arxiv: 2605.08507 · v1 · submitted 2026-05-08 · 🧮 math.AG · math.CV

Recognition: no theorem link

Equisingularity in families of double point curves

Manoel Messias da Silva J\'unior, Otoniel Nogueira da Silva

Pith reviewed 2026-05-12 01:15 UTC · model grok-4.3

classification 🧮 math.AG math.CV

keywords equisingularitydouble point curvesmap germsunfoldingscomplete intersection curvesWhitney conditionssingularity theory

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The pith

Equisingularity of a map unfolding does not imply equisingularity of its associated double point curve families.

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

The paper compares the equisingularity of one-parameter unfoldings of finitely determined map germs from the plane to three-space with the equisingularity of four related families of double point curves. It shows through explicit counterexamples that these equisingularity properties are not equivalent in general. The authors introduce Henry-type families of complete intersection curves that are topologically trivial yet fail Whitney equisingularity, and they generalize classical double point curve formulas to higher-dimensional map germs from n-space to (2n-1)-space.

Core claim

For a 1-parameter unfolding F = (f_t, t) of a finitely determined map germ f from (C^2, 0) to (C^3, 0), the equisingularity of F is compared to that of the families D(F), F(D(F)), D^2(F), and D^2(F)/S_2. The comparison reveals that equisingularity does not transfer between these objects in either direction, with counterexamples constructed. New families of complete intersection curves called Henry-type families are topologically trivial but not Whitney equisingular. Classical formulas for double point curves are generalized to map germs from (C^n, 0) to (C^{2n-1}, 0) for n at least 3, equipped with convenient analytic structures.

What carries the argument

The four families of double point curves D(F), F(D(F)), D^2(F), and D^2(F)/S_2 associated to the unfolding, together with the definitions of equisingularity and Whitney equisingularity applied to them.

If this is right

The equisingularity of the unfolding F does not determine or follow from the equisingularity of its double point curve families in general.
Topologically trivial families of complete intersection curves need not satisfy Whitney equisingularity conditions.
Classical double point curve formulas extend to the setting of map germs in dimensions n to 2n-1 for n greater than or equal to 3.

Where Pith is reading between the lines

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

This indicates that equisingularity must be verified independently for the map and for its double point loci when studying deformations in singularity theory.
The Henry-type families offer concrete test cases for distinguishing different notions of equisingularity in curve families.
Generalizing the formulas may enable similar comparisons for higher-dimensional mappings and their multiple point loci.

Load-bearing premise

The map germs must be finitely determined and the unfoldings must be one-parameter with the curves carrying convenient analytic structures for the comparisons and counterexamples to apply.

What would settle it

An explicit computation showing that every 1-parameter unfolding of a finitely determined map germ from the plane to space is equisingular precisely when its double point curve families are equisingular, or showing that no topologically trivial complete intersection curve families fail Whitney equisingularity.

Figures

Figures reproduced from arXiv: 2605.08507 by Manoel Messias da Silva J\'unior, Otoniel Nogueira da Silva.

**Figure 2.** Figure 2: The notion of a family of curves. (Whitney’s condition b): For any sequences of points (xn) ⊂ X\σ(T) and (tn) ⊂ σ(T)\ {0}, both converging to 0, and such that the sequence of lines (xntn) converges to a line l and the sequence of directions of tangent spaces TxnX, to X at xn, converges to a linear space H we have that the line l is contained in H. A classical result asserts that X is topologically trivial … view at source ↗

read the original abstract

In this paper, we provide a systematic comparison between the equisingularity of a 1-parameter unfolding F = (f_t, t) of a finitely determined map germ f: (\mathbb{C}^2, 0) \to (\mathbb{C}^3, 0) and the equisingularity of its associated families of double point curves: D(F), F(D(F)), D^2(F), and D^2(F)/S_2. We also construct explicit counterexamples to several natural questions concerning the equisingularity of these loci. As a key application, we introduce new families of complete intersection curves - referred to as Henry-type families - which are topologically trivial but fail to satisfy Whitney equisingularity conditions. Finally, we generalize classical double point curve formulas, originally established for map germs from (\mathbb{C}^2, 0) to (\mathbb{C}^3, 0), to the higher-dimensional setting of map germs from (\mathbb{C}^n, 0) to (\mathbb{C}^{2n-1}, 0) for n \geq 3, providing the associated curves with a convenient analytic structure.

