Non-associative Hilbert scheme and Thom polynomials

Abstract

Thom polynomial describes the cohomology class Poincar\'e dual to the locus of particular singularity of a generic holomorphic map. In this paper we derive a closed formula for the generating function of its coefficients. The method is based on a new construction of the embedding space of punctual Hibert scheme that we call the non-associative Hilbert scheme. The efficiency of the method is demonstrated on explicit computation of a number of Thom polynomials, including those associated with singularities of Thom-Boardman types~$\Sigma^{i,j}$, $\Sigma^{2,1,1}$, $\Sigma^{2,2,1}$, and~$\Sigma^{2,2,2}$.