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Irreducibility and singularities of some nested Quot schemes

This paper proves that nested Quot schemes parametrizing pairs of quotients of a vector bundle on a smooth projective curve of genus at least one are integral and local complete intersection schemes when the degrees of the quotients are sufficiently large and distinct.

Original authors: Parvez Rasul, Ronnie Sebastian

Published 2026-03-31

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CC BY 4.0

Original authors: Parvez Rasul, Ronnie Sebastian

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 an architect designing a massive, multi-story building. In the world of mathematics, specifically algebraic geometry, this "building" is a shape called a Quot Scheme.

Think of a Quot Scheme as a giant catalog or a map. It doesn't just list one thing; it lists every possible way you can take a complex object (like a vector bundle, which we can imagine as a flexible, multi-layered fabric) and cut it into smaller, specific pieces.

This paper, written by Parvez Rasul and Ronnie Sebastian, is about a special, more complicated version of this catalog called a Nested Quot Scheme.

Here is the breakdown of their discovery using simple analogies:

1. The Setup: The Russian Doll of Cuts

Imagine you have a long, colorful ribbon (this is your Vector Bundle $E$ ).

Standard Quot Scheme: You ask, "In how many ways can I cut a piece of this ribbon to make a scarf of a specific size?" The catalog lists all those ways.
Nested Quot Scheme: This paper asks a harder question: "In how many ways can I cut a big piece of ribbon (a scarf) AND then cut a smaller piece out of that scarf (a handkerchief)?"

You are looking for a chain of cuts: Ribbon $\to$ Scarf $\to$ Handkerchief.
The mathematicians want to know: Is this catalog of "Double Cuts" a single, solid, connected building? Or is it a messy pile of disconnected rooms? And is the building smooth, or does it have cracks and jagged edges (singularities)?

2. The Two Scenarios: "Big then Small" vs. "Small then Big"

The authors studied two different ways to order the sizes of the cuts (degrees $d_1$ and $d_2$ ):

Scenario A: The "Big to Small" Case ( $d_1 \gg d_2$ )
Imagine you cut a huge scarf first, and then cut a tiny handkerchief out of it.
- The Result: The authors proved that if the scarf is massive compared to the handkerchief, the catalog is perfect. It is one single, connected building (irreducible). It is solid with no holes (integral). It is smooth like polished marble (normal), and it has a very specific, predictable shape (local complete intersection).
- Analogy: If you have a giant sheet of paper and you cut a tiny dot out of it, there's only one logical way the "map" of all possible cuts looks. It's stable and predictable.
Scenario B: The "Small to Big" Case ( $d_1 \ll d_2$ )
Imagine you cut a tiny handkerchief first, and then try to find a huge scarf that contains it.
- The Result: This is trickier. The math gets messy because the "handkerchief" might have weird wrinkles (torsion) that make the "scarf" behave badly.
- The Discovery: The authors found that if the handkerchief is very small and the scarf is very large, the catalog is still a single, connected building! However, there is a catch: the building is only perfectly smooth if the "scarf" is significantly larger than the "handkerchief" in terms of rank (thickness). If they are too close in size, the building might have some jagged corners (singularities), but it's still one solid piece.

3. The "Smoothness" Guarantee

In math, "singularities" are like cracks, sharp points, or folds in a shape. You want your geometric building to be smooth so you can do calculus on it.

The paper proves that if you make the numbers (the sizes of the cuts) big enough, the "cracks" disappear or get pushed so far away that they don't matter.

The "Local Complete Intersection" term: This is a fancy way of saying the building is built from "standard Lego bricks." It's not a weird, twisted shape that breaks the rules of geometry. It's a very well-behaved structure.

4. Why Does This Matter?

You might ask, "Who cares about cutting ribbons?"

Moduli Spaces: These catalogs are actually maps to the "universe of all possible shapes." If we understand the shape of the catalog, we understand the shape of the universe of those objects.
Counting Problems: In physics and geometry, we often need to count how many ways something can happen. If the catalog is "connected" and "smooth," we can count these things accurately. If it's broken into pieces or has cracks, our counts will be wrong.
The "Nested" Connection: This work connects to Nested Hilbert Schemes, which are used in advanced physics (like string theory) and representation theory. By proving these nested catalogs are solid and smooth, the authors give physicists and other mathematicians a reliable tool to use in their own calculations.

