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Complements of caustics of the real $J_{10}$ singularities

V. A. Vassiliev

TL;DR

This work resolves the isotopy classification of Morse perturbations for the real parabolic $J_{10}$ singularities, enumerating 59 isotopy classes for $J_{10}^1$ and 56 for $J_{10}^3$. It develops a robust invariant framework based on virtual Morse functions, D-graphs, and an up-down involution, and leverages Lyashko–Looijenga coverings and the Gusein-Zade–A'Campo method to perform computer-assisted enumeration of virtual components. The authors realize many classes with explicit polynomials, analyze chirality via a cocycle on the virtual graph, and establish adjacencies to simple real singularities, yielding a detailed tabulation of realizable splittings and their Dynkin decompositions. The results extend the established program for $X_9$ and $P_8$ to the $J_{10}$ family, offering a complete combinatorial and geometric account of Morse perturbations in this parabolic setting.

Abstract

The isotopy classification of Morse perturbations of parabolic real function singularities is completed

Complements of caustics of the real $J_{10}$ singularities

TL;DR

This work resolves the isotopy classification of Morse perturbations for the real parabolic singularities, enumerating 59 isotopy classes for and 56 for . It develops a robust invariant framework based on virtual Morse functions, D-graphs, and an up-down involution, and leverages Lyashko–Looijenga coverings and the Gusein-Zade–A'Campo method to perform computer-assisted enumeration of virtual components. The authors realize many classes with explicit polynomials, analyze chirality via a cocycle on the virtual graph, and establish adjacencies to simple real singularities, yielding a detailed tabulation of realizable splittings and their Dynkin decompositions. The results extend the established program for and to the family, offering a complete combinatorial and geometric account of Morse perturbations in this parabolic setting.

Abstract

The isotopy classification of Morse perturbations of parabolic real function singularities is completed

Paper Structure

This paper contains 18 sections, 18 theorems, 31 equations, 25 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Main objects and definitions
  3. Invariants of isotopy classes of Morse polynomials
  4. Trivial invariant
  5. Set-valued invariant and virtual Morse functions
  6. D-graph invariant
  7. The up-down involution
  8. Enumeration of virtual components
  9. On the proofs of enumeration theorems
  10. Normalization of $D$-graphs
  11. Proof of Proposition \ref{['uniq']}
  12. Realization of some virtual Morse functions by polynomials
  13. $\Phi_1$
  14. $\Phi_3$
  15. Chirality of virtual Morse functions. Proof of Theorem \ref{['cher']}
  16. ...and 3 more sections

Key Result

Proposition 1

Each orbit of the action of the group ${\mathfrak G}$ on the space $\Phi_1$$($respectively, $\Phi_3)$ intersects the space of all polynomials of type $($vers1$)$$($respectively, $($vers3$))$ transversally at a single point.

Figures (25)

  • Figure 1: Standard systems of paths
  • Figure 2: Virtual Morse functions
  • Figure 3: $\Phi_1$, one maximum (122298)
  • Figure 4: $\Phi_1$, two maxima (26378)
  • Figure 5: 97702 (left) and 93489 (right)
  • ...and 20 more figures

Theorems & Definitions (39)

  • Definition 1
  • Proposition 1
  • Definition 2: cf. Jaw2, § 2
  • Theorem 2: see § \ref{['sectlast']}
  • Definition 3
  • Definition 4: see AGLV2, § V.3
  • Definition 5
  • Remark 3
  • Definition 6
  • Remark 4
  • ...and 29 more