Thom polynomials in $\mathcal{A}$-classification I: counting singular projections of a surface

TL;DR

This paper develops universal polynomials related to the $\

Contribution

It introduces systematic methods for counting singular projections of surfaces using Thom polynomials in the context of $\

Findings

Derived explicit formulas for counting singular projections.

Connected Thom polynomials to classical enumerative geometry.

Provided tools for enumerating lines with prescribed contact.

Abstract

We study universal polynomials of characteristic classes associated to the $\mathcal{A}$-classification (i.e. up to right-left equivalence) of holomorphic map-germs $(\mathbb{C}^2,0) \to (\mathbb{C}^n, 0)$ $(n=2,3)$. That enables us to systematically treat with classical enumerative problems of lines of prescribed contact with a given projective surface in $3$ and $4$-spaces.