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Truncated Grassmannians, blow-ups along Schubert varieties and collineations

This paper introduces truncated Grassmannians as orbit closures of abelian unipotent groups and demonstrates their role in describing the blow-ups of general flag varieties along Schubert subvarieties, specifically characterizing the fibers of the resulting projections onto Grassmannians via spaces of collineations.

Original authors: Evgeny Feigin

Published 2026-04-02

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Evgeny Feigin

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

The Big Picture: Fixing a Cracked Map

Imagine you are a cartographer trying to draw a map of a very complex, high-dimensional landscape called a Grassmannian. In math terms, this landscape represents all possible ways to choose a specific number of directions (a "subspace") inside a giant room of dimensions.

The problem is that this map has "cracks" or singularities. These cracks happen at specific locations called Schubert varieties. Think of these as dangerous cliffs or deep chasms in your landscape where the geometry gets messy and undefined.

Feigin's paper is about a new, clever way to "fix" these cracks. Instead of just patching them, he proposes a method to blow them up. In geometry, "blowing up" is like taking a sharp, singular point and replacing it with a whole new, smooth, multi-dimensional space (like replacing a sharp needle tip with a small, smooth dome).

The paper introduces three main characters to help us understand this process:

Truncated Grassmannians (The "Shadows")
Blow-ups (The "Fix")
Collineations (The "Connectors")

1. The Landscape: Grassmannians and the "Cracks"

The Grassmannian:
Imagine a giant room with $n$ dimensions. You want to pick a smaller room inside it with $d$ dimensions. The collection of all possible $d$ -dimensional rooms you could pick forms the Grassmannian. It's a smooth, beautiful shape, but it's very complex.

The Cracks (Schubert Varieties):
Now, imagine you have a specific rule: "Only pick rooms that touch a specific corner of the big room."

If you require the room to touch the corner at all, you get a specific shape.
If you require it to touch the corner deeply (in many dimensions), you get a smaller, more restrictive shape.

These restrictive shapes are the Schubert varieties. They are like "bad neighborhoods" in our map. If you try to do certain math operations right on these neighborhoods, the numbers break or become undefined. We need to fix them.

2. The "Shadows": Truncated Grassmannians

To fix the map, Feigin introduces a new tool: Truncated Grassmannians.

The Analogy: The Silhouette
Imagine you have a complex 3D sculpture (the Grassmannian). You shine a light on it from a specific angle. The shadow it casts on the wall is a 2D projection.

A Truncated Grassmannian is like a "partial shadow."
Instead of keeping every detail of the sculpture, we throw away the most complex, high-degree details. We keep only the "simpler" parts of the shape.
If we throw away almost everything, we get a simple flat wall (a Projective Space).
If we keep everything, we get the original sculpture (the Grassmannian).
The "Truncated" versions are the intermediate steps in between.

Feigin shows that these "shadows" aren't just random shapes; they are mathematically linked to the original landscape. They act as a bridge.

3. The "Fix": Blowing Up

Now, how do we fix the cracks? We use a Blow-up.

The Analogy: The Zoom Lens
Imagine you are looking at a pixelated, jagged image on a screen. You want to see the details clearly.

The Blow-up is like zooming in on the jagged edge.
When you zoom in, the jagged edge doesn't just get bigger; it reveals a whole new, smooth structure underneath.
In Feigin's paper, the "Blow-up" of the Grassmannian along a Schubert variety is a new, smoother shape that sits above the old one. It resolves the singularities.

The Big Discovery:
Feigin proves a surprising connection: The "Blow-up" is exactly the same thing as the "Graph" connecting the original Grassmannian to its "Truncated Shadow."

Think of it like this:

You have the original map (Grassmannian).
You have the shadow map (Truncated Grassmannian).
The "Blow-up" is a new, hybrid map that records both the original location and the shadow location simultaneously.
Where the original map was broken, this hybrid map is smooth because the shadow map fills in the missing information.

4. The "Connectors": Collineations

When you look at the "Blow-up" map, you might ask: "What does the new smooth space actually look like at the place where the crack used to be?"

This is where Collineations come in.

The Analogy: The Magic Mirror
Imagine you are standing in front of a mirror. If you move your hand, the reflection moves. But what if the mirror is broken?

