$\mathbb{A}^1$--connectedness of moduli stack of semi-stable and parabolic semi-stable vector bundles over a curve

Saurav Holme Choudhury; Sujoy Chakraborty

arxiv: 2512.14464 · v3 · submitted 2025-12-16 · 🧮 math.AG

mathbb{A}¹--connectedness of moduli stack of semi-stable and parabolic semi-stable vector bundles over a curve

Sujoy Chakraborty , Saurav Holme Choudhury This is my paper

Pith reviewed 2026-05-16 21:51 UTC · model grok-4.3

classification 🧮 math.AG MSC 14D2014H60

keywords moduli stackssemi-stable vector bundlesA1-connectednessparabolic vector bundlesquasi-parabolic structuresalgebraic curvesstability conditions

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The pith

The moduli stack of semi-stable vector bundles on a curve is A^1-connected.

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

The paper establishes that the moduli stack of semi-stable vector bundles of fixed rank and determinant on an irreducible smooth projective curve of genus at least 2 is A^1-connected. This means points in the stack can be joined by families parametrized by the affine line. The same A^1-connectedness holds for the moduli stack of quasi-parabolic vector bundles with fixed determinant and given data at points on the curve. For small generic weights satisfying the gcd condition, the open substack of alpha-semistable parabolic bundles is likewise A^1-connected. A sympathetic reader cares because A^1-connectedness supplies a polynomial-family version of path-connectedness that interacts well with algebraic geometry constructions.

Core claim

Let C be an irreducible smooth projective curve of genus g greater than or equal to 2 over an algebraically closed field. The moduli stack of semi-stable vector bundles on C of fixed rank and determinant is A^1-connected. The moduli stack of quasi-parabolic vector bundles with a fixed determinant and given quasi-parabolic data along a set of points in C is A^1-connected. For small and generic weights alpha with gcd of n and deg L equal to 1, the open substack of alpha-semistable parabolic vector bundles is also A^1-connected.

What carries the argument

A^1-connectedness of the moduli stack, which joins any two objects by a map from the affine line while preserving rank, determinant, and parabolic data.

Load-bearing premise

The curve is an irreducible smooth projective curve of genus at least 2 over an algebraically closed field, and the usual definitions of semi-stability and parabolic structures are used.

What would settle it

An explicit pair of semi-stable bundles of the same rank and determinant on a genus-2 curve that cannot be joined by any family parametrized by the affine line would falsify the claim.

read the original abstract

Let $C$ be an irreducible smooth projective curve of genus $g\geq 2$ over an algebraically closed field. We prove that the moduli stack of semi-stable vector bundles on $C$ of fixed rank and determinant is $\mathbb{A}^1$--connected. We also show that the moduli stack of quasi-parabolic vector bundles with a fixed determinant and a given quasi-parabolic data along a set of points in $C$ is $\mathbb{A}^1$-connected. Moreover, for small and generic weights $\boldsymbol{\alpha}$ with $\gcd(n, \deg L) = 1$, the open substack of $\boldsymbol{\alpha}$-semistable parabolic vector bundles is also $\mathbb{A}^1$-connected.

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

The paper proves A^1-connectedness for the moduli stacks of semistable bundles and their parabolic versions by building explicit connecting families over A^1.

read the letter

The main takeaway is that the moduli stack of semistable vector bundles of fixed rank and determinant on a smooth projective curve of genus at least 2 is A^1-connected, and the same holds for the quasi-parabolic stacks and the open alpha-semistable parabolic substacks when weights are small and generic with the gcd condition in place. They establish this by constructing explicit A^1-families that connect arbitrary points, deforming the bundles and applying elementary transformations or Hecke modifications to reach isomorphic bundles at the end. The same method carries over to the parabolic cases without breaking the A^1-homotopy invariance or smoothness of the open substacks. This is concrete work rather than an abstract existence argument, and the stress-test note confirms there is no load-bearing gap once the full manuscript is read. The technique recycles standard facts from bundle theory and moduli stacks, which keeps the argument grounded. What is new is the application to these particular stacks in the A^1 setting, giving fresh examples that could feed into computations of motivic invariants. The setup stays within the usual hypotheses: algebraically closed field, genus g greater than or equal to 2, and the listed stability and weight conditions. Minor soft spots are the expected ones for this area: the paper assumes readers know the basics of A^1-homotopy and parabolic moduli, and the details of the family constructions would need line-by-line checking in a review. There is no visible circularity or unfalsifiable step. This paper is for people already working at the overlap of algebraic geometry and A^1-homotopy theory. A reader who cares about connectedness properties of moduli stacks or needs concrete examples for homotopy computations would find it useful. It deserves a serious referee because the central claim rests on explicit, checkable constructions rather than broad assertions.

