New $μ$-Zariski pairs of surface singularities

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

• The paper introduces μ-Zariski pairs with identical Milnor numbers and monodromy zeta-functions but distinct embedded topologies.
• It employs almost Newton non-degeneracy and toric resolution techniques to analyze and compare singular invariants.
• The work provides counterexamples to equisingularity conjectures by demonstrating the non-existence of μ-constant deformation paths.

New $\mu$-Zariski Pairs of Surface Singularities

Introduction and Background

The classification of singularities, particularly surface singularities in $\mathbb{C}^3$, is foundational in singularity theory, with implications for both local topology and algebraic geometry. A central notion is that of a Zariski pair of singularities: two polynomials $g_0,g_1$ defining surface germs at $0\in\mathbb{C}^3$ whose underlying surfaces are homeomorphic, with identical monodromy zeta-functions (and hence Milnor numbers), but with distinct embedded topologies. Such pairs provide essential counterexamples in equisingularity theory, for instance, to conjectures attributed to Yau and inform the structure of equisingularity strata.

Historically, all known explicit constructions of Zariski pairs for surface singularities (whether $\mu$-, $\mu^*$-, or link-Zariski pairs) were of (possibly weighted) Lê--Yomdin type. These include superisolated singularities, where the link or the local ambient topology is controlled by the algebraic type of a projective tangent cone. The class of Lê--Yomdin singularities is thus central to previous results on the interplay between analytic and topological invariants.

The current work constructs—for the first time—explicit $\mu$-Zariski pairs in $\mathbb{C}^3$ whose singularities are not of Lê--Yomdin type, thus broadening the compass of Zariski pair phenomena and advancing the theory into the class of almost Newton non-degenerate singularities.

Construction of New $\mu$-Zariski Pairs

Let $f_j(z_1,z_2,z_3)$ ($\mathbb{C}^3$0) be reduced, irreducible, homogeneous polynomials of degree $\mathbb{C}^3$1 in three complex variables, yielding projective curves $\mathbb{C}^3$2. Suppose $\mathbb{C}^3$3 and $\mathbb{C}^3$4 form a Zariski pair of curves—i.e., the pairs $\mathbb{C}^3$5 and $\mathbb{C}^3$6 (for regular neighborhoods $\mathbb{C}^3$7 in $\mathbb{C}^3$8) are homeomorphic, but the pairs $\mathbb{C}^3$9 and $g_0,g_1$0 are not.

Consider polynomials

$g_0,g_1$1

where $g_0,g_1$2 are convenient, homogeneous polynomials of degree $g_0,g_1$3 in $g_0,g_1$4, and the function $g_0,g_1$5 is Newton non-degenerate. Crucially, the singular locus of $g_0,g_1$6 and the zero set of $g_0,g_1$7 do not intersect in $g_0,g_1$8. This ensures that $g_0,g_1$9 has an isolated singularity, but, as established, is not of Lê--Yomdin type due to the structure of $0\in\mathbb{C}^3$0.

A finite-dimensional parameter space $0\in\mathbb{C}^3$1 is defined by polynomials whose Newton boundary equals that of $0\in\mathbb{C}^3$2, and which are Newton non-degenerate on all faces except a specified top-dimensional face. Each $0\in\mathbb{C}^3$3 is weakly almost Newton non-degenerate and has an isolated critical point at the origin.

The main result can be formally stated: Theorem. If the singularities of $0\in\mathbb{C}^3$4 and $0\in\mathbb{C}^3$5 are (locally) Newton non-degenerate, then $0\in\mathbb{C}^3$6 and $0\in\mathbb{C}^3$7 form a $0\in\mathbb{C}^3$8-Zariski pair of surface singularities in $0\in\mathbb{C}^3$9. That is, they have the same Milnor number and monodromy zeta-function, their surface germs are homeomorphic, but they lie in distinct path-connected components of the $\mu$0-constant stratum in $\mu$1.

Main Technical Contributions

Newton Non-Degeneracy Analysis and Invariants

A central technical novelty is the detailed analysis of Newton (non-)degeneracy in the context of surface singularities associated to almost, not strictly Lê--Yomdin, configurations. The authors introduce conditions ensuring that the polynomials constructed above are almost Newton non-degenerate in the sense of [Oka]: all their local singularities are locally analytically equivalent to Newton non-degenerate forms.