A Collineation is a special kind of transformation that preserves "lines." It's a way of mapping one space to another while keeping straight lines straight.
Feigin discovers that the "new space" created by the blow-up (the part that replaced the crack) is actually a space of partial collineations.
In simple terms: The "fix" for the broken geometry is a collection of all possible ways to stretch, shrink, or map one set of directions to another, but only keeping the "straight" parts.

It's like saying: "To fix this broken corner of the universe, we replace it with a room full of all possible ways to draw straight lines between two points."

5. The General Case: From Specific to Universal

The paper starts with a specific example (Grassmannians) but then zooms out to a General Case.

The Specific: Fixing a specific type of room in a specific building.
The General: The author shows that this whole process works for any "Flag Variety" (a very general type of geometric shape used in advanced physics and math).
He defines a "Truncated Flag Variety" for any shape and proves that the "Blow-up" is always the closure of the path connecting the original shape to its truncated shadow.

Summary: The "Aha!" Moment

The paper is essentially saying:

"We have these complex geometric shapes with broken spots. Instead of trying to patch the holes, let's create a 'shadow' of the shape by ignoring the complicated parts. Then, let's build a new shape that records both the original and the shadow.

Surprise! This new shape is perfectly smooth. The 'broken' parts are replaced by a beautiful, structured space of 'line-mappings' (collineations). This gives us a universal recipe for smoothing out the most complex corners of mathematical geometry."

Why does this matter?
In physics and mathematics, smooth shapes are much easier to work with than broken ones. By understanding exactly how these shapes are related (via the truncated shadows and collineations), mathematicians can solve problems that were previously impossible, such as calculating the properties of quantum systems or understanding the structure of space-time in theoretical physics. Feigin has provided the "instruction manual" for turning jagged, broken geometry into smooth, usable forms.

1. Problem Statement

The paper addresses the geometric and representation-theoretic description of blow-ups of flag varieties along Schubert subvarieties. While the geometry of flag varieties and Schubert varieties is well-understood, the specific geometry of their blow-ups (particularly along singular Schubert subvarieties) is less explored.

The author focuses on the Grassmannian $Gr_d(V)$ (where $V$ is an $n$ -dimensional complex vector space) and investigates the blow-up $Bl_{S_r} Gr_d(V)$ along specific Schubert varieties $S_r$ . These varieties are defined by the condition $\dim(U \cap V_{n-d}) \geq r$ , where $V_{n-d}$ is a fixed subspace of dimension $n-d$ .

The core problem is to:

Construct these blow-ups explicitly using truncated Grassmannians (closures of orbits of abelian unipotent groups in truncated wedge powers).
Describe the fibers of the projection from the blow-up back to the original Grassmannian.
Generalize these findings to arbitrary flag varieties $G/B$ and mixed blow-ups.

2. Methodology

The author employs a combination of Lie theory, algebraic geometry, and representation theory:

Truncated Modules: The paper introduces "truncated" representations of the Lie algebra $\mathfrak{sl}_n$ . For the wedge power $\Lambda^d V$ , the author defines a quotient space $(\Lambda^d V)_r$ by modding out vectors corresponding to "high degree" indices (specifically, indices larger than $d$ ).
Orbit Closures: The Truncated Grassmannian $X_r$ is defined as the closure of the orbit of a cyclic vector under the action of the abelian unipotent group $\exp(\mathfrak{a}_d)$ , where $\mathfrak{a}_d$ is the opposite abelian unipotent radical of the parabolic subalgebra.
Koszul Resolutions: To establish the link between the blow-up and the truncated space, the author constructs a Koszul-type resolution for the ideal sheaf $\mathcal{I}_{S_r}$ of the Schubert variety. This involves computing cohomologies of vector bundles on the Grassmannian using the Borel-Weil-Bott theorem.
Graph Closures: The blow-up is realized as the closure of the graph of a birational map between the original Grassmannian and the truncated Grassmannian.
Collineations: The fibers of the projection maps are analyzed using the theory of partial collineations (varieties of linear maps with fixed rank conditions).

3. Key Contributions and Results

A. Truncated Grassmannians and Schubert Blow-ups (Theorem A)

The primary result for Grassmannians is an isomorphism between the blow-up of $Gr_d(V)$ along the Schubert variety $S_r$ and a specific orbit closure.