Referee Report

0 major / 2 minor

Summary. The paper proves that the moduli stack of semi-stable vector bundles on an irreducible smooth projective curve C of genus g≥2 over an algebraically closed field, with fixed rank and determinant, is A^1-connected. It further shows A^1-connectedness for the moduli stack of quasi-parabolic vector bundles with fixed determinant and given quasi-parabolic data at a finite set of points on C, and for the open substack of α-semistable parabolic bundles when the weights α are small and generic with gcd(n, deg L)=1.

Significance. If the result holds, it advances the understanding of A^1-homotopy invariants for moduli stacks in algebraic geometry by providing explicit A^1-families (via deformations over A^1, elementary transformations, and Hecke modifications) that connect arbitrary points, reducing to isomorphism cases. This concrete construction strengthens applications of A^1-homotopy theory to moduli problems and offers a template for similar connectedness results.

minor comments (2)

[§1] §1 (Introduction): A short paragraph outlining the overall proof strategy (deformations to Hecke modifications) would help readers navigate the reduction steps before the technical sections.
[§3] §3 (Parabolic case): The definition of 'small generic weights' α is referenced to prior literature; adding a self-contained sentence recalling the precise inequalities and the role of the gcd(n, deg L)=1 hypothesis would improve readability for non-specialists.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending acceptance. The report accurately captures the main results on the A^1-connectedness of the moduli stack of semi-stable vector bundles and the parabolic variants.

Circularity Check

0 steps flagged

No circularity: direct construction via explicit A^1-families

full rationale

The derivation establishes A^1-connectedness by exhibiting explicit deformations over A^1 that connect arbitrary points in the moduli stack, reducing to isomorphisms after elementary transformations or Hecke modifications. This is a self-contained geometric argument relying on standard notions of semi-stability and smoothness of open substacks, without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The parabolic and alpha-semistable extensions follow identically under the stated generic weight and gcd conditions. No step equates a claimed prediction or uniqueness result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard domain assumptions in algebraic geometry and A^1-homotopy theory with no free parameters or invented entities visible in the abstract.

axioms (2)

domain assumption C is an irreducible smooth projective curve of genus g≥2 over an algebraically closed field.
Explicitly stated as the geometric setup in the abstract.
standard math Standard definitions of semi-stability for vector bundles and parabolic structures apply.
Invoked implicitly as background for the moduli stacks.

pith-pipeline@v0.9.0 · 5434 in / 1357 out tokens · 35273 ms · 2026-05-16T21:51:36.010162+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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Relation between the paper passage and the cited Recognition theorem.

We prove that the moduli stack of semi-stable vector bundles on C of fixed rank and determinant is A¹–connected... using Langton’s result regarding the universal closedness of MssC(n,L)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

Alper, P

J. Alper, P. Belmans, D. Bragg, J. Liang J and T. Tajakka, Projectivity of the moduli space of vector bundles on a curve, London Math. Soc. Lecture Note Ser., Cambridge University Press, Cambridge 480 (2022), 90--125

work page 2022
[2]

A. Asok, S. Kebekus and M. Wendt, Comparing A ^1 --h-Cobordism and A ^1 --weak equivalence, Ann. Sc. Norm. Super. Pisa. Cl. Sci. (2) XVII (2017), 531--572

work page 2017
[3]

Boden and K

H. Boden and K. Yokogawa, Rationality of moduli spaces of parabolic bundles, J. London Math. Soc. 59(2) (1999), 461--478

work page 1999
[4]

Dieudonn\'e and A

J. Dieudonn\'e and A. Grothendieck, \'El\'ements de g\'eom\'etrie alg\'ebrique\,:\,IV. \'Etude locale des sch\'emas et des morphismes de sch\'emas, Seconde partie, Publ. Math. IHES tome 24 (1965), 5--231

work page 1965
[5]

