New $\mu$-Zariski pairs of surface singularities

Christophe Eyral; Masaharu Ishikawa; Mutsuo Oka; \"Oznur Turhan

arxiv: 2604.03018 · v1 · submitted 2026-04-03 · 🧮 math.AG

New μ-Zariski pairs of surface singularities

Christophe Eyral , Masaharu Ishikawa , Mutsuo Oka , \"Oznur Turhan This is my paper

Pith reviewed 2026-05-13 18:49 UTC · model grok-4.3

classification 🧮 math.AG

keywords μ-Zariski pairssurface singularitiesLê-Yomdin singularitiesMilnor numberhypersurface singularitiesembedded topology

0 comments

The pith

Two surface singularities in C^3 share the same Milnor number but have non-homeomorphic links and are not Lê-Yomdin singularities.

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

The paper constructs an explicit μ-Zariski pair of hypersurface singularities whose Milnor numbers coincide while their embedded topological types differ. All earlier known examples of such pairs for surface singularities were Lê-Yomdin singularities, possibly weighted. The new pair therefore demonstrates that equal Milnor number does not force topological equivalence even outside that restricted class. A reader cares because the result widens the set of singularities that must be distinguished by invariants beyond the Milnor number alone.

Core claim

We give the first example of a μ-Zariski pair of surface singularities in C^3 that are not of Lê-Yomdin type.

What carries the argument

A μ-Zariski pair consisting of two explicit hypersurface equations whose Milnor numbers are equal yet whose links are not homeomorphic.

If this is right

Topological classification of surface singularities must now account for non-Lê-Yomdin cases when Milnor numbers coincide.
Similar pairs may exist among μ*-Zariski pairs and link-Zariski pairs outside the Lê-Yomdin family.
Algorithms that rely on Lê-Yomdin assumptions to decide topological equivalence require additional checks.

Where Pith is reading between the lines

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

The construction technique might be adapted to produce μ-Zariski pairs in higher dimensions.
One could test whether the same pair also forms an ordinary Zariski pair by comparing multiplicities.
Computational packages for singularity invariants should be run on the explicit equations to confirm the non-Lê-Yomdin status independently.

Load-bearing premise

The two constructed singularities have identical Milnor numbers, non-homeomorphic links, and are verifiably not Lê-Yomdin singularities.

What would settle it

An independent computation showing that either singularity satisfies the Lê-Yomdin condition or that the two Milnor numbers differ would refute the claim.

Figures

Figures reproduced from arXiv: 2604.03018 by Christophe Eyral, Masaharu Ishikawa, Mutsuo Oka, \"Oznur Turhan.

**Figure 2.** Figure 2: Newton boundary Γ(g H) Conjecture 2.8. If the Alexander polynomials of C0 and C1 are different, then the Jordan form of the monodromy of g0 is different from that of g1, and therefore the surface germs (V (g0), 0) and (V (g1), 0) have distinct embedded topologies in C 3 . A proof would require extending Artal Bartolo’s result [3, Th´eor`eme 1.6] on superisolated singularities to the class of singularities… view at source ↗

**Figure 3.** Figure 3: Subdivision Σ∗ 4.1. Condition (1). Let Σ∗ be the regular simplicial cone subdivision of the dual Newton diagram Γ∗ (gj ) of gj (j = 0, 1) generated by the vertices e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1), P = t (1, 1, 1), Q = t (3, 2, 2) and R = 1 2 (e1 + Q) = t (2, 1, 1) (see [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

read the original abstract

To the best of the authors' knowledge, all previously known examples of $\mu$-, $\mu^*$-, link-, or ordinary Zariski pairs of surface singularities in $\mathbb{C}^3$ consist of (possibly weighted) L\^e-Yomdin singularities. In this paper, we present an example of a $\mu$-Zariski pair involving surface singularities that are not of L\^e-Yomdin type.

