Thom series in negative relative codimension

Abstract

We develop a theory of Thom series for contact function singularities. Quadratic stabilization $σ_q$ of a contact singularity $η\in J^k(n,1)$ gives contact singularities $σ_q^{i}η\in J^k(n+i,1)$. We study the stable Thom polynomials (i.e.\ the Thom polynomial in quotient variables) of $σ_q^{i}η$ as $i$ increases. To this end, we define the Thom series of a contact function singularity $η$, which for any given $i$ is a linear combination of Schur polynomials $s_λ$. For large enough $i$ the value of the Thom series at $1-(n+i)$ is the stable Thom polynomial of $σ_q^iη$ and for small $i$ it determines its coefficients outside the kernel of a specialization map. We prove that the Thom series has two main properties: 1) as $i$ increases, the partitions $λ$ follow a simple stabilization pattern, and there is a finite set of $λ$ which generates the support of the entire Thom series via this stabilization; 2) the coefficients of the $s_λ$ are polynomials in $i$ with explicit degree bounds. We describe the precise relationship between unstable and stable Thom polynomials of contact function singularities and Legendre Thom polynomials and we carry out computations of all three, as follows. We compute the complete family of stable Thom polynomials for function singularities with $γ\leq 6$. We compute unstable and Legendre Thom polynomials for several families of binary and ternary singularities. We define a class of multi-binary singularities, and compute their Thom polynomials. We discuss second order Thom-Boardman classes, and indicate difficulties that arise beyond function singularities. The results of the paper will be used in a companion paper, where Thom polynomials will be applied to problems in enumerative geometry.