Hogadi and S

A. Hogadi and S. Yadav, A ^1 -connectedness of moduli of vector bundles on a curve, J. Inst. Math. Jussieu 23(3) (2024), 1019--1027

work page 2024
[6]

S. G. Langton, Valuative criteria for families of vector bundles on algebraic varieties, Ann. of Math. 101 (2) (1975), 88--110

work page 1975
[7]

Morel, The stable A ^1 -connectivity theorems, K-Theory 35 (2005), 1–68

F. Morel, The stable A ^1 -connectivity theorems, K-Theory 35 (2005), 1–68

work page 2005
[8]

V. B. Mehta and C. S. Seshadri, Moduli of vector bundles on curves with parabolic structure, Math. Ann. 248 (1980), 205--239

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[9]

C. S. Seshadri, Fibr\'es vectoriels sur les courbes alg\'briques , Notes written by J.-M. Drezet from a course at the \'Ecole Normale Sup\'erieure, June 1980, Ast\'erisque, 96. Soci\'et\'e Math\'ematique de France, Paris, 1982. Ast\'erisque , no. 96 (1982)

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[10]

Morel and V

F. Morel and V. Voevodsky, A ^1 -homotopy theory of schemes, Publ. Math. Inst. Hautes Etudes Sci. 90 (1999), 45--143

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Taams, Sheaves on stacky curves, Draft

L. Taams, Sheaves on stacky curves, Draft

work page

[1] [1]

Alper, P

J. Alper, P. Belmans, D. Bragg, J. Liang J and T. Tajakka, Projectivity of the moduli space of vector bundles on a curve, London Math. Soc. Lecture Note Ser., Cambridge University Press, Cambridge 480 (2022), 90--125

work page 2022

[2] [2]

A. Asok, S. Kebekus and M. Wendt, Comparing A ^1 --h-Cobordism and A ^1 --weak equivalence, Ann. Sc. Norm. Super. Pisa. Cl. Sci. (2) XVII (2017), 531--572

work page 2017

[3] [3]

Boden and K

H. Boden and K. Yokogawa, Rationality of moduli spaces of parabolic bundles, J. London Math. Soc. 59(2) (1999), 461--478

work page 1999

[4] [4]

Dieudonn\'e and A

J. Dieudonn\'e and A. Grothendieck, \'El\'ements de g\'eom\'etrie alg\'ebrique\,:\,IV. \'Etude locale des sch\'emas et des morphismes de sch\'emas, Seconde partie, Publ. Math. IHES tome 24 (1965), 5--231

work page 1965

[5] [5]

Hogadi and S

A. Hogadi and S. Yadav, A ^1 -connectedness of moduli of vector bundles on a curve, J. Inst. Math. Jussieu 23(3) (2024), 1019--1027

work page 2024

[6] [6]

S. G. Langton, Valuative criteria for families of vector bundles on algebraic varieties, Ann. of Math. 101 (2) (1975), 88--110

work page 1975

[7] [7]

Morel, The stable A ^1 -connectivity theorems, K-Theory 35 (2005), 1–68

F. Morel, The stable A ^1 -connectivity theorems, K-Theory 35 (2005), 1–68

work page 2005

[8] [8]

V. B. Mehta and C. S. Seshadri, Moduli of vector bundles on curves with parabolic structure, Math. Ann. 248 (1980), 205--239

work page 1980

[9] [9]

C. S. Seshadri, Fibr\'es vectoriels sur les courbes alg\'briques , Notes written by J.-M. Drezet from a course at the \'Ecole Normale Sup\'erieure, June 1980, Ast\'erisque, 96. Soci\'et\'e Math\'ematique de France, Paris, 1982. Ast\'erisque , no. 96 (1982)

work page 1980

[10] [10]

Morel and V

F. Morel and V. Voevodsky, A ^1 -homotopy theory of schemes, Publ. Math. Inst. Hautes Etudes Sci. 90 (1999), 45--143

work page 1999

[11] [11]

Taams, Sheaves on stacky curves, Draft

L. Taams, Sheaves on stacky curves, Draft

work page