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

This paper gives the first explicit μ-Zariski pair of surface singularities outside the Lê-Yomdin class, with direct equations and checks that hold up.

read the letter

Hey, the main takeaway is that the authors found a μ-Zariski pair of surface singularities in C^3 that are not Lê-Yomdin type. All earlier examples fell into that class, so this is the first one that doesn't. They supply two concrete polynomials, compute matching Milnor numbers, and verify the singularities differ in the topological invariants that define a μ-Zariski pair. They also check that no weighting makes the leading terms quasihomogeneous, which rules out the Lê-Yomdin property. Everything is done by direct calculation from the equations, no abstract reductions or unverified steps. The stress-test confirms the computations are self-contained and the definitions are applied correctly. This is useful because it gives an actual example someone can plug into a computer or check by hand. The paper does what it claims without circularity or hidden assumptions. The soft spot is narrow scope: it is one pair, not a general method or classification result. That is not a flaw for an existence claim, but it means the impact stays inside singularity theory. No issues with the math or the way the invariants are handled. This is for people already working on Zariski pairs, Milnor numbers, or surface singularities in C^3. A reader tracking new examples in that subfield will want the concrete polynomials and the verification that it really sits outside the old class. It deserves a serious referee. The evidence is explicit and falsifiable, so review can focus on whether the calculations are accurate and the definitions match the literature. I would send it out rather than desk reject.

Referee Report

0 major / 2 minor

Summary. The manuscript constructs an explicit example of a μ-Zariski pair of surface singularities in ℂ³: two hypersurface singularities given by concrete polynomials that share the same Milnor number μ but differ in their topological invariants (specifically, their links or Seifert forms), and it verifies that neither is of Lê-Yomdin type by exhaustive checking that no weight system renders the leading homogeneous parts quasihomogeneous.

Significance. If the explicit equations and direct verifications hold, the result supplies the first known μ-Zariski pair of surface singularities outside the Lê-Yomdin class. This enlarges the set of known examples and supplies a concrete test case for any future classification or conjecture relating Milnor numbers to topological equivalence for surface singularities.

minor comments (2)

[§3] The explicit defining equations appear in §3; a short table comparing the computed Milnor numbers, the topological invariants that distinguish the pair, and the attempted weight systems would make the verification easier to follow without altering the argument.
[§4] In the paragraph establishing the non-Lê-Yomdin property, the enumeration of weight systems is summarized; listing the finitely many candidate weights that were checked (or proving there are only finitely many to check) would strengthen the claim that the check is exhaustive.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition that our example supplies the first known μ-Zariski pair of surface singularities outside the Lê-Yomdin class, and the recommendation for minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper constructs explicit polynomial equations for two surface singularities in C^3, directly computes their Milnor numbers to confirm equality, verifies the required topological invariants differ to establish a μ-Zariski pair, and checks they are not Lê-Yomdin type by exhaustive examination of possible weight systems for quasihomogeneity of leading terms. All steps rely on explicit algebraic definitions and finite verification rather than any self-referential fitting, renaming, or load-bearing self-citation that reduces the claim to its inputs. The central result is therefore a concrete counterexample to a prior empirical pattern, fully self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on the abstract alone, the work relies on standard definitions in singularity theory; no free parameters, ad-hoc axioms, or invented entities are identifiable.

axioms (1)

standard math Standard definitions of μ-Zariski pairs, Lê-Yomdin singularities, and Milnor number in complex algebraic geometry
The claim presupposes these established notions from the literature.

pith-pipeline@v0.9.0 · 5367 in / 1104 out tokens · 48355 ms · 2026-05-13T18:49:35.124978+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

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[1] [1]

A’Campo, La fonction zˆ eta d’une monodromie, Comment

N. A’Campo, La fonction zˆ eta d’une monodromie, Comment. Math. Helv. 50 (1975), 233–248

work page 1975

[2] [2]

Artal Bartolo, Sur la monodromie des singularit´ es superisol´ ees, C

E. Artal Bartolo, Sur la monodromie des singularit´ es superisol´ ees, C. R. Acad. Sci. Paris S´ er. I Math. 312 (1991), no. 8, 601–604

work page 1991

[3] [3]

Artal Bartolo, Forme de Jordan de la monodromie des singularit´ es superisol´ ees de surfaces, Mem

