Interpolation characterization of higher Thom polynomials

TL;DR

This work presents a symmetry-driven, interpolation-based method to compute $SSM$-Thom polynomials, extending Ohmoto's framework by leveraging stable-envelope-inspired axioms constrained by a degree bound. It reframes Thom polynomials as universal invariants that, via embeddings into equivariant cohomology and the ratio of Chern classes, yield the full Segre-Schwartz-MacPherson data for singularity loci. The main contribution is an interpolation theorem that reduces the computation to a finite set of prototypes determined by local algebras and their symmetries, enabling automatic degree-bounded calculations and explicit examples. The approach supports practical applications in enumerative geometry, the analysis of singularity hierarchies, and the comparison of complex vs real singularities, while offering new algebraic insight through expansions in Schur-type bases and exploring positivity and stabilization phenomena.

Abstract

Thom polynomials provide universal formulas for the fundamental class of singularity loci in terms of characteristic classes. Ohmoto extended this notion to SSM-Thom polynomials, which refine this description by capturing the richer Segre-Schwartz-MacPherson (SSM) class of singularity loci. While previous methods for computing SSM-Thom polynomials relied on intricate geometric arguments, we introduce a more efficient approach that depends solely on the symmetries of singularities. Our method is inspired by connections to Geometric Representation Theory, particularly the interpolation properties of Maulik-Okounkov stable envelopes. By formulating SSM analogs of these axioms within a degree-bounded framework, we obtain new computational tools for SSM-Thom polynomials. We also present explicit examples of SSM-Thom polynomials, and illustrate their applications in enumerative geometry and singularity theory.

Figures (5)

• Figure 1: Classification of contact singularities for $\ell=0$ in the nice dimensions. Edges indicate the covering relation of the adjacency hierarcy.
• Figure 2: Classification and hierarchy of contact singularities for $\ell=1$ in the nice dimensions. Notation: $b_{ij}=III_{ij}$, $b'_{ij}=I_{ij}$, $c_1=(x^2,xy^2,y^3)$, $c_2=(x^2,y^3)$, $c_3=(x^2+y^3,xy^2,y^4)$, $c_4=(x^2+y^3,xy^2)$, $c_5=(x^2,xy^2,y^4)$, $d_1=(x^2+y^2+z^2,xy,xz,yz)$, $d_2=(x^2,y^2,z^2,xy+xz)$, $d_3=(x^2-y^2+z^3,xy,xz,yz)$, $d_4=(x^2+yz,xz,y^2,z^2)$.
• Figure 3: Classification of contact singularities for $\ell=2$ up to codimension 15. Here $D$ stands for $(x^2-y^2,x^2-z^2,xy,xz,yz)$. The classification is finite up to codimension $M(3)=20$. The classification of those with codimension$\in[16,20]$ are not shown.
• Figure 4: SSM-Thom polynomials for $l=0$ up to degree 5
• Figure 5: SSM-Thom polynomials for $l=1$ up to degree 6

Theorems & Definitions (27)

• Remark 3.1
• Theorem 3.2
• Remark 3.3
• Example 3.4
• Example 3.5
• Example 3.6
• Example 3.7
• Example 3.8
• Theorem 3.9
• Theorem 4.1: and Definition