Movable curve classes and slope stability on Deligne-Mumford stacks

Sebastian Casalaina-Martin; Shend Zhjeqi

Movable curve classes and slope stability extend from varieties to smooth proper DM stacks with projective coarse moduli spaces, yielding a Bogomolov-Gieseker inequality.

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2026-06-29 20:22 UTC pith:OLL4WKLI

load-bearing objection Extends movable curve classes, slope stability, and a Bogomolov-Gieseker inequality from varieties to smooth proper DM stacks with projective coarse space, as the second paper in a hyperbolicity series.

arxiv 2605.26101 v1 pith:OLL4WKLI submitted 2026-05-25 math.AG

Movable curve classes and slope stability on Deligne-Mumford stacks

Sebastian Casalaina-Martin , Shend Zhjeqi This is my paper

classification math.AG

keywords Deligne-Mumford stacksmovable curve classesslope stabilityBogomolov-Gieseker inequalitycoherent sheavesmoduli spacesalgebraic stacks

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper generalizes theorems on movable curve classes and the slope stability of coherent sheaves, previously known for smooth projective varieties, to the broader setting of smooth proper Deligne-Mumford stacks that have projective coarse moduli spaces. This extension directly produces the Bogomolov-Gieseker inequality for sheaves on the stacks. The move matters because many natural moduli spaces appear as DM stacks rather than varieties, so the new statements open access to stability and positivity results in those cases. The work forms the second part of a series that aims to carry Viehweg hyperbolicity results to this stack setting.

Core claim

Movable curve classes and slope stability of coherent sheaves on smooth projective varieties extend to smooth proper DM stacks admitting projective coarse moduli spaces; the resulting slope function then implies the Bogomolov-Gieseker inequality on these stacks.

What carries the argument

The extension of the movable cone of curve classes and the associated slope function for coherent sheaves from varieties to DM stacks.

Load-bearing premise

The Deligne-Mumford stacks under consideration are smooth and proper and possess projective coarse moduli spaces.

What would settle it

Exhibiting a coherent sheaf on a smooth proper DM stack with projective coarse moduli space whose Chern classes violate the Bogomolov-Gieseker inequality would falsify the central claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

The Bogomolov-Gieseker inequality holds for coherent sheaves on all such stacks.
Slope stability with respect to movable classes behaves as it does on varieties.
The inequality supplies a new positivity tool for sheaves on moduli stacks.
The results prepare the ground for hyperbolicity statements on KSBA moduli spaces viewed as stacks.

Where Pith is reading between the lines

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

The same extension technique may apply to other classes of algebraic stacks beyond DM stacks.
The inequality could be used to bound the geometry of moduli spaces that arise as coarse spaces of these stacks.
Analogous statements might hold when the coarse space is only quasi-projective rather than projective.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Extends movable curve classes, slope stability, and a Bogomolov-Gieseker inequality from varieties to smooth proper DM stacks with projective coarse space, as the second paper in a hyperbolicity series.

read the letter

This paper carries over some results on movable curve classes and slope stability of coherent sheaves from smooth projective varieties to smooth proper DM stacks that admit projective coarse moduli spaces, then derives a Bogomolov-Gieseker inequality on those stacks. It is positioned as the second paper in a series aimed at generalizing Popa-Schnell and Wei-Wu type Viehweg hyperbolicity statements to DM stacks, especially certain KSBA moduli spaces.

The contribution is the direct extension itself. The hypotheses are stated plainly in the abstract, and the work supplies the stack versions of the tools that were already available on varieties. For researchers who need positivity and stability statements on moduli problems that naturally live on stacks rather than schemes, this fills a gap that will likely be cited in follow-up papers.

The main thing to check is whether the extension really holds under exactly the listed stack hypotheses or whether hidden adjustments are needed that the variety proofs do not require. The abstract gives no sign of internal inconsistency, and the stress-test found no load-bearing flaw in the argument structure, but the proofs will have to be read carefully. Citations look standard and point to the variety literature without circularity.

This is for algebraic geometers working on moduli of stacks and on hyperbolicity questions in the KSBA setting. A reader already tracking that program will get immediate use from the generalized statements. Outside that niche the paper is narrower.

It deserves a serious referee because the target application to KSBA moduli spaces is active and the tools are the kind that get reused.

Referee Report

0 major / 3 minor

Summary. The manuscript generalizes results from the literature on movable curve classes and slope stability of coherent sheaves on smooth projective varieties to the setting of smooth proper Deligne-Mumford stacks that admit projective coarse moduli spaces. As an application it establishes a Bogomolov-Gieseker inequality on such stacks. The work is presented as the second paper in a series whose ultimate goal is to extend results of Popa-Schnell and Wei-Wu on Viehweg hyperbolicity to DM stacks, in particular to certain KSBA moduli spaces.