Summary in a Nutshell

The authors took a very complex, nested mathematical puzzle (cutting a piece of fabric, then cutting a piece of that piece) and proved that if the pieces are big enough, the solution isn't a chaotic mess. Instead, it forms a single, solid, smooth, and predictable structure.

They showed that no matter how you look at it (whether you cut big-then-small or small-then-big), as long as you have enough "material" (high degrees), the mathematical universe of these cuts is one connected, well-behaved world.

1. Problem Statement

The paper investigates the geometric properties of nested Quot schemes over a smooth projective curve $C$ of genus $g \geq 1$ .

Context: Let $E$ be a vector bundle on $C$ of rank $r$ and degree $e$ . The standard Quot scheme $Q^k_d(E)$ parametrizes quotients $E \twoheadrightarrow F$ where $F$ has rank $k$ and degree $d$ .
Object of Study: The nested Quot scheme $Q^{k_1, k_2}_{d_1, d_2}(E)$ parametrizes pairs of quotients $[E \twoheadrightarrow F_1 \twoheadrightarrow F_2]$ such that $F_i$ has rank $k_i$ and degree $d_i$ , with $r > k_1 > k_2 > 0$ .
Goal: To determine conditions under which these schemes are irreducible, integral, normal, and local complete intersections (LCI), and to analyze their singularities.
Cases Considered: The authors analyze two asymptotic regimes for the degrees:
1. $d_1 \gg d_2 \gg 0$ (High degree for the first quotient).
2. $0 \ll d_1 \ll d_2$ (High degree for the second quotient, relative to the first).

2. Methodology

The authors employ a combination of deformation theory, dimension counting, and relative Quot scheme techniques.

A. Relative Quot Schemes and Families

A central tool is the study of the relative Quot scheme $Q^k_d(\mathcal{A}) \to T$ , where $\mathcal{A}$ is a vector bundle on $C \times T$ flat over an irreducible base $T$ .

Theorem 3.16 (Generalization of [PR03]): The authors prove that for sufficiently large degree $d$ , the relative Quot scheme is irreducible, flat, and a local complete intersection morphism over $T$ . They establish that the fibers are irreducible of the expected dimension and that the singular locus has high codimension if $T$ is reduced.

B. Stratification by Torsion

To handle cases where the intermediate sheaves are not locally free (which prevents direct application of the relative Quot theorems), the authors stratify the nested Quot scheme based on the torsion subsheaves of the quotients.

They define loci $Y_{\delta, \mu}$ where the torsion lengths of $F_1$ and $F_2$ are $\delta$ and $\mu$ , respectively.
The "good" locus $U$ consists of points where $F_1$ (in the $d_1 \ll d_2$ case) or $F_2$ (in the $d_1 \gg d_2$ case) is locally free.

C. Dimension Bounds

The core of the irreducibility proofs involves showing that the "bad" strata (where torsion exists) have dimension strictly less than the expected dimension of the scheme.

Expected Dimension:
$\text{expdim}(d_1, d_2) = [d_1r - k_1e + k_1(r-k_1)(1-g)] + [d_2k_1 - d_1k_2 + k_2(k_1-k_2)(1-g)]$
Strategy:
1. Identify an open dense subset $U$ (where the relevant sheaf is locally free) and prove it is irreducible of dimension $\text{expdim}$ .
2. Construct maps from auxiliary schemes (involving relative Quot schemes of kernels and torsion sheaves) to the strata $Y_{\delta, \mu}$ .
3. Compute upper bounds for $\dim(Y_{\delta, \mu})$ .
4. Show that for sufficiently large degrees, $\dim(Y_{\delta, \mu}) < \text{expdim}$ . This implies that no component of the nested Quot scheme can be supported entirely on the singular/torsion locus, forcing the scheme to be irreducible with the generic point lying in $U$ .

D. Normality and Singularities

Normality: Proven by verifying Serre's conditions ( $R_0$ and $R_1$ ). Since the schemes are shown to be Cohen-Macaulay (via LCI property), showing the singular locus has codimension $\geq 2$ is sufficient for normality.
Singularities: The authors analyze the codimension of the singular locus by relating it to the singularities of the base Quot scheme and the vanishing of $H^1$ groups (obstruction spaces).

