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Higher order double point formulas via SSM-Thom polynomials

This paper extends classical double point formulas by computing Segre-Schwartz-MacPherson (SSM) classes for double point loci and $A_0, A_1$ singularities using interpolation techniques for SSM-Thom polynomials, thereby providing a cohomological deformation with universal corrections that yield refined geometric insights into these singularity loci.

Original authors: Reese Lance

Published 2026-01-27

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Reese Lance

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a cartographer trying to map a complex journey. You have a starting point (a shape called M) and a destination (a shape called N). You draw a path, or a map (F), connecting them.

Sometimes, this path gets messy. It might loop back on itself, crossing over the same spot in the destination multiple times. In mathematics, these "crossing points" are called double points (where the path crosses itself once) or multiple points (where it crosses itself many times).

For decades, mathematicians had a "classic rulebook" (the Fulton-Laksov formula) to count these crossing points. It worked like a simple calculator: if you knew the size and shape of your starting and ending maps, you could plug in the numbers and get the total count of crossings. It was a universal law, much like gravity, that didn't care about the specific details of the journey, only the general dimensions.

The Problem:
The classic rulebook gave you the total count, but it was a bit blunt. It treated the crossing points as a single, flat number. It didn't tell you about the texture of the crossing. Is the crossing a simple "X"? Is it a messy knot? Does the path wiggle differently at the crossing? The classic formula missed these subtle, higher-order details.

The New Discovery:
Reese Lance, in this paper, introduces a "super-calculator" called SSM-Thom polynomials. Think of this as upgrading from a basic ruler to a high-definition 3D scanner.

The "One-Parameter Deformation":
Imagine the classic formula is a black-and-white photo. Lance's new formula is a color photo with depth. It takes that same basic count but adds a "knob" (a parameter) that lets you see the layers underneath.
- The Leading Term: If you turn the knob all the way down, you get the classic, black-and-white result (the old formula).
- The Higher Terms: As you turn the knob up, you see "corrections." These are extra terms that describe the fine geometry of the crossing. They tell you not just how many times the map crossed itself, but how it crossed, revealing hidden geometric information that the old formula ignored.
The "Interpolation" Trick:
How did the author calculate these complex new formulas? He used a technique called interpolation.
- The Analogy: Imagine you want to know the exact shape of a mysterious, invisible mountain. You can't see the whole thing, but you can send out drones to measure the height at a few specific, known points (called "prototypes" or "test cases").
- The Method: The author found a few specific, simple maps where he knew exactly what the crossing points looked like. He calculated the "SSM class" (the high-definition measurement) for these simple cases.
- The Magic: Because these formulas are "universal" (they work the same way for all maps of a certain type), he could use the data from these few simple test cases to "fill in the blanks" and reconstruct the formula for any complex map. It's like guessing the shape of a whole mountain range just by measuring three specific peaks.
The Results:
The author successfully calculated these new, detailed formulas for the most common types of crossings (called singularities A₀, A₁, and A₂₀).
- He showed that for a specific range of complexity, these formulas are precise and complete.
- He provided a list of these new "correction terms" (the higher-degree parts of the formula) for different dimensions.
The Application (The "So What?"):
The paper doesn't just stop at the math; it uses these new formulas to answer a specific geometric question: Can these crossing points form a "Complete Intersection"?
- The Analogy: Imagine a sculpture. A "complete intersection" is a sculpture made by simply slicing a block of cheese with a few flat knives. It's a very specific, clean type of shape.
- The Finding: Using the new, high-definition formulas, the author proved that for certain types of maps, the crossing points cannot be simple "knife-slice" shapes. They are too complex and "wiggly" to be formed that way. The new formulas provided the extra data needed to prove this impossibility, which the old, blunt formula could not have detected.

In Summary:
This paper takes an old, reliable rule for counting self-crossing paths and upgrades it into a high-resolution tool. By using a clever "guess-and-check" method on simple test cases, the author derived new formulas that reveal the hidden, detailed geometry of these crossings. This new level of detail allows mathematicians to prove that certain crossing patterns are too complex to be formed by simple geometric cuts, refining our understanding of how shapes interact in higher dimensions.

Technical Summary: Higher order double point formulas via SSM–Thom polynomials

Problem Statement
The paper addresses the problem of describing the geometry of double point loci for maps $F: M \to N$ between complex manifolds. Classically, the fundamental class of the double point locus $\Delta(F)$ (the set of points where the fiber has exactly two preimages) is expressed via the Fulton-Laksov formula in terms of global invariants (Chern classes) of the source, target, and the map. The central question is whether this classical formula can be extended to capture higher-order geometric information. Specifically, the author seeks to compute the Segre-Schwartz-MacPherson (SSM) class of the double point locus. Unlike the fundamental class, the SSM class is an inhomogeneous cohomology class whose leading term is the fundamental class, with higher-degree terms providing "universal corrections" that encode refined topological data, such as Euler characteristics of linear sections.

