🔢 mathematics

Hessenberg varieties of codimension one in the flag variety

This paper investigates the geometric and topological properties of codimension one Hessenberg varieties in type A flag varieties, providing formulas for their Poincaré polynomials, characterizing their irreducibility, proving they are reduced schemes with Hessenberg singular loci, and extending these findings to all codimension one Schubert varieties.

Original authors: Laura Escobar, Martha Precup, John Shareshian

Published 2026-04-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Laura Escobar, Martha Precup, John Shareshian

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 standing in a vast, multi-dimensional city called the Flag Variety. This city is made up of all possible ways to arrange a stack of nested boxes (like Russian dolls) inside a larger room. Each "building" in this city represents a specific arrangement of these boxes.

Now, imagine a mysterious force, represented by a matrix $x$ (think of it as a specific rule or a "lens" that distorts the view of the room). We are interested in a special neighborhood within this city called a Hessenberg Variety. This neighborhood contains only the arrangements of boxes that obey a specific rule set by our matrix $x$ and a "safety zone" called $H$ .

The authors of this paper are focusing on a very specific type of neighborhood: one that is exactly one dimension smaller than the whole city. Think of the city as a 3D volume; this neighborhood is a 2D surface floating inside it.

Here is what the authors discovered, translated into everyday concepts:

1. The "Point Counting" Trick (The Magic Mirror)

One of the biggest headaches in math is figuring out the shape and "holes" of these complex neighborhoods. Usually, you have to do incredibly difficult calculus or topology.

The authors found a magical shortcut. They realized that instead of calculating the shape in the complex, infinite world of numbers, you can shrink the problem down to a tiny, finite world (like a video game with a limited number of pixels).

The Analogy: Imagine you want to know the surface area of a giant, complex sculpture. Instead of measuring it directly, you build a tiny, pixelated version of it using Legos. If you count how many Legos you need, you can mathematically predict the exact surface area of the giant sculpture.
The Result: They proved that if you count the number of points in this "finite Lego version" of the neighborhood, you can instantly calculate the Poincaré polynomial. This polynomial is like a "DNA report" for the shape; it tells you exactly how many holes, tunnels, and loops the shape has.

2. When is the Neighborhood One Piece? (Irreducibility)

Sometimes, a neighborhood might be two separate islands floating next to each other, rather than one connected landmass. The authors wanted to know: When is this neighborhood all one piece?

The Discovery: They found that the neighborhood is broken into pieces (reducible) only if the matrix $x$ is very "lopsided." Specifically, if $x$ acts like a machine that squashes almost everything down to a single line (rank one), the neighborhood splits apart.
The Metaphor: Imagine a rubber sheet. If you pull it evenly, it stays one piece. But if you pinch it so hard that it almost tears into two separate strips, the shape changes. The authors gave the exact formula for when that "pinch" happens.

3. The "Smoothness" Check (Singularities)

In these mathematical cities, some spots are smooth (like a polished marble floor), while others are jagged or sharp (like a crumpled piece of paper). These jagged spots are called singularities.

The Discovery: The authors mapped out exactly where the jagged spots are. They found that the "rough" part of the neighborhood is actually a smaller, simpler version of the same type of neighborhood!
The Metaphor: If the main city is a mountain range, the jagged spots are just a smaller, specific valley within that range. They proved that if you zoom in on the "rough" part, it looks exactly like another Hessenberg variety with slightly different rules. This is a huge deal because it means we can study the rough parts using the same tools we use for the smooth parts.

4. The "No Cracks" Guarantee (Reduced Schemes)

In advanced math, sometimes a shape can be "fuzzy" or "double-layered" in a way that isn't visible to the naked eye but exists in the equations. This is called being "non-reduced." It's like a ghost building that exists in the math but not in reality.

The Discovery: The authors proved that for these specific codimension-one neighborhoods, there are no ghosts. The mathematical equations perfectly describe a solid, real shape.
The Metaphor: It's like proving that a blueprint for a house doesn't have any "phantom rooms" that only exist on paper. The house is exactly as solid as it looks.

