A tropical framework for using Porteous formula

This paper establishes a tropical analogue of Porteous' formula by developing characteristic classes for tropical vector bundles on rational polyhedral spaces with boundary, utilizing a splitting principle and the boundary framework to ensure degeneracy loci achieve their expected codimension.

The Gist

Imagine you are an architect trying to understand the shape of a city. In the classical world (standard math), cities are made of smooth, curved roads and perfect buildings. But in the Tropical World, the city is made entirely of straight lines, sharp corners, and flat planes. It's like a city built out of cardboard boxes and rulers.

This paper, written by Andrew R. Tawfeek, is a guidebook for doing advanced geometry in this "cardboard city." Specifically, it solves a puzzle about how to count and measure specific "problem areas" in the city where things go wrong.

Here is the breakdown using simple analogies:

1. The Setting: The "Cardboard City" (Tropical Geometry)

In normal math, numbers are like water—they flow smoothly. In Tropical Geometry, numbers are like a grid of streets.

• Addition becomes taking the maximum (like choosing the highest floor in a building).
• Multiplication becomes addition (like stacking floors).
• The Boundary: The paper introduces a special feature: the edge of the city. In normal math, if you walk off the edge, you fall into nothingness. In this tropical city, the edge is a special "sedentary" zone. If you walk there, your tools (matrices) might break or lose power. This is crucial because it allows things to "fail" in a controlled way, creating the specific shapes the author wants to study.

2. The Tools: Tropical Vector Bundles

Think of a Vector Bundle as a giant, multi-story parking garage attached to every building in the city.

• Classical View: The garage has smooth ramps and perfect floors.
• Tropical View: The garage is made of straight, angular ramps.
• Sections: A "section" is like a delivery truck driving through the garage. The author focuses on trucks that stay within certain limits (bounded sections).

3. The Problem: The "Traffic Jam" (Degeneracy Loci)

Imagine you have a map (a morphism) that tells a truck how to move from one garage (Bundle E) to another (Bundle F).

• Usually, the truck moves smoothly.
• But sometimes, at certain spots in the city, the map breaks. The truck gets stuck, or the engine dies. In math terms, the "rank" of the map drops.
• The Degeneracy Locus is simply the collection of all these "stuck" spots. It's the "Traffic Jam Zone."

In classical math, we have a famous rule called Porteous' Formula that tells us exactly how big this Traffic Jam Zone is, just by looking at the blueprints of the garages (the bundles). The author asks: Does this rule work in our cardboard city?

4. The Solution: The "Magic Mirror" (The Splitting Principle)

The author proves that yes, the rule works, but it requires a special trick.

• The Problem: The garages in the cardboard city are complex and twisted. It's hard to calculate the size of the Traffic Jam directly.
• The Trick (Splitting Principle): The author builds a "Magic Mirror" (a new space called $Y$). When you look at the city through this mirror, every complex garage magically splits apart into simple, straight, single-lane garages (Line Bundles).
• Why it helps: It is much easier to calculate traffic jams in simple, straight garages. Once the author solves the puzzle in the mirror world, they use a "projection" to bring the answer back to the real city. Because the mirror is a perfect reflection (mathematically speaking), the answer is valid for the real city too.

5. The Big Result: The Tropical Porteous Formula

The paper proves a specific formula (Theorem 1.0.2) that acts like a calculator for Traffic Jams.

• Input: You give it the blueprints of the two garages (their "Chern classes," which are like the garage's ID card).
• Process: The formula uses a specific type of math determinant (a Sylvester determinant) to crunch the numbers.
• Output: It tells you the exact "volume" or "size" of the Traffic Jam Zone.

The Catch: This only works perfectly for the "Rank 0" case. This means the formula calculates the size of the zone where the map breaks completely (the truck stops moving entirely). The author sets the stage for future work to handle cases where the map only partially breaks (Rank 1, Rank 2, etc.).

6. Why Does This Matter? (The "Brill-Noether" Connection)

Why should anyone care about cardboard cities and traffic jams?