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 offers concrete counterexamples and new Henry-type families to separate topological and Whitney equisingularity for double point curves, plus a higher-dimensional extension of the formulas.

read the letter

The main thing to know is that this paper constructs explicit counterexamples showing that equisingularity of a 1-parameter unfolding does not always pass to its associated double point curve families, and it defines Henry-type families of curves that are topologically trivial yet fail Whitney equisingularity. It also extends some classical formulas for double point curves to the case of map germs from C^n to C^{2n-1} when n is at least 3. The systematic comparison of equisingularity conditions across D(F), F(D(F)), D^2(F), and the quotient by S_2 is helpful for organizing the different loci. The counterexamples are valuable because they address natural questions directly with concrete instances. The Henry-type families serve as a good example to separate topological and Whitney notions, which is a useful distinction in the field. The generalization adds some new ground beyond the two-dimensional case. What the paper does well is deliver these constructions explicitly. In singularity theory, examples often clarify the theory more than general statements do, and here they seem to do that. The soft spots are not major. The work relies on the standard assumptions of finite determinacy for the map germs and convenient analytic structures for the curves in the families. These are common, but they do limit how far the results reach without further checks. The higher-dimensional part is presented as a generalization, but it could use more illustration to show the changes in the formulas. This is aimed at a small group of specialists in algebraic geometry focused on singularities and equisingularity. Readers outside that area will likely find it too technical and narrow. I would send it to peer review. The new families and counterexamples provide enough substance for experts to evaluate and build upon.

Referee Report

0 major / 3 minor

Summary. The manuscript provides a systematic comparison between the equisingularity of a 1-parameter unfolding F = (f_t, t) of a finitely determined map germ f: (C^2, 0) → (C^3, 0) and the equisingularity of its associated families of double point curves D(F), F(D(F)), D^2(F), and D^2(F)/S_2. It constructs explicit counterexamples to several natural questions concerning these equisingularities, introduces new families of complete intersection curves called Henry-type families that are topologically trivial but fail Whitney equisingularity, and generalizes classical double point curve formulas to map germs from (C^n, 0) to (C^{2n-1}, 0) for n ≥ 3, endowing the associated curves with a convenient analytic structure.

Significance. If the comparisons, counterexamples, and generalizations hold, the work advances singularity theory by clarifying relationships among equisingularity notions for double-point loci in unfoldings, supplying concrete examples that separate topological triviality from Whitney equisingularity, and extending classical formulas to higher-dimensional map germs. The Henry-type families and the higher-dimensional analytic structures are particularly useful for future studies of equisingularity in families of curves.

minor comments (3)

In the section introducing Henry-type families, state explicitly which Whitney equisingularity conditions fail and provide the local equations or defining ideals of at least one example to facilitate independent verification.
In the generalization to n ≥ 3, clarify the precise meaning of 'convenient analytic structure' on the double-point curves and indicate how it is constructed from the map germ (e.g., via Fitting ideals or Fitting ideals of the appropriate module).
Ensure that the counterexamples in the comparison section include the explicit 1-parameter unfoldings and the computed invariants (e.g., Milnor numbers or multiplicities) that demonstrate the failure of equisingularity equivalence.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript, the accurate summary of its contributions, and the recommendation for minor revision. We are pleased that the comparisons between equisingularity notions, the counterexamples, the introduction of Henry-type families, and the higher-dimensional generalizations are viewed as advancing the field.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's core contributions consist of explicit comparisons between equisingularity notions for unfoldings and their double-point loci, construction of counterexamples to naive equivalences, introduction of Henry-type families as concrete distinctions, and a generalization of existing double-point formulas to higher dimensions. These rest on standard hypotheses (finite determinacy of map germs and convenient analytic structures on loci) and are carried out via direct constructions rather than any derivation that reduces by definition or self-citation to its own inputs. No equations or fitted parameters are presented as predictions; the work is self-contained through algebraic-geometric constructions and does not invoke load-bearing self-citations or uniqueness theorems from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; no specific free parameters, axioms, or invented entities are identifiable. The work relies on standard notions such as finitely determined map germs and Whitney equisingularity from prior literature in algebraic geometry.

