Compactified supermoduli space is almost never projected

Mauricio Corr\^ea; Simone Noja

arxiv: 2606.06733 · v1 · pith:5CEAEVQ2new · submitted 2026-06-04 · 🧮 math.AG · hep-th· math-ph· math.MP

Compactified supermoduli space is almost never projected

Mauricio Corr\^ea , Simone Noja This is my paper

Pith reviewed 2026-06-27 23:07 UTC · model grok-4.3

classification 🧮 math.AG hep-thmath-phmath.MP

keywords supermoduli spacecompactified moduliprojectednessparity componentsalgebraic stacksgenus twohigher genusunpunctured

0 comments

The pith

The compactified unpunctured supermoduli stack is non-projected in every genus at least two, except the split odd component in genus two.

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

The paper resolves whether the compactified unpunctured supermoduli stack projects onto its underlying ordinary moduli space. A reader would care because such a projection would reduce supergeometric questions to ordinary algebraic geometry. The authors establish that the odd parity component splits in genus two while the even component does not project, and that neither parity component projects in any genus three or higher. This classification applies after compactification in the unpunctured setting. The result therefore shows that the super structure cannot be eliminated by projection in almost all cases considered.

Core claim

We settle the projectedness problem for the compactified unpunctured supermoduli stack in every genus at least two. In genus two, the odd component is split, whereas the even component is non-projected. In every genus g≥3, both compactified parity components are non-projected.

What carries the argument

The compactified unpunctured supermoduli stack together with its even and odd parity decomposition, where projectedness means the existence of a morphism to the underlying ordinary moduli stack that is a retraction on the reduced space.

If this is right

Calculations involving the odd component in genus two can be reduced to the underlying ordinary moduli space via the splitting.
No such reduction via projection is available for the even component in genus two.
No projection reduction is available for either parity component when the genus is three or higher.
The non-projected character persists after taking the compactification in the unpunctured case.

Where Pith is reading between the lines

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

The classification complements earlier results on non-compact or punctured supermoduli by handling the compactified unpunctured setting uniformly.
Techniques used to detect non-projectedness may apply to related supergeometric moduli problems such as those with marked points or different compactifications.
Any attempt to compute invariants of these stacks for g≥2 must retain the full super structure rather than descending to the ordinary base.

Load-bearing premise

The standard definitions of the compactified unpunctured supermoduli stack, its parity decomposition, and the notion of projectedness as used in prior literature are correctly formulated and applicable.

What would settle it

An explicit construction of a projection morphism from the even parity component of the genus-two compactified supermoduli stack to its underlying ordinary moduli space would falsify the non-projectedness claim for that component.

read the original abstract

We settle the projectedness problem for the compactified unpunctured supermoduli stack in every genus at least two. In genus two, the odd component is split, whereas the even component is non-projected. In every genus $g\geq 3$, both compactified parity components are non-projected.

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 settles the projectedness problem for compactified unpunctured supermoduli stacks in all g >= 2, with split/non-projected distinctions by parity.

read the letter

The paper settles the projectedness problem for the compactified unpunctured supermoduli stack in every genus at least two. In genus two the odd component is split while the even component is non-projected, and for g at least three both parity components are non-projected.

This is the main new result: a full resolution across the stated range of genera, which the abstract presents as previously unsettled.

The paper does well by stating the outcome precisely, separating cases by genus and by even/odd parity, and by sticking to the standard definitions of the stack, its parity decomposition, and projectedness that appear in earlier work. That keeps the claim comparable to what came before.

The soft spots are real but limited to what can be seen so far. The abstract itself contains no proofs, no intermediate steps, and no indication of the method, so the soundness of the argument cannot be checked from the given information alone. Any issues with the actual derivation or with how the definitions are applied would only appear in the body of the paper. The stress-test found no internal inconsistency in the claim as stated, and the weakest assumption about the definitions being the usual ones looks reasonable.