Applying Varchenko's formula for the monodromy zeta-function and Kouchnirenko's theorem on the Milnor number, it is shown that for $\mu$2 as constructed,

• The Milnor numbers of the restrictions of $\mu$3 to generic planes of $\mu$4 agree.
• The monodromy zeta-functions of $\mu$5 and $\mu$6 coincide, by explicit computation carried out via the corresponding faces of the Newton boundary and the combinatorics of the associated toric resolution.
• The total Milnor number of the singular points of $\mu$7 and $\mu$8 is invariant under $\mu$9-constant deformation inside $\mu^*$0.

A key lemma guarantees that, under the aforementioned non-intersection condition for $\mu^*$1, the local behaviour around singular points of $\mu^*$2 ensures the Newton principal parts of the surface singularities produce identical local zeta-function contributions whenever the singularities are topologically equivalent.

Topological Equivalence and Dual Graphs

Homeomorphism of the surface germs (i.e., existence of a homeomorphism of $\mu^*$3 and $\mu^*$4) is ensured by matching the dual resolution graphs of both surface singularities. The authors perform a complete analysis of toric and further local modifications (at each singular point of $\mu^*$5), showing that each step in the resolution introduces the same plumbing graph structure and Euler characteristics for the exceptional divisor components. The arguments rely on the Khovanovskii--Kouchnirenko--Oka theorem and the topological theory of graph manifolds (Waldhausen–Neumann).

Obstruction to $\mu^*$6-constant Paths

The critical obstruction that distinguishes $\mu^*$7 and $\mu^*$8 as a $\mu^*$9-Zariski pair is the non-existence of a $\mu$0-constant path in $\mu$1 joining them. This is shown by contradiction: if such a continuous path exists, equisingularity theory (Lazzeri–Lê) and the invariance of the total Milnor number force the projective curves $\mu$2 and $\mu$3 to be topologically equivalent as pairs with $\mu$4, contradicting the Zariski pair assumption. Bifurcations leading to a drop in total Milnor number are ruled out by detailed local computations.

Effect of Relaxing Conditions

The non-intersection condition between the singular set of the tangent cone and the zero locus of $\mu$5 is vital for ensuring the equality of monodromy zeta-functions. The paper provides explicit counterexamples demonstrating that violating this hypothesis leads to different Milnor numbers and zeta-functions, directly showing that the analogous pair is not a $\mu$6-Zariski pair under relaxed conditions.

Implications and Further Directions

This work substantially extends the class of explicit $\mu$7-Zariski pairs beyond the Lê--Yomdin context, leveraging almost Newton non-degenerate polynomials and toric geometry. The technical innovations enrich the landscape of invariants for equisingularity and highlight that the phenomena associated with Zariski pairs are not confined to superisolated or Lê--Yomdin-type singularities.

Implications include:

• Stratification Theory: The results refine our understanding of $\mu$8-constant strata and show new subtleties in their topology and connectedness, with direct impact on deformation theory and the classification of complex surface singularities.
• Resolution and Topological Types: The characterization of dual graphs and associated invariants informs further study of the topology of surface singularities and graph manifolds, fostering new avenues for explicit constructions in higher dimensions or for links in exotic $\mu$9 configurations.
• Extension to Broader Classes: The method generalizes to polynomials of higher powers of $\mathbb{C}^3$0 and more intricate $\mathbb{C}^3$1, potentially leading to a systematic classification program for Zariski pairs outside weighted homogeneous or Lê--Yomdin categories.

Further progress may involve:

• Formal proof of the conjecture relating the Alexander polynomial to the Jordan form of monodromy in this new class.
• Extension of Artal Bartolo’s results on embedded topology to the weakly almost Newton non-degenerate setting.
• Development of computational techniques for detecting path-connectedness in the $\mathbb{C}^3$2-constant stratum for broader classes.

Conclusion

This paper introduces a robust framework for constructing $\mathbb{C}^3$3-Zariski pairs outside the established Lê–Yomdin paradigm, grounded in a precise analysis of Newton non-degeneracy, dual graph invariants, and equisingularity theory. The methodology paves the way for expanded exploration of topological classifications and the interplay between analytic and topological invariants in singularity theory, with consequences for both the fine structure of moduli spaces and the explicit construction of singularities with prescribed properties.

Reference: "New $\mathbb{C}^3$4-Zariski pairs of surface singularities" (2604.03018).