pith-pipeline@v0.9.0 · 5508 in / 1161 out tokens · 33386 ms · 2026-05-12T01:15:25.483549+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages

[1]

R.; Pe˜ nafort-Sanchis, G.; Or´ efice-Okamoto, B.; Tomazella, J

Borges Zampiva, J. R.; Pe˜ nafort-Sanchis, G.; Or´ efice-Okamoto, B.; Tomazella, J. N.:Double points and image of reflection maps. Proc. Roy. Soc. Edinburgh Sect. A., (2025), 1–45

work page 2025
[2]

Brasselet, J.-P.; Ruas, M. A. S.; Thuy, N. T. B.:Local bi-Lipschitz classification of semialgebraic surfaces. Dalat Univ. J. Sci.,15(3) (2025), 76–97

work page 2025
[3]

Brian¸ con, J.; Galligo, A.; Granger, M.:Deformations ´ equisinguli` eres des germes de courbes gauches r´ eduites. M´ em. Soc. Math. Fr. (N.S.),2(1980), 1–69

work page 1980
[4]

Brian¸ con, J.; Speder, J.-P.:Les conditions de Whitney impliquentµconstant. Ann. Inst. Fourier (Grenoble),26(2) (1976), 153—163

work page 1976
[5]

Manuscripta Math.,70(1991), 93–114

Br¨ ucker, C.; Greuel, G.-M.:Deformationen isolierter Kurvensingularit¨ aten mit eigebetteten Komponenten. Manuscripta Math.,70(1991), 93–114

work page 1991
[6]

O.; Greuel, G.-M.:The Milnor Number and Deformations of Complex Curve Singularities

Buchweitz, R. O.; Greuel, G.-M.:The Milnor Number and Deformations of Complex Curve Singularities. Invent. Math.,58(1980), 241–281

work page 1980
[7]

Callejas Bedregal, R.; Houston, K.; Ruas, M. A. S.:Topological triviality of families of singular surfaces. Preprint: arXiv:math/0611699v1 (2006)

work page arXiv 2006
[8]

Nonlinearity,5(2) (1992), 373–412

Damon, J.:Topological triviality and versality for subgroups ofAandK.II.Sufficient conditions and applications. Nonlinearity,5(2) (1992), 373–412

work page 1992
[9]

Fern´ andez de Bobadilla, J.; Pe Pereira, M.:Equisingularity at the Normalisation. J. Topol.,1(4) (2008), 879–909. 22

work page 2008
[10]

E.:Topological invariants and Milnor fibre for A-finite germsC 2 →C 3

Fern´ andez de Bobadilla, J.; Pe˜ nafort-Sanchis, G.; Sampaio, J. E.:Topological invariants and Milnor fibre for A-finite germsC 2 →C 3. Dalat Univ. J. Sci.,12(2) (2022), 19–25

work page 2022
[11]

Topology,32(1993), 185–223

Gaffney, T.Polar multiplicities and equisingularity of map germs. Topology,32(1993), 185–223

work page 1993
[12]

da:On reflection maps from then-space to then+ 1-space

Gama, M.B.; Silva, O.N. da:On reflection maps from then-space to then+ 1-space. Rev. Mat. Iberoam.,42(2), 441–476, (2026)

work page 2026
[13]

G.; Wirthm¨ uller, K.; du Plessis, A

Gibson, C. G.; Wirthm¨ uller, K.; du Plessis, A. A.; Looijenga, E. J. N.:Topological stability of smooth mappings. Springer LNM.,552(1976), 71–88

work page 1976
[14]

