arxiv: 2604.21303 · v1 · submitted 2026-04-23 · 🧮 math.AG

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Birational Geometry of Quot Schemes on smooth projective curves via Stable Pairs

Chandranandan Gangopadhyay , Atsushi Ito

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Pith reviewed 2026-05-09 21:05 UTC · model grok-4.3

classification 🧮 math.AG

keywords Quot schemesstable pairsMori dream spacesbirational geometrysmooth projective curvesdeterminant morphismPicard varietyQ-factorial modifications

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The pith

Stable pairs induce small Q-factorial modifications that turn the fiber of the Quot scheme into a Mori dream space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper studies the birational geometry of Quot schemes that parametrize quotients of a fixed vector bundle on a smooth projective curve of genus two or more. For sufficiently large degree n, the moduli space of stable pairs is used to produce small Q-factorial modifications of the Quot scheme and of its fiber over the Picard variety of degree n line bundles. The authors then compute the nef, movable, and effective cones on these modified spaces. This leads to the conclusion that the fiber is a Mori dream space and that the determinant map to the Picard variety is a Mori dream morphism.

Core claim

Let C be a smooth projective curve of genus g ≥ 2 over ℂ, and let E⁰ be a vector bundle on C. We investigate the birational geometry of the Quot scheme Quot_C(E⁰, k, n), which parametrizes quotients of E⁰ of rank k and degree n, and its fiber Q_L over Picⁿ(C) for n ≫ 0. Our main tool is the moduli space of stable pairs, which yields small ℚ-factorial modifications (SQMs) of Quot_C(E⁰, k, n) and Q_L. We explicitly describe the nef, movable, and effective cones of each SQM. Consequently, we prove that Q_L is a Mori dream space and that the determinant morphism Quot_C(E⁰, k, n) → Picⁿ(C) is a Mori dream morphism.

What carries the argument

The moduli space of stable pairs on the curve, serving as the source of small ℚ-factorial modifications (SQMs) for the Quot scheme and its fiber Q_L.

If this is right

The nef cone of each small Q-factorial modification is explicitly described.
The movable cone of each SQM is explicitly described.
The effective cone of each SQM is explicitly described.
The fiber Q_L is a Mori dream space.
The determinant morphism from the Quot scheme to Picⁿ(C) is a Mori dream morphism.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The cone computations may allow direct determination of the Cox ring generators in these cases.
The method relies on the curve being one-dimensional, so the same stable-pair construction may not produce SQMs in higher-dimensional settings.
For n not large, the Quot scheme and its fiber may require different modifications or fail to be Mori dream spaces.

Load-bearing premise

The assumption that n is sufficiently large so that the moduli space of stable pairs produces the desired small Q-factorial modifications of the Quot scheme and its fiber.

What would settle it

An explicit computation for a specific curve, bundle E⁰, and large n showing that the Cox ring of an SQM of Q_L is not finitely generated or that one of the nef, movable, or effective cones differs from the description obtained from the stable pair moduli space.

Figures

Figures reproduced from arXiv: 2604.21303 by Atsushi Ito, Chandranandan Gangopadhyay.

**Figure 2.** Figure 2: The slice of the movable cone Mov(QL) for k ≥ 2 We further show that det : Quot(E 0 , k, n) → Picn (C) is a Mori dream morphism, which is the relative version of Mori dream spaces as introduced in [Oht22]. We also study MC,δ(E0, s, d) and MC,δ(E0, s, d)L for s > r and obtain similar results. This paper is organized as follows. In §2, we recall the notion of δ-stability. In §3, we consider MC,δ(E0, s, d) fo… view at source ↗

**Figure 3.** Figure 3: The slice of Mov(Mi,L) = Eff(Mi,L) for s ≥ r + 2 In the rest of this section, we prove this theorem [PITH_FULL_IMAGE:figures/full_fig_p039_3.png] view at source ↗

**Figure 4.** Figure 4: The slice of Mov(Mi,L) = R≥0θ0 + R≥0γ ′ and Eff(Mi,L) = R≥0θ0 + R≥0γ for s = r + 1 ≥ 2 9.1. The case when the (relative) Picard number is one. Proof of Theorem 9.1, the case s = 1, or s = r + 1 and κ♯ < δ < δ♯. If s = 1, MC,δ(E0, 1, d) = QuotC(E 0 , r − 1, d + e) for any δ > 0. By Lemma 3.2 and Remark 3.3, πδ : MC,δ(E0, 1, d) → MC(1, d) = Picd (C), [(E, α)] 7→ [E] is a projective bundle and hence MC,δ(E0, … view at source ↗

read the original abstract

Let $C$ be a smooth projective curve of genus $g \geq 2$ over $\mathbb C$, and let $E^0$ be a vector bundle on $C$. We investigate the birational geometry of the Quot scheme ${\rm Quot}_C(E^0, k, n)$, which parametrizes quotients of $E^0$ of rank $k$ and degree $n$, and its fiber $\mathcal Q_L$ over ${\rm Pic}^n(C)$ for $n \gg 0$. Our main tool is the moduli space of stable pairs, which yields small $\mathbb Q$-factorial modifications (SQMs) of ${\rm Quot}_C(E^0, k, n)$ and $\mathcal Q_L$. We explicitly describe the nef, movable, and effective cones of each SQM. Consequently, we prove that $\mathcal Q_L$ is a Mori dream space and that the determinant morphism ${\rm Quot}_C(E^0, k, n) \to {\rm Pic}^n(C)$ is a Mori dream morphism.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper uses stable pairs to produce small Q-factorial modifications of Quot scheme fibers on curves for large n, then gives explicit cone descriptions that establish the Mori dream space and morphism properties.

