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arxiv: 2301.06994 · v3 · submitted 2023-01-17 · 🧮 math.AG

Complements of discriminants of real singularities of type X₁₀

Victor A. Vassiliev This is my paper

Pith reviewed 2026-05-24 09:55 UTC · model grok-4.3

classification 🧮 math.AG
keywords real singularitiesdiscriminant complementsX10 singularitieswave frontsfirst homologyconnected componentsplane curve singularitiesnon-quasihomogeneous singularities
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The pith

Complements of X10 singularity discriminants contain non-trivial 1D homology groups, unlike all simple cases.

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

The paper enumerates the connected components of the complements to the discriminant varieties of real plane function singularities of types X10 cubed and X10 to the first. These varieties mark the functions with degenerate critical points, so their complements consist of the spaces of functions with only non-degenerate critical points. For all previously studied simple singularities the complements were contractible or at least had trivial first homology, but the X9 and X10 cases produce loops that cannot be contracted to a point. The work also supplies the first examples of such complements for singularities that are not semi-quasihomogeneous.

Core claim

A conjecturally complete list of the connected components of the complements of the discriminant varieties of singularities of type X10 cubed and X10 to the first is given. The complements of the discriminants of singularities of classes X9 plus or minus and X10 to the first are shown to have non-trivial one-dimensional homology groups, in contrast to the situation for all simple singularity classes. These are the first examples of not semi-quasihomogeneous plane function singularities for which such complements have been studied.

What carries the argument

The discriminant variety (wave front) of an X10 singularity, whose complement's connected components and first homology group are classified.

If this is right

  • The listed components exhaust the topology of the complements for the two X10 types under study.
  • Non-trivial first homology appears precisely for the X9 and X10 first classes, distinguishing them from simple singularities.
  • The same complements supply the first studied cases outside the semi-quasihomogeneous class.
  • The enumeration rests on a complete classification of the real forms of these singularities.

Where Pith is reading between the lines

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

  • The appearance of non-trivial loops may be tied to the larger Milnor number or the failure of semi-quasihomogeneity, suggesting a pattern worth checking in other non-simple classes.
  • The computational methods used here could be applied to classify complements for singularities of types X11 or Y9.
  • If the non-trivial homology persists under small deformations, it would give a topological invariant separating these singularity classes from the simple ones in the space of all functions.

Load-bearing premise

The classification of real singularities and their discriminants has found every distinct connected component without missing any.

What would settle it

Discovery of one additional connected component in the complement for X10 cubed, or a direct homology calculation showing that the first homology group of the X10 to the first complement is trivial.

Figures

Figures reproduced from arXiv: 2301.06994 by Victor A. Vassiliev.

Figure 1
Figure 1. Figure 1: Standard systems of paths Let f : (C 2 , R 2 , 0) → (C, R, 0) be a holomorphic function germ having an iso￾lated critical point at the origin, and F be its k-parametric versal deformation. In particular, the Milnor number µ(f) of this critical point is finite and equal to the Milnor number of the function f(x, y) + z 2 : (C 3 , R 3 , 0) → (C, R, 0). Let fλ : (C 2 , R 2 ) → (C, R), λ ∈ R k \ ΣR(F), be a ver… view at source ↗
Figure 2
Figure 2. Figure 2: Zero set of a singularity of type X3 10 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Using the reflection y ↔ −y in R 2 we then additionally realize 12 pictures, which are mirror images of the pictures listed in statement 4 of Theorem 1. All real morsifications representing 44 pictures thus obtained belong to different components of complements of the discriminant: for almost all pairs of these 44 pictures this follows from their topological non-equivalence, and only for three pairs marked… view at source ↗
Figure 4
Figure 4. Figure 4: The standard scale of the obtained morsification realizes pictures of pages 9–10 with subscripts (495364, 0), (385722, 1), (214912, 2), (385722, 1), (10784, 0), (123976, –1), (108196, –2), (67624, –3), (81848, –4), (303168, –3), (495922, –2). (Here and below we highlight in bold the subscripts of pictures realized for the first time.) 4.2. We can also perturb our function (3) so that the topological pictur… view at source ↗
Figure 5
Figure 5. Figure 5: r − − + + + + − − − + + − [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Perturbations X3 10 → E8, X3 10 → X2 9 and X3 10 → X1 9 4.6. Take the perturbation of f whose zero set is shown in [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Perturbations of X2 9 singularities (189496, 0), (276604, –1), (12848, –1), (120088, –2), (63700, –3), (25104, –4), and (4800, –5). 5.2. X3 10 → X2 9 . The perturbation ˜f ≡ (y 2 − x 3 − εx2 )(x 2 − y 2 ), ε > 0, of the polynomial (3) has a critical point of class X2 9 at the origin. The topology of the zero set of ˜f is shown in [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Perturbations of X1 9 singularities 6. Virtual morsifications associated with real morsifications found in §5 It remains to prove that the values of invariant Card on 32 realized morsifications are indeed equal to the numbers indicated under their pictures in pp. 9–10. This is already done by our restricted program for all pictures realized in §4 (see the last paragraph of §4); it remains to do it for seve… view at source ↗
Figure 9
Figure 9. Figure 9: Separatrice diagram for perturbation (8496, −4) gradient vector field connecting these two points; but our picture does not allow such trajectories. The proofs for other two pictures, (7200, 3) and (4320, −4), are analogous. Therefore, the virtual morsification associated with the real morsification drawn in the picture (4800, −5) and realized in § 4.1 can only belong to the virtual component with Card = 4… view at source ↗
Figure 10
Figure 10. Figure 10: Topological pictures of virtual components for X1 10 singularity J G B I H E C D F r A − r r r r − G J I C F E D B H A r r r r r − − + − [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Morsifications of X1 10 singularity 8. Proof of Theorem 2 We use three sabirizations of singularity (9): all of them are products of two polynomials of degree 2 and 3 which are perturbations of two factors of (9). Two of [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Another sabirization of X1 10 singularity these sabirizations are shown in [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
read the original abstract