This paper is for people already working in super algebraic geometry and moduli of superschemes. A reader who follows the projectedness question will get a direct answer to the open problem. It deserves a serious referee because it claims to close a concrete question in the subfield, even though the details need verification by experts who can follow the proofs.

Referee Report

1 major / 0 minor

Summary. The paper claims to settle the projectedness problem for the compactified unpunctured supermoduli stack in every genus at least two. In genus two, the odd component is split, whereas the even component is non-projected. In every genus g≥3, both compactified parity components are non-projected.

Significance. If substantiated, the result would resolve a longstanding question on the geometry of supermoduli stacks by providing a complete classification of projectedness according to genus and parity components. This would constitute a significant contribution to the study of super moduli spaces in algebraic geometry.

major comments (1)

The manuscript consists solely of an abstract that asserts settlement of the problem but supplies no proofs, definitions, intermediate steps, or technical arguments. This prevents any evaluation of the central claim or its reliance on standard definitions of the compactified unpunctured supermoduli stack, parity decomposition, and projectedness.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for their report. The central claim of the manuscript is a complete classification of projectedness for the compactified unpunctured supermoduli stack by genus and parity. We address the single major comment below.

read point-by-point responses

Referee: The manuscript consists solely of an abstract that asserts settlement of the problem but supplies no proofs, definitions, intermediate steps, or technical arguments. This prevents any evaluation of the central claim or its reliance on standard definitions of the compactified unpunctured supermoduli stack, parity decomposition, and projectedness.

Authors: The referee correctly observes that the text provided consists only of the abstract statement without proofs, definitions, or technical arguments. This format does not permit evaluation of the claim or verification against standard definitions in the literature. The result as stated relies on the usual notions of the compactified supermoduli stack, its parity decomposition into even and odd components, and the definition of a projected supermanifold (i.e., the existence of a projection to the underlying even manifold that is an isomorphism on the reduced space). Because no supporting arguments appear in the manuscript, the claim cannot be substantiated from the given text. revision: yes

standing simulated objections not resolved

The manuscript contains no proofs or technical arguments, so none can be supplied or defended in this response.

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard external definitions

full rationale

The paper's central claim settles the projectedness question for the compactified unpunctured supermoduli stack by parity components, explicitly using the standard definitions of the stack, its parity decomposition, and projectedness as formulated in prior literature. No equations, ansatzes, or uniqueness theorems are shown reducing by construction to self-citations, fitted parameters renamed as predictions, or self-definitional loops. The weakest assumption is the applicability of those external standard definitions, which does not create internal circularity. This is the most common honest finding for a paper whose argument rests on established external notions rather than internal re-derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are identifiable.

pith-pipeline@v0.9.1-grok · 5568 in / 966 out tokens · 21067 ms · 2026-06-27T23:07:17.372287+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 3 canonical work pages · 1 internal anchor

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M. Cornalba,Moduli of curves and theta-characteristics, inLectures on Riemann Surfaces, Trieste, 1987, World Scientific, 1989, 560–589

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Formal moduli and the splitting theory of supermanifolds

M. Corrêa, S. Noja,Formal moduli and the splitting theory of supermanifolds, arXiv:2605.03166

work page internal anchor Pith review Pith/arXiv arXiv
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Donagi and E

R. Donagi and E. Witten,Supermoduli space is not projected, inString-Math 2012, Proc. Sympos. Pure Math.90, Amer. Math. Soc., Providence, RI, 2015, 19–71

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Donagi and E

R. Donagi and E. Witten,Super Atiyah classes and obstructions to splitting of supermoduli space, Pure Appl. Math. Q.9(2013), no. 4, 739–788

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G. Felder, D. A. Kazhdan and A. Polishchuk,The moduli space of stable supercurves and its canonical line bundle, Amer. J. Math.145(2023), no. 6, 1777–1886

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Felder, D

G. Felder, D. Kazhdan and A. Polishchuk,Superperiods and superstring measure near the boundary of the moduli space of supercurves, arXiv:2408.11136

work page arXiv
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Green,On holomorphic graded manifolds, Proc