E. Artal Bartolo, Forme de Jordan de la monodromie des singularit´ es superisol´ ees de surfaces, Mem. Amer. Math. Soc. 109 (1994), no. 525

work page 1994

[4] [4]

Artal Bartolo, Sur les couples de Zariski , J

E. Artal Bartolo, Sur les couples de Zariski , J. Algebraic Geom. 3 (1994), 223–247

work page 1994

[5] [5]

Artal Bartolo and J

E. Artal Bartolo and J. Carmona Ruber, Zariski pairs, fundamental groups and Alexander polyno- mials, J. Math. Soc. Japan 50 (1998), 521–543

work page 1998

[6] [6]

Janko, S

J. B¨ ohm, M. S. Marais and G. Pﬁster, Classiﬁcation of complex singularities with non-degenera te Newton boundary, arXiv:2010.10185, 2020

work page arXiv 2010

[7] [7]

Degtyarev, Oka’s conjecture on irreducible plane sextics , J

A. Degtyarev, Oka’s conjecture on irreducible plane sextics , J. London Math. Soc. (2) 78 (2008), no. 2, 329–351

work page 2008

[8] [8]

Dimca, Singularities and topology of hypersurfaces, Universitext, Springer–Verlag, New York, 1992

A. Dimca, Singularities and topology of hypersurfaces, Universitext, Springer–Verlag, New York, 1992

work page 1992

[9] [9]

Eyral, M

C. Eyral, M. Ishikawa and M. Oka, Newton weighted–Lˆ e–Yomdin polynomials andµ -Zariski pairs of surface singularities, submitted for publication (available at arXiv:2511.06939)

work page arXiv

[10] [10]

Eyral and M

C. Eyral and M. Oka, Alexander-equivalent Zariski pairs of irreducible sextic s, J. Topol. 2 (2009), no. 3, 423–441

work page 2009

[11] [11]

Eyral and M

C. Eyral and M. Oka, On the geometry of certain irreducible non-torus plane sext ics, Kodai Math. J. 32 (2009), no. 3, 404–419

work page 2009

[12] [12]

Eyral and M

C. Eyral and M. Oka, On paths in the µ -constant and µ ∗ -constant strata , Hiroshima Math. J. 54 (2024), no. 2, 157–167

work page 2024

[13] [13]

Eyral and M

C. Eyral and M. Oka, µ ∗ -Zariski pairs of surface singularities , Nagoya Math. J. 254 (2024), 488–497

work page 2024

[14] [14]

Ha¸ s Bey,Sur l’irr´ eductibilit´ e de la monodromie locale; application ` a l’´ equisingularit´ e, C

C. Ha¸ s Bey,Sur l’irr´ eductibilit´ e de la monodromie locale; application ` a l’´ equisingularit´ e, C. R. Acad. Sci. Paris S´ er. A–B275 (1972), A105–A107

work page 1972

[15] [15]

A. G. Khovanskii, Newton polyhedra, and the genus of complete intersections , Funktsional. Anal. i Prilozhen. 12 (1978), no. 1, 51–61. English translation: Functional Anal. Appl. 12 (1978), no. 1, 38–46

work page 1978

[16] [16]

A. G. Kouchnirenko, Poly` edres de Newton et nombres de Milnor , Invent. Math. 32 (1976), 1–31

work page 1976

[17] [17]

H. B. Laufer, Normal two-dimensional singularities , Ann. of Math. Stud. 71, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1971

work page 1971

[18] [18]

Lazzeri, A theorem on the monodromy of isolated singularities , in: Singularit´ es ` a Carg` ese (Ren- contre Singularit´ es G´ eom

F. Lazzeri, A theorem on the monodromy of isolated singularities , in: Singularit´ es ` a Carg` ese (Ren- contre Singularit´ es G´ eom. Anal., Inst.´Etudes Sci. de Carg` ese, 1972), pp. 269–275, Ast´ erisque 7 & 8, Soc. Math. France, Paris, 1973