• The author hints at a connection to a famous unsolved problem in classical math called the Brill-Noether Conjecture. This problem is about counting how many ways you can arrange points on a curve (like beads on a string).
• In the classical world, mathematicians solved this by turning the problem into a "Traffic Jam" problem and using Porteous' Formula.
• The Goal: The author hopes that by mastering this Tropical Porteous Formula, they can solve the tropical version of this bead-counting problem. If they can, it might give new insights into the classical problem too.

Summary Analogy

Imagine you are trying to predict where a flock of birds will get stuck in a storm.

1. Classical Math: You use smooth fluid dynamics to predict the storm.
2. Tropical Math: You model the storm as a grid of wind tunnels.
3. The Paper: The author shows that even in this grid world, you can predict exactly where the birds will get stuck (the degeneracy locus) by looking at the shape of the wind tunnels (the bundles).
4. The Method: They use a "Magic Mirror" to turn the complex wind tunnels into simple straight tubes, solve the puzzle there, and translate the answer back.
5. The Future: They hope this method will help solve a giant puzzle about how birds migrate (the Brill-Noether conjecture).

In short, this paper builds the mathematical machinery to measure "failures" in a geometric world made of straight lines and sharp corners, paving the way to solve deep problems in geometry.

Technical Summary

Here is a detailed technical summary of the paper "A Tropical Framework for Using Porteous Formula" by Andrew R. Tawfeek.

1. Problem Statement

In classical algebraic geometry, Porteous' Formula provides a method to calculate the fundamental class of a degeneracy locus $D_k(\phi)$—the set of points where a morphism of vector bundles $\phi: E \to F$ drops rank to $k$ or below. The formula expresses this class as a determinantal polynomial (Sylvester determinant) in the Chern classes of $E$ and $F$.

The paper addresses the challenge of adapting this theory to tropical geometry. Key obstacles include:

• Lack of Boundary in Standard Tropical Geometry: In standard tropical affine space $\mathbb{R}^n$, continuous functions cannot "drop" to $-\infty$ in a way that creates degeneracy loci of the expected codimension without artificial constraints.
• Rank Definition: Defining the rank of a tropical matrix and ensuring it behaves semicontinuously is non-trivial.
• Chern Classes: Establishing a robust theory of Chern classes for tropical vector bundles that supports intersection theory and splitting principles.

The author aims to construct a framework where degeneracy loci naturally acquire the expected codimension and where Porteous' formula holds, specifically proving the rank-0 case (where the morphism becomes identically $-\infty$).

2. Methodology and Framework

The paper builds its theory on the Rational Polyhedral Space framework developed by Mikhalkin and Rau (MR18), which extends tropical geometry to the extended tropical affine space $\mathbb{T}^n = (\mathbb{R} \cup \{-\infty\})^n$.

A. The Boundary Mechanism

The core innovation is utilizing the sedentary strata (boundary faces where coordinates are $-\infty$) of $\mathbb{T}^n$.

• Sedentarity: Points $p$ have a sedentarity order based on how many coordinates are $-\infty$.
• Rank Drop: A morphism $\phi: E \to F$ is represented locally by a matrix of regular invertible functions. As one approaches a sedentary stratum, these functions can degenerate to $-\infty$. This causes the tropical rank of the matrix to drop, creating degeneracy loci with positive codimension naturally, without artificial truncation.

B. Tropical Vector Bundles

The author defines tropical vector bundles of rank $r$ over a rational polyhedral space $X$ using:

• Transition Functions: Maps $M_{ij}: U_i \cap U_j \to G(r)$, where $G(r)$ is the group of invertible tropical matrices (permutation matrices with entries in $\mathbb{R}$ and $-\infty$ elsewhere).
• Bounded Rational Sections: The theory focuses on bundles admitting bounded rational sections. This restriction is crucial for defining well-behaved Chern classes and ensuring the intersection product is well-defined.
• Splitting: The paper establishes that every short exact sequence of tropical vector bundles splits (Proposition 3.2.5), a property that simplifies the construction of characteristic classes.