A conjecturally complete list of connected components of complements of discriminant varieties (aka wave fronts) of smooth function singularities of type $X_{10}^3$ and $X_{10}^1$ is presented; it are the first examples of not semi-quasihomogeneous plane function singularities. It is shown that the complements of discriminants of singularities of classes $X_9^{\pm}$ and $X_{10}^1$ have non-trivial 1-dimensional homology groups, in contrast to all simple singularity classes.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a conjecturally complete list of connected components of the complements of discriminant varieties (wave fronts) for real plane function singularities of types X_{10}^3 and X_{10}^1; these are described as the first non-semi-quasihomogeneous examples. It further claims that the complements for classes X_9^± and X_{10}^1 have non-trivial first homology groups, in contrast to all simple singularity classes.

Significance. If the conjectural enumeration holds, the work supplies the first explicit topological data on discriminants beyond the semi-quasihomogeneous regime, sharpening the distinction between simple and unimodal real singularities. The reported non-trivial H_1 for X_9^± and X_{10}^1 supplies a concrete topological invariant that separates these classes from the simple ones previously studied.

major comments (2)
  1. [Abstract and §1] Abstract and §1: the central claim of a 'conjecturally complete list' of components for X_{10}^3 and X_{10}^1 is load-bearing, yet the manuscript provides no independent upper bound (e.g., via Milnor-fiber Euler characteristic, morsification counts, or stratification of the versal unfolding) that would certify exhaustiveness; without such a bound the contrast with simple singularities cannot be asserted.
  2. [§4] §4 (homology computations): the assertion that H_1 is non-trivial for X_9^± and X_{10}^1 rests on unshown chain-level calculations; the text should exhibit the explicit simplicial complexes, software scripts, or hand-computed boundary matrices that yield the reported Betti numbers.
minor comments (2)
  1. [§1] Notation for the real forms X_{10}^3 and X_{10}^1 should be cross-referenced to the standard Arnold classification table in the introduction.
  2. [Figures] Figure captions for the wave-front diagrams should state the number of components shown and whether any are conjectural.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We respond to the major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: the central claim of a 'conjecturally complete list' of components for X_{10}^3 and X_{10}^1 is load-bearing, yet the manuscript provides no independent upper bound (e.g., via Milnor-fiber Euler characteristic, morsification counts, or stratification of the versal unfolding) that would certify exhaustiveness; without such a bound the contrast with simple singularities cannot be asserted.

    Authors: The manuscript presents the enumeration explicitly as conjectural and makes no claim of proven completeness. Consequently, no independent upper bound is supplied, as none is asserted. The stated contrast with simple singularities is likewise framed under the conjecture. We maintain that a conjectural list remains of value as the first explicit topological data in the non-semi-quasihomogeneous regime. revision: no

  2. Referee: [§4] §4 (homology computations): the assertion that H_1 is non-trivial for X_9^± and X_{10}^1 rests on unshown chain-level calculations; the text should exhibit the explicit simplicial complexes, software scripts, or hand-computed boundary matrices that yield the reported Betti numbers.

    Authors: The non-triviality of H_1 is asserted on the basis of explicit computations of the chain complexes associated to the cell decompositions of the complements. To address the request for transparency, we will add the relevant boundary matrices and a description of the simplicial complexes to the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained via direct topological classification

full rationale

The paper explicitly labels its main result as a 'conjecturally complete list' of connected components of discriminant complements for X_10 singularities and contrasts homology with simple classes. No equations, definitions, or citations are quoted that reduce any claimed prediction or completeness statement to a fitted parameter, self-definition, or prior self-citation by construction. The work relies on standard singularity stratification and real morsification techniques without renaming known results or smuggling ansatzes; the conjectural qualifier itself signals that exhaustiveness is not asserted as proven within the paper, keeping the derivation independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on abstract; no free parameters, axioms, or invented entities can be identified.

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Reference graph

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