N.; Snoussi, J.:On tangency in equisingular families of curves and surfaces

Giles-Flores, A.; Silva, O. N.; Snoussi, J.:On tangency in equisingular families of curves and surfaces. Quart. J. Math.,71(2020), 485–505

work page 2020
[15]

N.; Snoussi, J.:On the fifth Whitney cone of a complex analytic curve

Giles-Flores, A.; Silva, O. N.; Snoussi, J.:On the fifth Whitney cone of a complex analytic curve. J. Singul.,24 (2022), 96–118

work page 2022
[16]

Springer, Berlin, (2007)

Greuel, G.-M.; Lossen, C.; Shustin, E.:Introduction to singularities and deformations, Springer Monographs in Mathematics. Springer, Berlin, (2007)

work page 2007
[17]

Hironaka, H.:On the Arithmetic Genera and the Effective Genera of Algebraic Curves, Mere. Coll. Sci., Univ. of Kyoto 30, 177-195 (1957). [18]Institute IMAGINARY. SURFER– Software for visualization of algebraic surfaces. [Computer Software]. Avail- able at: https://imaginary.org/program/surfer. (Accessed: December 2025)

work page 1957
[18]

H.; Nu˜ no-Ballesteros, J

Jorge-P´ erez, V. H.; Nu˜ no-Ballesteros, J. J.:Finite determinacy and Whitney equisingularity of map germs from Cn toC 2n−1. Manuscripta Math.,128(3) (2009), 389–410

work page 2009
[19]

L.; Nu˜ no-Ballesteros, J

Marar, W. L.; Nu˜ no-Ballesteros, J. J.:Slicing corank 1 map germs fromC 2 toC 3. Quart. J. Math.,65(2014), 1375–1395

work page 2014
[20]

L.; Nu˜ no-Ballesteros, J

Marar, W. L.; Nu˜ no-Ballesteros, J. J.; Pe˜ nafort-Sanchis, G.:Double point curves for corank 2 map germs from C2 toC 3.Topology Appl.,159(2012), 526–536

work page 2012
[21]

L.; Mond, D

Marar, W. L.; Mond, D. :Multiple point schemes for corank 1 maps. J. London Math. Soc.,39(1989), 553–567

work page 1989
[22]

N.:Stability ofC ∞ mappings

Mather, J. N.:Stability ofC ∞ mappings. VI: The nice dimensions. Proceedings of Liverpool Singularities- Symposium, I, (1969/70). Lecture Notes in Math., Springer, Berlin,192(1971), 207–253

work page 1969
[23]

D.; Weber, C.:Sur le comportement des polaires associ´ ees aux germes de courbes planes

Michel, F.; Tr` ang, L. D.; Weber, C.:Sur le comportement des polaires associ´ ees aux germes de courbes planes. Compositio Math.,72(1989), 87–113

work page 1989
[24]

Mond, D.:Some remarks on the geometry and classification of germs of maps from surfaces to 3-space.Topology, 26(1987), 361–383

work page 1987
[25]

London Math

Mond, D.:On the classification of germs of maps fromR 2 toR 3.Proc. London Math. Soc.,50(3) (1985), 333–369

work page 1985
[26]

Mond, D.:The number of vanishing cycles for a quasihomogeneous mapping fromC 2 toC 3. Quart. J. Math.,42 (1991), 335–345

work page 1991
[27]

J.:Singularities of Mappings

Mond, D.; Nu˜ no-Ballesteros, J. J.:Singularities of Mappings. volume 357 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York, first edition, (2020)

work page 2020
[28]

In: Algebraic Geometry and Complex Analysis

Mond, D.; Pellikaan, R.:Fitting ideals and multiple points of analytic mappings. In: Algebraic Geometry and Complex Analysis. Lecture Notes in Math., Springer-Verlag,1414(1987), 107–161

work page 1987
[29]

D.; Pichon, A.:Lipschitz geometry of complex curves

Neumann, W. D.; Pichon, A.:Lipschitz geometry of complex curves. J. Singul.,10(2014), 225–234

work page 2014
[30]