C. Chern Classes and Intersection Theory

• Definition: Chern classes $c_k(F)$ are defined via the intersection product of bounded rational sections with subcycles.
• Properties: The author proves the Tropical Whitney Formula ($c(E \oplus F) = c(E)c(F)$) and functoriality properties.
• Splitting Principle: A Tropical Splitting Principle (Theorem 4.0.10, Corollary 5.1.9) is proven. By pulling back a bundle $E$ to a space $Y$ (constructed via iterated projectivization), $E$ splits into a direct sum of line bundles. This allows computations with "virtual Chern roots" in $A(Y)$, which descend to $A(X)$ due to the injectivity of the pullback map.

3. Key Contributions and Results

A. Tropical Porteous Formula (Rank 0)

The main result is Theorem 6.4.1, which establishes the Porteous formula for the case where the rank drops to 0 (the morphism becomes the zero matrix, i.e., all entries are $-\infty$).

• Setup: Let $\phi: E \to F$ be a morphism of tropical vector bundles of ranks $e$ and $f$ over $X$.
• Degeneracy Locus: $D_0(\phi) = \{p \in X \mid \text{troprank}(\phi_p) = 0\}$.
• Formula: If $D_0(\phi)$ has the expected codimension $ef$, then its fundamental class is:
  $$[D_0(\phi)] = \Delta_e^f \left( \frac{c[t](F)}{c[t](E)} \right)$$
  where $\Delta_e^f$ is the Sylvester determinant of the ratio of the Chern polynomials $c[t](F)$ and $c[t](E)$.

B. Proof Strategy

The proof relies on three pillars:

1. Hom Bundle Identification: The morphism $\phi$ is identified with a bounded rational section $\sigma_\phi$ of the bundle $E^\vee \otimes F$. The locus $D_0(\phi)$ corresponds to the zero locus of this section.
2. Chern Class Calculation: The class of the zero locus is given by the top Chern class $c_{ef}(E^\vee \otimes F)$.
3. Splitting Principle: Using the splitting principle, $E$ and $F$ are treated as sums of line bundles with Chern roots $\alpha_i$ and $\beta_j$. The top Chern class of the tensor product becomes the product $\prod_{i,j} (\beta_j - \alpha_i)$, which is shown to equal the Sylvester determinant via Lemma 6.1.2.

C. Tropical Projectivization

The paper constructs the tropical projectivization $TP(E)$ of a bundle $E$. This space serves as the tropical analogue of the projective bundle in classical geometry. It is used to prove the Splitting Principle by iteratively reducing the rank of the bundle until it splits into line bundles.

4. Significance and Future Directions

• Foundational Step: This work provides the first rigorous derivation of a Porteous-type formula in tropical geometry, bridging the gap between tropical intersection theory and classical enumerative geometry.

• Mechanism for Codimension: It demonstrates that the "boundary" in the Mikhalkin–Rau framework is not just a technicality but the essential mechanism allowing degeneracy loci to have the correct dimension.

• Applications:

  • Brill-Noether Theory: The author suggests that this formula could be applied to the Tropical Brill-Noether Conjecture (Conjecture 7.2.1). In classical geometry, Brill-Noether loci are degeneracy loci of evaluation maps; a tropical Porteous formula would allow computing their fundamental classes.
  • Enumerative Geometry: It opens the door to computing counts of tropical curves and other objects defined by rank conditions.

• Future Work (Higher Rank): The paper outlines the path to the full Porteous formula for $D_k(\phi)$ where $k > 0$. This requires constructing a Tropical Grassmann Bundle (analogous to the classical Grassmannian bundle) to reduce the $k$-rank case to the $0$-rank case. The author identifies the construction of this bundle as the primary remaining obstacle.

Conclusion

Andrew R. Tawfeek successfully adapts the machinery of vector bundles, Chern classes, and the splitting principle to the tropical setting using rational polyhedral spaces with boundaries. The resulting Tropical Porteous Formula (Rank 0) confirms that tropical geometry can replicate deep structural results from classical algebraic geometry, provided the correct framework (incorporating sedentary strata) is used. This lays the groundwork for solving complex enumerative problems in tropical geometry, such as the tropical Brill-Noether problem.