Methodology
The paper employs the theory of SSM-Thom polynomials and interpolation techniques developed recently by Koncki, Nekarda, Ohmoto, and Rimányi. The methodology proceeds as follows:

Framework of Multisingularities: The double point locus is treated as a specific "multisingularity" type, denoted $A_0^2$ (or $A_0 A_0$ ), representing the concatenation of two simple singularities ( $A_0$ ). The study is restricted to $C_\ell$ -maps (target-locally stable maps with finite fibers), where $\ell = \dim(N) - \dim(M)$ .
Thom Principle: The author utilizes the Thom principle, which posits that the SSM class of a singularity locus for any stable map is determined by evaluating a universal power series (the SSM-Thom polynomial) on the characteristic classes of the map. For multisingularities, this polynomial depends on two sets of variables: Chern classes ( $c$ ) and Landweber-Novikov classes ( $s$ ).
Interpolation: To compute these universal polynomials, the paper uses an interpolation theorem (Proposition 3.1 from [KR25]). This theorem states that an SSM-Thom polynomial is uniquely determined, up to a specific cohomological degree (the Mather bound), by its values on a finite set of "prototype" maps. These prototypes are specific germs representing the singularity types.
Explicit Computation: The author constructs prototypes for the singularities $A_0$ , $A_0^2$ , and $A_1$ . By evaluating the proposed polynomial formulas on these prototypes and verifying that the equality holds in equivariant cohomology, the coefficients of the SSM-Thom polynomials are determined.

Key Contributions and Results

Deformation of the Double Point Formula: The primary result (Theorem 4.5) provides an explicit computation of the SSM-Thom polynomial for the double point locus ( $A_0^2$ ) in a large cohomological degree range ( $\ell \le \text{deg} < 2\ell$ ).
- The leading term of the computed SSM class recovers the classical Fulton-Laksov double point formula ( $s_\emptyset - c_\ell$ ).
- The higher-degree terms provide explicit, universal corrections involving Chern classes ( $c_i$ ) and Landweber-Novikov classes ( $s_\lambda$ ).
- The paper lists explicit formulas for $\ell = 1, 2, 3, 4$ , showing the structure of these corrections.
Computation of Related Singularities: In the same degree range, the paper computes the SSM-Thom polynomials for the simple singularity $A_0$ (Theorem 4.6) and the singularity $A_1$ (Corollary 4.7).
- A key proposition (4.1) establishes that for $C_\ell$ -maps within this degree bound, the sum of the SSM classes of $A_0$ , $A_0^2$ , and $A_1$ loci equals the fundamental class of the domain (normalized to 1). This allows the determination of one polynomial from the other two.
Application to Projective Spaces: The paper applies these deformed formulas to maps between projective spaces $F: \mathbb{P}^{\ell+1} \to \mathbb{P}^{2\ell+1}$ of degree $d$ .
- Using a theorem by Aluffi and Ohmoto (Theorem 5.1), which relates the SSM class of a variety to the Euler characteristics of its general linear sections, the author computes the topological Euler characteristics ( $\chi$ ) and degrees of the singularity loci $A_0(F)$ , $A_0^2(F)$ , and $A_1(F)$ .
- The results are expressed explicitly in terms of the degree $d$ and relative dimension $\ell$ .
Constraints on Complete Intersections: As a specific geometric application, the paper investigates whether the double point locus $A_0^2(F)$ can be a complete intersection.
- By comparing the computed Euler characteristic and degree of the locus against the theoretical constraints for complete intersections, the author demonstrates (Example 5.5) that for $\ell=2$ and $d > 1$ , the locus cannot be a complete intersection. The system of equations required for it to be a complete intersection is shown to be inconsistent.

Significance and Claims
The paper claims to extend the classical enumerative geometry of double points by incorporating the machinery of SSM classes. The significance lies in:

Universality: The derived formulas are "universal," meaning they apply to any map with controlled singularity types ( $C_\ell$ -maps) regardless of the specific source and target manifolds, depending only on their characteristic classes.
Refined Geometry: The higher-order terms in the SSM-Thom polynomials provide geometric information (Euler characteristics of linear sections) that is inaccessible via the classical fundamental class alone.
Algorithmic Approach: The work demonstrates the efficacy of the interpolation technique for multisingularities, moving beyond monosingularities to compute complex universal polynomials.
Geometric Constraints: The application to projective spaces yields concrete constraints on the geometry of singularity loci, specifically ruling out the complete intersection property for certain double point loci, thereby offering new insights into the structure of these self-intersection sets.

The author notes that while the interpolation method allows for the computation of terms up to the Mather bound (approximately $6\ell + 7$ ), it does not currently yield a closed formula for the full power series beyond this bound. The results presented are the first explicit higher-order deformations of the double point formula in this cohomological range.

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