Higher order double point formulas via SSM-Thom polynomials

Abstract

We study the geometry of double point loci of maps $F:M\to N$ of complex manifolds through the lens of Segre-Schwartz-MacPherson (SSM) classes. Classical double point formulas express the fundamental class of the closure of the double point locus of $F$ in terms of global invariants of source and target spaces, as well as $F$. In this paper we extend these results by computing a one-parameter cohomological deformation of the double point formula given by the SSM class. We compute the SSM class of the double point locus in a large cohomological degree range. The leading term in our new formulas recovers the classical double point formula of Fulton and Laksov, while higher-degree terms provide explicit universal corrections. Our approach uses interpolation techniques for SSM-Thom polynomials of multisingularities, recently developed by Koncki, Nekarda, Ohmoto and Rim\'anyi. We also compute SSM-Thom polynomials for the singularities $A_0$ and $A_1$ in the same range. As an application, we show how the deformed formulas yield refined geometric information about those singularity loci through a theorem of Aluffi and Ohmoto, including constraints on when such loci can arise as complete intersections.