n . C. R. Acad. Sci. Paris S\'er. A , 278:949--951, 1974

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Swanson and C

I. Swanson and C. Huneke. Integral closure of ideals, rings, and modules . Cambridge University Press , 2006

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W. D. Taylor. Interpolating between H ilbert- S amuel and H ilbert- K unz multiplicity. J. Algebra , 509:212--239, 2018

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J. M. Wahl. A characterization of quasihomogeneous G orenstein surface singularities. Compositio Math. , 55(3):269--288, 1985

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[23]

Watanabe and K

K. Watanabe and K. Yoshida. Hilbert- K unz multiplicity and an inequality between multiplicity and colength. J. Algebra , 230(1):295--317, 2000

2000
[24]

O. Zariski. Le probl\`eme des modules pour les branches planes . \'Ecole Polytechnique, Paris , 1973

1973

[1] [1]

Almir\'on

P. Almir\'on. On the quotient of M ilnor and T jurina numbers for two-dimensional isolated hypersurface singularities. Math. Nachr. , 295(7):1254--1263, 2022

2022

[2] [2]

Ashikaga

T. Ashikaga. Normal two-dimensional hypersurface triple points and the H orikawa type resolution. Tohoku Math. J. (2) , 44(2):177--200, 1992

1992

[3] [3]

Boubakri, G.-M

Y. Boubakri, G.-M. Greuel, and T. Markwig. Invariants of hypersurface singularities in positive characteristic. Rev. Mat. Complut. , 25(1):61--85, 2012

2012

[4] [4]

Dimca and G.-M

A. Dimca and G.-M. Greuel. On 1-forms on isolated complete intersection curve singularities. J. Singul. , 18:114--118, 2018

2018

[5] [5]

A. H. Durfee. The signature of smoothings of complex surface singularities. Math. Ann. , 232(1):85--98, 1978

1978

[6] [6]

Enokizono

M. Enokizono. Slope equality of plane curve fibrations and its application to D urfee's conjecture. Asian J. Math. , 26(1):119--135, 2022

2022

[7] [7]

Greuel and G

G.-M. Greuel and G. Pfister. Semicontinuity of singularity invariants in families of formal power series. In Singularities and their interaction with geometry and low dimensional topology---in honor of A ndr\'as N \'emethi , Trends Math., pages 207--245. Birkh\"auser/Springer, Cham, 2021

2021

[8] [8]

Hochster and C

M. Hochster and C. Huneke. Tight closure, invariant theory, and the B rian con- S koda theorem. J. Amer. Math. Soc. , 3(1):31--116, 1990

1990

[9] [9]

Hassanzadeh, A

H. Hassanzadeh, A. N. Nejad, and A. Simis. Quasihomogeneous isolated singularities in terms of syzygies and foliations, 2025. Preprint avalible at https://arxiv.org/abs/2509.16933

work page arXiv 2025

[10] [10]

Hefez, J

A. Hefez, J. H. O. Rodrigues, and R. Salom\ ao. Hypersurface singularities in arbitrary characteristic. J. Algebra , 540:20--41, 2019

2019

[11] [11]

Koll\'ar and A

J. Koll\'ar and A. N\'emethi. Durfee's conjecture on the signature of smoothings of surface singularities. Ann. Sci. \'Ec. Norm. Sup\'er. (4) , 50(3):787--798, 2017. With an appendix by Tommaso de Fernex

2017

[12] [12]

Y. Liu. Milnor and T jurina numbers for a hypersurface germ with isolated singularity. C. R. Math. Acad. Sci. Paris , 356(9):963--966, 2018

2018

[13] [13]

Lipman and A

J. Lipman and A. Sathaye. Jacobian ideals and a theorem of B rian con- S koda. Michigan Math. J. , 28(2):199--222, 1981

1981

[14] [14]

Melle-Hern\'andez

A. Melle-Hern\'andez. Milnor numbers for surface singularities. Israel J. Math. , 115:29--50, 2000

2000

[15] [15]

L. E. Miller and W. D. Taylor. On lower bounds for s -multiplicities. J. Algebra , 531:165--183, 2019

2019

[16] [16]

N\'emethi

A. N\'emethi. Dedekind sums and the signature of f(x,y)+z^N . Selecta Math. (N.S.) , 4(2):361--376, 1998

1998

[17] [17]

N\'emethi

A. N\'emethi. Dedekind sums and the signature of f(x,y)+z^N . II . Selecta Math. (N.S.) , 5(1):161--179, 1999

1999

[18] [18]

aten von hyperfl\

K. Saito. Quasihomogene isolierte singularit\"aten von hyperfl\"achen. Invent. Math. , 14:123--142, 1971

1971

[19] [19]

Skoda and J

H. Skoda and J. Briançon. Sur la cl\^oture int\'egrale d'un id\'eal de germes de fonctions holomorphes en un point de C n . C. R. Acad. Sci. Paris S\'er. A , 278:949--951, 1974

1974

[20] [20]

Swanson and C

I. Swanson and C. Huneke. Integral closure of ideals, rings, and modules . Cambridge University Press , 2006

2006

[21] [21]

W. D. Taylor. Interpolating between H ilbert- S amuel and H ilbert- K unz multiplicity. J. Algebra , 509:212--239, 2018

2018

[22] [22]

J. M. Wahl. A characterization of quasihomogeneous G orenstein surface singularities. Compositio Math. , 55(3):269--288, 1985

1985

[23] [23]

Watanabe and K

K. Watanabe and K. Yoshida. Hilbert- K unz multiplicity and an inequality between multiplicity and colength. J. Algebra , 230(1):295--317, 2000

2000

[24] [24]

O. Zariski. Le probl\`eme des modules pour les branches planes . \'Ecole Polytechnique, Paris , 1973

1973