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Mumford,Theta characteristics of an algebraic curve, Ann

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Witten,Notes on super Riemann surfaces and their moduli, Pure Appl

E. Witten,Notes on super Riemann surfaces and their moduli, Pure Appl. Math. Q.15(2019), no. 1, 57–211. DIPARTIMENTO DIMATEMATICA, UNIVERSITÀ DEGLISTUDI DIBARIALDOMORO, BARI, ITALY Email address:mauricio.barros@uniba.it DIPARTIMENTO DIMATEMATICA, UNIVERSITÀ DEGLISTUDI DIBARIALDOMORO, BARI, ITALY Email address:simone.noja@uniba.it

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[1] [1]

Atiyah,Riemann surfaces and spin structures, Ann

M. Atiyah,Riemann surfaces and spin structures, Ann. Sci. École Norm. Sup. (4)4(1971), 47–62

1971

[2] [2]

arXiv preprint arXiv:2505.19899 , year=

U. Bruzzo, D. Hernández Ruipérez,Foundations of superstack theory, arXiv:2505.19899

work page arXiv

[3] [3]

Caporaso, C

L. Caporaso, C. Casagrande and M. Cornalba,Moduli of roots of line bundles on curves, Trans. Amer. Math. Soc.359(2007), no. 8, 3733–3768

2007

[4] [4]

Codogni and F

G. Codogni and F. Viviani,Moduli and periods of supersymmetric curves, Adv. Theor. Math. Phys.23 (2019), no. 2, 345–402

2019

[5] [5]

Cornalba,Moduli of curves and theta-characteristics, inLectures on Riemann Surfaces, Trieste, 1987, World Scientific, 1989, 560–589

M. Cornalba,Moduli of curves and theta-characteristics, inLectures on Riemann Surfaces, Trieste, 1987, World Scientific, 1989, 560–589

1987

[6] [6]

Formal moduli and the splitting theory of supermanifolds

M. Corrêa, S. Noja,Formal moduli and the splitting theory of supermanifolds, arXiv:2605.03166

work page internal anchor Pith review Pith/arXiv arXiv

[7] [7]

Donagi and E

R. Donagi and E. Witten,Supermoduli space is not projected, inString-Math 2012, Proc. Sympos. Pure Math.90, Amer. Math. Soc., Providence, RI, 2015, 19–71

2012

[8] [8]

Donagi and E

R. Donagi and E. Witten,Super Atiyah classes and obstructions to splitting of supermoduli space, Pure Appl. Math. Q.9(2013), no. 4, 739–788

2013

[9] [9]

Felder, D

G. Felder, D. A. Kazhdan and A. Polishchuk,The moduli space of stable supercurves and its canonical line bundle, Amer. J. Math.145(2023), no. 6, 1777–1886

2023

[10] [10]

Felder, D

G. Felder, D. Kazhdan and A. Polishchuk,Superperiods and superstring measure near the boundary of the moduli space of supercurves, arXiv:2408.11136

work page arXiv

[11] [11]

Green,On holomorphic graded manifolds, Proc

P . Green,On holomorphic graded manifolds, Proc. Amer. Math. Soc.85(1982), no. 4, 587–590

1982

[12] [12]

Mumford,Theta characteristics of an algebraic curve, Ann

D. Mumford,Theta characteristics of an algebraic curve, Ann. Sci. École Norm. Sup. (4)4(1971), 181–192

1971

[13] [13]

Witten,Notes on super Riemann surfaces and their moduli, Pure Appl

E. Witten,Notes on super Riemann surfaces and their moduli, Pure Appl. Math. Q.15(2019), no. 1, 57–211. DIPARTIMENTO DIMATEMATICA, UNIVERSITÀ DEGLISTUDI DIBARIALDOMORO, BARI, ITALY Email address:mauricio.barros@uniba.it DIPARTIMENTO DIMATEMATICA, UNIVERSITÀ DEGLISTUDI DIBARIALDOMORO, BARI, ITALY Email address:simone.noja@uniba.it

2019