Theorem A: The blow-up $Bl_{S_r} Gr_d(V)$ is isomorphic to the closure of the graph of the birational map $X_{r-1} \dashrightarrow Gr_d(V)$ .
Embedding: The blow-up admits a closed embedding into the product $Gr_d(V) \times X_{r-1}$ . The image is the closure of the $\exp(\mathfrak{a}_d)$ -orbit of the point $V_d \times [w_{r-1}]$ , where $w_{r-1}$ is the cyclic vector in the truncated module.
Symmetry: These varieties admit an action of the opposite parabolic subgroup $P^-_d$ .

B. Fiber Description via Collineations (Theorem B)

The paper provides a precise description of the fibers of the projection $\pi: Bl_{S_r} Gr_d(V) \to Gr_d(V)$ .

For a point $U \in Gr_d(V)$ , let $m = \dim(U \cap V_{n-d})$ .
The fiber $\pi^{-1}(U)$ is isomorphic to a partial collineation variety $Col_{m-r+1}(m)$ .
Specifically, this is the closure of the image of the map $\phi \mapsto \Lambda^{m-r+1}\phi$ for linear maps $\phi$ between specific subspaces derived from $U$ .
Smoothness: The "complete mixed blow-up" (blowing up along all $S_2, \dots, S_d$ simultaneously) is proven to be smooth and serves as a desingularization for all other partial blow-ups in the family.

C. Generalization to Flag Varieties (Theorem C)

The construction is generalized to arbitrary flag varieties $G/B$ for a simple simply-connected Lie group $G$ .

Setup: For a regular dominant weight $\lambda$ and a Weyl group element $\sigma$ , let $S_\sigma$ be the opposite Schubert variety.
Truncated Flag Variety: Define the truncated module $L_\sigma(\lambda) = L(\lambda) / D(\sigma\lambda)$ , where $D(\sigma\lambda)$ is the Demazure module. Let $X_{\sigma(\lambda)}$ be the closure of the $B^-$ -orbit in $\mathbb{P}(L_\sigma(\lambda))$ .
Theorem C: The blow-up $Bl_{S_\sigma}(G/B)$ embeds into $G/B \times X_{\sigma(\lambda)}$ as the closure of the $B^-$ -orbit of the product of highest weight lines.
Conjecture: The author conjectures that the "complete mixed blow-up" (involving all Weyl group elements) is smooth.

D. Resolution of Ideal Sheaves

A significant technical contribution is the explicit construction of a resolution for the ideal sheaf $\mathcal{I}_{S_r}$ (Section 3 and Appendix A).

The author derives a complex of sheaves resolving $\mathcal{I}_{S_r}$ using Schur functors and the Borel-Weil-Bott theorem.
This resolution proves that the space of global sections $H^0(Gr_d(V), \mathcal{I}_{S_r}(1))$ is isomorphic to the dual of the truncated wedge power $(\Lambda^d V)_r^*$ . This isomorphism is the key to establishing the embedding of the blow-up into the product space.

4. Significance and Implications

Unification of Geometric Objects: The paper unifies several seemingly distinct geometric objects:
- Classical Grassmannians.
- Projective spaces (as $X_1$ ).
- Spaces of collineations.
- Blow-ups of Schubert varieties.
  All are shown to be members of a single family of birationally isomorphic varieties defined by orbit closures.
Desingularization: The identification of the complete mixed blow-up as a smooth variety provides a new, explicit desingularization for singular Schubert varieties and their partial blow-ups. This is valuable for computing intersection theory and cohomology rings.
Representation-Theoretic Insight: The work bridges the gap between the geometry of flag varieties and the representation theory of Lie algebras (specifically Demazure modules and truncated modules). It suggests that the geometry of blow-ups is governed by the structure of these truncated modules.
Connection to Existing Literature: The results connect to:
- Kausz Compactifications: Related to blow-ups along zero schemes of Plücker coordinates.
- Landsberg-Manivel Maps: The case $r=d$ recovers the Landsberg-Manivel map from affine space to the Grassmannian.
- Graph Closures: The blow-up is explicitly realized as the graph closure of a rational map, a technique useful in moduli problems.

In summary, Feigin provides a powerful framework where the blow-up of a flag variety along a Schubert variety is realized as an orbit closure in a product of the variety and a truncated flag variety, with fibers described by collineation varieties. This offers a concrete, computable, and symmetric approach to studying singularities in flag manifolds.