3. Key Contributions and Results

A. The Case $d_1 \gg d_2 \gg 0$ (Theorem 4.1)

Result: There exist thresholds $d(E, k_2)$ $d (E, k_{2})$ and $\xi(E, k_1, k_2, d_2)$ $ξ (E, k_{1}, k_{2}, d_{2})$ such that if $d_2 \geq d(E, k_2)$ $d_{2} \geq d (E, k_{2})$ and $d_1 - d_2 \geq \xi$ $d_{1} - d_{2} \geq ξ$ , the nested Quot scheme $Q^{k_1, k_2}_{d_1, d_2}(E)$ $Q_{d_{1}, d_{2}}^{k_{1}, k_{2}} (E)$ is:
- Irreducible of dimension $\text{expdim}(d_1, d_2)$ .
- A local complete intersection (LCI).
- Integral and Normal.
Method: The scheme is viewed as a relative Quot scheme over the standard Quot scheme $Q^{k_2}_{d_2}(E)$ . Since $Q^{k_2}_{d_2}(E)$ is well-behaved for large $d_2$ , and the kernel of the universal quotient is locally free, Theorem 3.16 applies directly.

B. The Case $0 \ll d_1 \ll d_2$ (Theorems 5.26, 5.27)

This case is more difficult because the universal quotient $F_1$ over $Q^{k_1}_{d_1}(E)$ is not locally free everywhere, so Theorem 3.16 cannot be applied directly to the whole space.

Theorem 5.26 (The $k_1 + k_2 > r$ case):
- If $k_1 + k_2 > r$ $k_{1} + k_{2} > r$ and $k_1 - k_2 \geq 2$ $k_{1} - k_{2} \geq 2$ , there exist thresholds $\gamma$ $γ$ and $\beta$ $β$ such that for $d_1 \geq \gamma$ $d_{1} \geq γ$ and $d_2 \geq \beta$ $d_{2} \geq β$ :
  1. $Q^{k_1, k_2}_{d_1, d_2}(E)$ is irreducible of dimension $\text{expdim}$ .
  2. The map to $Q^{k_1}_{d_1}(E)$ is an LCI morphism.
  3. The scheme is integral and normal.
- Significance: The condition $k_1 + k_2 > r$ ensures that the "bad" strata (where torsion exists) have sufficiently high codimension to be negligible.
Theorem 5.27 (General Case):
- The result is extended to all $k_1, k_2$ (without the $k_1+k_2 > r$ restriction) by embedding the problem into a larger bundle $E' = E \oplus \mathcal{O}_C^{\oplus l}$ .
- By increasing the rank of the bundle, the condition $k'_1 + k'_2 > r'$ is satisfied, allowing the application of Theorem 5.26. A surjective map from the nested scheme of the larger bundle to the original scheme proves irreducibility and integrality.

4. Significance and Impact

Generalization of Known Results: The paper extends classical results on standard Quot schemes (e.g., [PR03], [BDW96]) to the nested setting, which is significantly more complex due to the interplay between the two quotient maps.
Resolution of Singularities Questions: It addresses open questions regarding the singularities of nested Hilbert/Quot schemes (referencing [RS23]), proving that under high-degree conditions, these schemes are not only irreducible but also have mild singularities (normal, LCI).
Technical Innovation: The development of a robust framework for handling relative Quot schemes over non-locally-free sheaves (via the stratification and dimension bounding of torsion loci) provides a template for studying other nested moduli problems.
Applications: Nested Quot schemes are fundamental in:
- Enumerative Geometry: Counting maps to Grassmannians.
- Moduli of Sheaves: Understanding the geometry of moduli spaces.
- Representation Theory: Connections to Lie algebras and vertex algebras (via nested Hilbert schemes on surfaces, which this work parallels for curves).
Virtual Fundamental Classes: The results support the construction of virtual fundamental classes (as mentioned in [MR25]) by ensuring the underlying spaces have the expected dimension and good geometric properties.

In summary, the paper establishes that nested Quot schemes on curves behave "nicely" (being integral, normal, and LCI) when the degrees of the quotients are sufficiently large, providing a solid geometric foundation for further enumerative and moduli-theoretic investigations.