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

The complete list of connected components of the set of Morse functions in the deformations of function singularities of class $J_{10}$ is given. Thus, the isotopy classification of Morse perturbations of parabolic real function singularities is finished.

💡 Research Summary

The paper completes the isotopy classification of Morse perturbations of real parabolic function singularities by treating the remaining class J₁₀ (and its sub‑class J₃₀). A real function f : ℝ²→ℝ has a J₁₀ (or J₃₀) singularity at the origin if, after a suitable change of coordinates, it can be written as (x² ± y⁴)(x − γy²) with Milnor number 10; the parameter γ runs over ℝ for J₁₀ and over (−1, 1) for J₃₀. The authors consider the 16‑dimensional spaces Φ₁ and Φ₃ of all real polynomials whose principal quasihomogeneous part belongs to J₁₀ or J₃₀, respectively. A six‑parameter group G of affine‑quadratic diffeomorphisms of ℝ² acts on these spaces, and each G‑orbit meets the normal‑form families (1) and (2) transversally in a single point. Consequently, the connected components of the Morse‑function subsets of Φ₁ and Φ₃ are in one‑to‑one correspondence with the connected components of the parameter spaces of the normal forms.

To distinguish these components the authors introduce a combinatorial invariant based on a graph. Vertices encode the “passport” of a Morse function – the numbers of minima, saddles and maxima (m₋, mₓ, m₊) – which, because of index constraints, can be reduced to the pair (m₊, M) where M=m₋+mₓ+m₊≤10. Edges correspond to six elementary surgeries (s₁–s₆) that model the standard local topological changes of critical values (collision of two real values, collision of a complex‑conjugate pair, birth/death of a pair, etc.). By removing those edges that can change the isotopy class, the graph splits into subgraphs; each subgraph represents a distinct isotopy class.

A crucial tool is the Lyashko–Looijenga map, which sends a complexified polynomial to the unordered set of its critical values, i.e. a point of Sym¹⁰(ℂ). On the locus of polynomials with ten distinct critical values this map is a covering of the configuration space B(ℂ, 10). Restricting to the real subspaces (1) and (2) yields maps whose images are invariant under complex conjugation. Near generic points these maps are local diffeomorphisms, allowing the authors to translate the topology of the Morse‑function spaces into combinatorial data of point configurations and to read off the graph invariant.

The main results are Theorems 14–16, which state that there are exactly 59 isotopy classes for J₁₀ and 56 for J₃₀. For each class the authors exhibit a concrete polynomial representative. They also provide a “splitting table” (Table 2) that records which pairs of simple singularities (Ξ, ˜Ξ) with µ(Ξ)+µ(˜Ξ)=10 can be realized as a degeneration of J₁₀ or J₃₀; this is the real analogue of the complex classification in