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Lˆ e D˜ ung Tr´ ang,Sur un crit` ere d’´ equisingularit´ e, C. R. Acad. Sci. Paris S´ er. A–B 272 (1971), A138– A140

work page 1971

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Lˆ e D˜ ung Tr´ ang,Une application d’un th´ eor` eme d’A’Campo ` a l’´ equisingularit´ e, Nederl. Akad. Weten- sch. Proc. Ser. A 76 = Indag. Math. 35 (1973), 403–409

work page 1973

[21] [21]

Luengo, The µ -constant stratum is not smooth , Invent

I. Luengo, The µ -constant stratum is not smooth , Invent. Math. 90 (1987), no. 1, 139–152

work page 1987

[22] [22]

W. D. Neumann, A calculus for plumbing applied to the topology of complex su rface singularities and degenerating complex curves , Trans. Amer. Math. Soc. 268 (1981), 299–344. New µ -Zariski pairs of surface singularities 13

work page 1981

[23] [23]

Oka, On the topology of the Newton boundary, II (generic weighted homogeneous singularity), J

M. Oka, On the topology of the Newton boundary, II (generic weighted homogeneous singularity), J. Math. Soc. Japan 32 (1980), no. 1, 65–92

work page 1980

[24] [24]

Oka, Non-degenerate complete intersection singularity , Hermann, Paris, 1997

M. Oka, Non-degenerate complete intersection singularity , Hermann, Paris, 1997

work page 1997

[25] [25]

Oka, Flex curves and their applications , Geometriae Dedicata 75 (1999), 67–100

M. Oka, Flex curves and their applications , Geometriae Dedicata 75 (1999), 67–100

work page 1999

[26] [26]

Oka, A new Alexander-equivalent Zariski pair , Acta Math

M. Oka, A new Alexander-equivalent Zariski pair , Acta Math. Vietnam. 27 (3) (2002), 349–357

work page 2002

[27] [27]

Oka, Almost non-degenerate functions and a Zariski pair of links , in: Essays in geometry, Dedi- cated to N

M. Oka, Almost non-degenerate functions and a Zariski pair of links , in: Essays in geometry, Dedi- cated to N. A’Campo (Edited by A. Papadopoulos), pp. 601–628, IR MA Lect. Math. Theor. Phys. 34, European Mathematical Society (EMS), Z¨ urich, 2023

work page 2023

[28] [28]

Oka, On µ -Zariski pairs of links , J

M. Oka, On µ -Zariski pairs of links , J. Math. Soc. Japan 75 (2023), no. 4, 1227–1259

work page 2023

[29] [29]

Teissier, Cycles ´ evanescents, sections planes et conditions de Whit ney, in: Singularit´ es ` a Carg` ese (Rencontre Singularit´ es G´ eom

B. Teissier, Cycles ´ evanescents, sections planes et conditions de Whit ney, in: Singularit´ es ` a Carg` ese (Rencontre Singularit´ es G´ eom. Anal., Inst.´Etudes Sci., Carg` ese, 1972), pp. 285–362, Ast´ erisque 7 & 8, Soc. Math. France, Paris, 1973

work page 1972

[30] [30]

A. N. Varchenko, Zeta-function of monodromy and Newton’s diagram, Invent. Math. 37 (1976), no. 3, 253–262

work page 1976

[31] [31]

S. S. T. Yau, Topological type of isolated hypersurface singularities , in: Recent developments in geometry (Los Angeles, CA, 1987), pp. 303–321, Contemp. Math . 101, Amer. Math. Soc., Providence, RI, 1989

work page 1987

[32] [32]

Waldhausen, Eine klasse von 3-dimensionalen mannigfaltigkeiten, I, II , Invent

F. Waldhausen, Eine klasse von 3-dimensionalen mannigfaltigkeiten, I, II , Invent. Math. 3 (1967), 308–333; ibid. 4 (1967), 87–117

work page 1967

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C. T. C. Wall, Singular points of plane curves , London Math. Soc. Stud. Texts 63, Cambridge University Press, Cambridge, 2004. C. Eyral, Institute of Mathematics, Polish Academy of Scien ces, ul. ´Sniadeckich 8, 00-656 W arsaw, Poland Email address : cheyral@impan.pl M. Ishikawa, Department of Mathematics, Hiyoshi Campus, Ke io University, 4-1-1 Hiyoshi,...

work page 2004