Significance. If the central generalization is valid, the paper supplies foundational tools (movable classes, slope stability, and the resulting Bogomolov-Gieseker inequality) that are needed for the hyperbolicity program on stacks. The hypotheses are stated explicitly and the application follows directly from the generalized statements, which is a strength of the manuscript.

minor comments (3)

[Introduction] §1 (Introduction): the precise statements being generalized from the variety case should be recalled with equation or theorem numbers so that the reader can immediately see which hypotheses are relaxed and which remain unchanged.
The notation for the coarse moduli space and the stacky structure should be fixed consistently throughout; occasional switches between X and its coarse space X create minor ambiguity in the statements of slope stability.
A short comparison table or paragraph contrasting the new statements with the corresponding results on varieties would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, its significance for the hyperbolicity program on stacks, and the recommendation of minor revision. No specific major comments appear in the report, so we have no points requiring point-by-point rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a direct generalization of results from the literature on movable curve classes and slope stability from smooth projective varieties to smooth proper DM stacks with projective coarse moduli spaces, with the Bogomolov-Gieseker inequality as an application. The abstract explicitly frames the work as extending external prior results (Popa-Schnell, Wei-Wu) under clearly stated hypotheses, and the provided text shows no self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the central claims to the inputs by construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; all technical content is deferred to the full manuscript.

pith-pipeline@v0.9.1-grok · 5606 in / 1040 out tokens · 26323 ms · 2026-06-29T20:22:19.277240+00:00 · methodology

0 comments

read the original abstract

We generalize some results in the literature on movable curve classes and slope stability of coherent sheaves on smooth projective varieties to the case of smooth proper DM stacks admitting projective coarse moduli spaces. As an application, we establish a Bogomolov-Gieseker inequality on such stacks. This paper is the second in a series aiming to generalize results of Popa-Schnell and Wei-Wu on Viehweg hyperbolicity to the setting of DM stacks, and in particular, to certain KSBA moduli spaces.

discussion (0)

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Moduli spaces of snc klt KSBA stable pairs are naturally of log general type
math.AG 2026-07 conditional novelty 7.0

Smooth proper DM stacks carrying maximal-variation families of relative snc klt KSBA stable pairs are of log general type (K_X+Δ big), extending Wei-Wu's result from varieties to stacks.
Removing the Torsion-free Hypothesis in a Positivity Theorem on Deligne-Mumford Stacks
math.AG 2026-07 accept novelty 3.5

Every coherent quotient of a tensor power of the log cotangent sheaf on a smooth proper DM stack has pseudo-effective first Chern class once a big-determinant subsheaf exists, without needing the quotient to be torsion-free.

Reference graph

Works this paper leans on

10 extracted references · cited by 2 Pith papers

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[AGV08] Dan Abramovich, Tom Graber, and Angelo Vistoli,Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math.130(2008), no. 5, 1337–1398. MR 2450211 [Alp13] Jarod Alper,Good moduli spaces for Artin stacks, Ann. Inst. Fourier (Grenoble)63(2013), no. 6, 2349–2402. MR 3237451 [BM08] Edward Bierstone and Pierre D. Milman,Functoriality in resolution of...

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MR 1644323 [Gie79] D. Gieseker,On a theorem of Bogomolov on Chern classes of stable bundles, Amer. J. Math.101(1979), no. 1, 77–85. MR 527826 [GKP14] Daniel Greb, Stefan Kebekus, and Thomas Peternell,Reflexive differential forms on singular spaces. Geometry and cohomology, J. Reine Angew. Math.697(2014), 57–89. MR 3281652 [GKP16] ,Movable curves and semis...

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2024

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det” and “Div

MR 4691930 [KM76] Finn Faye Knudsen and David Mumford,The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”, Math. Scand.39(1976), no. 1, 19–55. MR 437541 [Kol90] J ´anos Koll´ar,Projectivity of complete moduli, J. Differential Geom.32(1990), no. 1, 235–268. MR 1064874 [Kre09] Andrew Kresch,On the geometry of Deligne-M...

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Part 1, Proc. Sympos. Pure Math., vol. 80, Part 1, Amer. Math. Soc., Providence, RI, 2009, pp. 259–271. MR 2483938 [KT23] Andrew Kresch and Yuri Tschinkel,Birational geometry of deligne-mumford stacks,

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London Math

[KV04] Andrew Kresch and Angelo Vistoli,On coverings of Deligne-Mumford stacks and surjectivity of the Brauer map, Bull. London Math. Soc.36(2004), no. 2, 188–192. MR 2026412 [Laz04] Robert Lazarsfeld,Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete

2004

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2004

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8, Cambridge University Press, Cambridge, 1989, Translated from the Japanese by M

MR 1771927 [Mat89] Hideyuki Matsumura,Commutative ring theory, second ed., Cambridge Studies in Advanced Mathemat- ics, vol. 8, Cambridge University Press, Cambridge, 1989, Translated from the Japanese by M. Reid. MR 1011461 [Miy77] Yoichi Miyaoka,On the Chern numbers of surfaces of general type, Invent. Math.42(1977), 225–237. MR 460343 [Miy87] ,The Cher...

1989

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J.161(2012), no

[Tem12] Michael Temkin,Functorial desingularization of quasi-excellent schemes in characteristic zero: the nonembedded case, Duke Math. J.161(2012), no. 11, 2207–2254. MR 2957701 [Tem18] ,Functorial desingularization overQ: boundaries and the embedded case, Israel J. Math.224(2018), no. 1, 455–504. MR 3799764 [Tot04] Burt Totaro,The resolution property fo...

2012