5. Connecting to the "Schubert" Neighborhoods

Finally, they connected their findings to another famous type of neighborhood in this city called Schubert varieties. Mathematicians have long wondered which Schubert neighborhoods look exactly like Hessenberg neighborhoods.

The Result: They solved a puzzle that had been partially solved before. They showed that for most "edge" neighborhoods in the city, the answer is "No, they aren't the same." They only match up in very specific, rare cases. This helps mathematicians organize the entire city map more accurately.

Summary

In short, this paper is like a surveyor's guide for a very specific, thin slice of a complex mathematical city. The authors:

Gave a shortcut to measure the shape's complexity (using finite counting).
Told us when the shape breaks apart.
Mapped the rough spots and showed they are just smaller versions of the same shape.
Confirmed the shape is solid and real, not a mathematical ghost.
Clarified how this shape relates to its famous neighbors, the Schubert varieties.

Their work turns a chaotic, high-dimensional mess into a clean, understandable map, allowing other mathematicians to navigate this part of the city with confidence.

1. Problem Statement and Context

The paper investigates Hessenberg varieties within the type A flag variety ( $B = GL_n(\mathbb{C})/B$ ). Specifically, the authors focus on varieties of codimension one.

Definitions:
- Let $G = GL_n(\mathbb{C})$ and $B$ be the Borel subgroup of upper triangular matrices.
- A Hessenberg space $H \subseteq \mathfrak{g} = \mathfrak{gl}_n(\mathbb{C})$ is a $B$ -invariant subspace containing the Borel subalgebra $\mathfrak{b}$ . These are determined by a Hessenberg vector $m = (m(1), \dots, m(n))$ where $m(i) \geq i$ and $m$ is weakly increasing.
- The Hessenberg variety $B(x, H)$ for $x \in \mathfrak{g}$ is defined as $\{gB \in B \mid g^{-1}xg \in H\}$ .
The Specific Case: The authors concentrate on the case where $H = H(m_{\max})$ with $m_{\max} = (n-1, n, \dots, n)$ . This is the unique maximal proper $B$ -invariant subspace of $\mathfrak{g}$ .
The Problem: When $x$ $x$ is not a scalar matrix, $B(x, H(m_{\max}))$ $B (x, H (m_{m a x}))$ has codimension one in $B$ $B$ . The paper aims to characterize the geometric and topological properties of these specific varieties, including:
1. Computing their Poincaré polynomials.
2. Determining conditions for irreducibility.
3. Identifying their singular loci.
4. Establishing whether they are reduced schemes.
5. Determining which codimension-one Schubert varieties are isomorphic to these Hessenberg varieties.

2. Methodology

The authors employ a blend of combinatorial geometry, algebraic geometry, and finite field arithmetic.

The Point Count Heuristic (Main Tool):
The central innovation is Theorem 1, which establishes a link between the topology of a complex Hessenberg variety and the arithmetic of a corresponding variety over a finite field.
- They utilize Tymoczko's result that Hessenberg varieties admit an affine paving (decomposition into affine cells), implying their cohomology is concentrated in even degrees.
- They prove that for a matrix $x$ with integer entries in "highest form and permuted Jordan form" (HFPJF), and for a sufficiently large prime $p$ , the number of points in the variety over $\mathbb{F}_p$ equals the Poincaré polynomial evaluated at $p^{1/2}$ :
  $|B_p(\bar{x}, \bar{H}(m))| = \text{Poin}(B(x, H(m)); p^{1/2})$
- This allows the computation of topological invariants (Betti numbers) by simply counting points in finite fields, avoiding complex case-by-case topological analysis.
Affine Patch Analysis:
To study singularities and scheme-theoretic properties, the authors analyze local affine patches.
- They use the unipotent radical $U_-$ of the opposite Borel subgroup to define open neighborhoods $N_g \cong \mathbb{C}^{\binom{n}{2}}$ .
- The intersection of the Hessenberg variety with these patches is shown to be isomorphic to an affine hypersurface defined by the vanishing of a specific determinant, $\det(A_g)$ .
- Smoothness is determined by the Jacobian criterion applied to this determinant.
Combinatorial Enumeration:
For the codimension-one case, they explicitly enumerate flags in finite fields satisfying specific linear constraints imposed by the matrix $x$ and the Hessenberg space $H(m_{\max})$ .