read the letter

The punchline is that for n large the moduli space of stable pairs gives small Q-factorial modifications of the Quot scheme and its fiber Q_L. From there the work describes the nef, movable and effective cones explicitly and concludes that Q_L is a Mori dream space and the determinant map to the Picard variety is a Mori dream morphism. What the paper does well is to make this connection concrete on curves. The explicit cone descriptions stand out because they turn the abstract existence of modifications into something one can actually work with. The setup with a general vector bundle E^0 and the fiber over Pic^n(C) keeps the results applicable beyond special cases. The soft spots are minor. Everything rests on n being sufficiently large, which is the usual condition to make stability and birational maps behave nicely. The stress-test found no internal contradictions, so the logic seems sound from the abstract. If the full paper has complete proofs of the correspondence between stable pairs and the Quot scheme, that should be fine; otherwise the cone calculations might need more verification. This is for people in algebraic geometry who care about birational properties of moduli spaces of sheaves on curves. A reader who knows Quot schemes and has seen stable pairs before will see the value in the specific statements here. It deserves serious referee time because the claims are precise and the method is a reasonable extension of prior techniques. I would recommend sending it to peer review.

Referee Report

0 major / 4 minor

Summary. The paper studies the birational geometry of the Quot scheme Quot_C(E^0, k, n) on a smooth projective curve C of genus g ≥ 2 and its fiber Q_L over Pic^n(C) for n ≫ 0. Using the moduli space of stable pairs, it constructs small ℚ-factorial modifications (SQMs) of Quot_C(E^0, k, n) and Q_L, explicitly describes the nef, movable, and effective cones on these SQMs, and concludes that Q_L is a Mori dream space while the determinant morphism Quot_C(E^0, k, n) → Pic^n(C) is a Mori dream morphism.

Significance. If the results hold, the work supplies explicit cone descriptions and establishes the Mori dream property for these Quot schemes and their fibers in the large-n regime. This advances the birational geometry of moduli spaces of sheaves on curves by providing a stable-pairs route to SQMs and cone computations, which may serve as a model for similar problems on higher-dimensional varieties or other moduli spaces.

minor comments (4)

The introduction should include a brief roadmap of how the stable-pairs moduli space is used to produce the SQMs and why the large-n hypothesis guarantees that the birational maps are small.
Notation for the moduli space of stable pairs and the precise stability condition employed should be fixed at the first appearance rather than introduced piecemeal.
Figure captions or diagrams illustrating the wall-crossing or the chamber structure for the stability parameter would improve readability of the cone descriptions.
A short comparison paragraph with existing results on the birational geometry of Quot schemes (e.g., via GIT or other compactifications) would help situate the contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the clear summary of our results, and the recommendation for minor revision. As the major comments section contains no specific points, we have nothing to address point by point.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds by invoking the moduli space of stable pairs as an independent construction that supplies small Q-factorial modifications of Quot_C(E^0, k, n) and its fiber Q_L when n ≫ 0. From these modifications the paper then computes the nef, movable, and effective cones explicitly and deduces the Mori-dream-space and Mori-dream-morphism statements. None of these steps reduces by definition or by self-citation to the final claims; the stable-pairs moduli space is treated as an external geometric tool whose properties are not presupposed to be the cones or the dream-space property. The large-n hypothesis is the standard asymptotic regime in which stability conditions yield small birational maps, not a fitted parameter. No load-bearing self-citation chain or ansatz smuggling is required for the central argument.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the existence of the moduli space of stable pairs and its ability to produce small Q-factorial modifications of the Quot scheme when n is large; these are treated as known from prior literature in algebraic geometry.

axioms (2)

domain assumption The moduli space of stable pairs on a curve yields small Q-factorial modifications of the Quot scheme Quot_C(E^0, k, n) for n sufficiently large.
Invoked in the abstract as the main tool without further justification.
standard math Standard facts about nef, movable, and effective cones on Q-factorial varieties and the definition of Mori dream spaces.
Used to conclude that Q_L is a Mori dream space once the cones are described.

pith-pipeline@v0.9.0 · 5487 in / 1422 out tokens · 29001 ms · 2026-05-09T21:05:07.965601+00:00 · methodology

discussion (0)

Reference graph

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