J.; Tomazella, J

Nu˜ no-Ballesteros, J. J.; Tomazella, J. N.:Equisingularity of families of map germs between curves, Math. Z.,272 (2012), 349–360. 23

work page 2012
[31]

Parusi´ nski, A.; P˘ aunescu, L.:Lipschitz stratification of complex hypersurfaces in codimension 2. J. Eur. Math. Soc.,25(5) (2023), 1743–1781

work page 2023
[32]

Ann.,378(2020), 559–598

Pe˜ nafort Sanchis, G.:Reflection maps.Math. Ann.,378(2020), 559–598

work page 2020
[33]

PhD Thesis, Universitat de Val` encia, (2016)

Pe˜ nafort-Sanchis, G.:Multiple point spaces of finite holomorphic maps. PhD Thesis, Universitat de Val` encia, (2016)

work page 2016
[34]

Centre Math

Pham, F.; Teissier, B.:Fractions lipschitziennes d’une alg` ebre analytique complexe et saturation de Zariski. Centre Math. l’ ´Ecole Polytech., Paris (1969)

work page 1969
[35]

Appendice in: Briancon, J.; Galligo, A.; Granger, M.: D´ eformations ´ equisingulieres des germes de courbes gauches r´ eduites

Pham, F.; Teissier, B.:Saturation des alg` ebres analytiques locales de dimension un. Appendice in: Briancon, J.; Galligo, A.; Granger, M.: D´ eformations ´ equisingulieres des germes de courbes gauches r´ eduites. M´ em. Soc. Math. France, deuxi` eme s´ erie, tome 1,1(1980) , 1–69

work page 1980
[36]

Ruas, M. A. S.; Silva, O. N.:Whitney equisingularity of families of surfaces inC 3. Math. Proc. Cambridge Philos. Soc.,166(2) (2019), 353–369

work page 2019
[37]

E.:Bi-Lipschitz homeomorphic subanalytic sets have bi-Lipschitz homeomorphic tangent cones

Sampaio, J. E.:Bi-Lipschitz homeomorphic subanalytic sets have bi-Lipschitz homeomorphic tangent cones. Se- lecta Math. (N.S.),22(2016), 553–559

work page 2016
[38]

N.:Surfaces with non-isolated singularities

Silva, O. N.:Surfaces with non-isolated singularities. PhD thesis, S˜ ao Carlos, Universidade de S˜ ao Paulo, (2017), Available at: http://www.teses.usp.br/teses/disponiveis/55/55135/tde-10052017-085440/pt-br.php. (Ac- cessed: December 2025)

work page 2017
[39]

N.:On the topology of the transversal slice of a quasi-homogeneous map germ

Silva, O. N.:On the topology of the transversal slice of a quasi-homogeneous map germ. Math. Proc. Cambridge Philos. Soc.,176(2) (2024), 339–359

work page 2024
[40]

N.; Silva Jr, M

Silva, O. N.; Silva Jr, M. M.:Transverse slices, Ruas’ conjecture, and Zariski’s multiplicity conjecture for quasi- homogeneous surfaces. Preprint: arXiv:2509.01634 (2025)

work page arXiv 2025
[41]

Silva, O.N.:On Invariants of Generic Slices of Weighted Homogeneous Corank 1 Map Germs from the Plane to 3-Space. Bull. Braz. Math. Soc.,52, 663–677, (2021)

work page 2021
[42]

Wall, C. T. C.:Finite determinacy of smooth map germs. Bull. London Math. Soc.,13(1981), 481–539

work page 1981
[43]

Whitney, H.:Local properties of analytic varieties. Differ. and Combinat. Topology, Sympos. Marston Morse, Princeton (1965), 205–244

work page 1965
[44]

A computer algebra system for polyno- mial computations

Wolfram, D.; Greuel, G.-M.; Pfister, G.; Sch¨ onemann, H.:Singular 4-0-2. A computer algebra system for polyno- mial computations. 2015:https://www.singular.uni-kl.de. 24

work page 2015