3. Key Contributions and Results

A. Poincaré Polynomial Formula (Theorem 3)

The authors derive a closed-form formula for the Poincaré polynomial of $B(x, H(m_{\max}))$ based on the eigenvalues of $x$ .

If $x$ has $\ell$ distinct eigenvalues $\lambda_1, \dots, \lambda_\ell$ with multiplicities $d_1, \dots, d_\ell$ , the polynomial is:
$\text{Poin}(B(x, H(m_{\max})); q) = [n-2]_{q^2}! \left( [n]_{q^2}[n-2]_{q^2} + q^{2n-4} \sum_{j=1}^\ell [d_j]_{q^2} \right)$
where $[k]_q = 1 + q + \dots + q^{k-1}$ and $[k]_q!$ is the $q$ -factorial.
Significance: This formula is significantly simpler than previous methods which required analyzing multiple cases based on the Jordan structure of $x$ .

B. Irreducibility Characterization (Theorem 4)

The paper provides a necessary and sufficient condition for the variety to be reducible.

Result: $B(x, H(m_{\max}))$ is reducible if and only if there exists $\lambda \in \mathbb{C}$ such that $x - \lambda I_n$ has rank one.
Implication: This occurs if $x$ is nilpotent with Jordan type $(2, 1^{n-2})$ or if $x$ is semisimple with exactly two distinct eigenvalues (one with multiplicity $n-1$ ).

C. Singular Locus (Theorem 5)

The authors characterize the singular points of the variety when $x$ is nilpotent.

Result: The singular locus of $B(x, H(m_{\max}))$ is exactly the smaller Hessenberg variety $B(x, H((1, n-1, \dots, n-1, n)))$ .
Method: This is proven by analyzing the linear terms of the defining determinant $\det(A_g)$ in the affine patches.

D. Reduced Scheme Structure (Theorem 6)

Result: For $n \geq 3$ , the scheme $B(x, H(m_{\max}))$ is reduced for any $x$ .
Method: They prove that the determinant defining the local hypersurface generates a radical ideal. This relies on a technical lemma (Lemma 8.3) regarding the initial terms of determinants of specific matrix forms under lexicographic term orders.
Note: The result fails for $n=2$ if $x$ is nilpotent (the scheme is non-reduced).

E. Relationship with Schubert Varieties (Corollary 7)

The authors resolve a question regarding the isomorphism between codimension-one Schubert varieties and Hessenberg varieties.

Context: Codimension-one Schubert varieties are of the form $X_{s_i w_0}$ . It was known that $X_{s_1 w_0}$ and $X_{s_{n-1} w_0}$ are Hessenberg varieties.
Result: For $i \in \{2, n-2\}$ , the Schubert variety $X_{s_i w_0}$ is not isomorphic to any Hessenberg variety.
Proof Strategy: They compare the Euler characteristics of the singular loci. The singular locus of $X_{s_i w_0}$ (for $i \in \{2, n-2\}$ ) has a different Euler characteristic than the singular locus of the corresponding rank-two nilpotent Hessenberg variety, proving they are not isomorphic.

4. Significance

Unification of Methods: The "point count heuristic" (Theorem 1) is a powerful general tool. It demonstrates that for type A Hessenberg varieties, topological invariants can be computed via elementary finite field counting, bypassing complex geometric constructions.
Completeness of Classification: The paper provides a complete topological and geometric description of the codimension-one case, a regime that is often a boundary case in algebraic geometry.
Resolution of Open Problems: It settles the isomorphism question for codimension-one Schubert varieties, extending previous work by Tymoczko and the authors themselves.
Scheme-Theoretic Insight: The proof that these varieties are reduced (for $n \geq 3$ ) clarifies the scheme-theoretic nature of these objects, distinguishing them from cases where non-reduced structures might appear.

In summary, the paper successfully bridges combinatorial representation theory and algebraic geometry, providing explicit formulas and structural characterizations for a specific, important class of Hessenberg varieties using a novel finite-field counting technique.