Higher order double point formulas via SSM-Thom polynomials
Reese Lance
TL;DR
The paper develops a one-parameter deformation of the classical Fulton–Laksov double point formula by computing the Segre-Schwartz-MacPherson (SSM) class of the double point locus, yielding explicit universal higher-order corrections in a large cohomological range. It systematizes interpolation for SSM-Thom polynomials of multisingularities and computes the polynomials for $A_0^2$, $A_0$, and $A_1$, expressing them in terms of quotient Chern data $\underline{c}$ and Landweber–Novikov classes $\underline{s}$; the leading term recovers the classical formula while higher terms provide new universal corrections. The authors then apply these results to maps of projective spaces, using the Thom principle and Aluffi–Ohmoto’s linear-section theory to extract refined geometric information about singularity loci, including Euler characteristics and degrees, and deduce constraints showing that certain singular loci cannot be complete intersections. Overall, the work provides a method to obtain higher-order, universal information about singularity loci of maps, with concrete computations that yield new geometric constraints and invariants. The approach is algorithmic in the sense of the interpolation framework, enabling systematic determination of SSM data for stable maps in the specified degree range.
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 Rimányi. 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.