A cellular (co)homology computation for $\overline{M_{0,n}}$

Jan Hennig

arxiv: 2606.21272 · v1 · pith:I4UFNW3Unew · submitted 2026-06-19 · 🧮 math.AG · math.AT· math.KT

A cellular (co)homology computation for overline{M_(0,n)}

Jan Hennig This is my paper

Pith reviewed 2026-06-26 13:13 UTC · model grok-4.3

classification 🧮 math.AG math.ATmath.KT

keywords moduli spacestable curvescellular decompositionhomologyChow groupssingular cohomologygenus zero curves

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The pith

A cellular decomposition computes the homology of the moduli space of stable genus zero curves.

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

This paper sets up a method to calculate the (co)homology of algebraic spaces using a cellular decomposition when the coefficients are strictly A1-invariant sheaves. The central example is the moduli space of stable genus zero curves with n marked points, which the paper shows has the required cellular structure. The method captures classical invariants including Chow groups and the singular cohomology of the real points of the space. It further permits extending enumerative results from algebraically closed fields to arbitrary fields. A reader would care because this provides a single computational tool for several different kinds of invariants on these spaces.

Core claim

The paper establishes that because the moduli space of stable genus zero curves with n marked points admits a cellular structure via a stratification by cohomologically trivial spaces, its (co)homology with values in strictly A1-invariant sheaves is computed by the associated cellular chain complex, which in turn encapsulates the Chow groups and the singular cohomology of the real points.

What carries the argument

The cellular structure, a stratification of the space into cohomologically trivial pieces that permits homology computation from the strata alone.

If this is right

The Chow groups of the moduli space are determined by the cellular decomposition.
The singular cohomology of the real points of the moduli space is obtained as a special case of the computation.
Enumerative geometry arguments extend from algebraically closed fields to general fields using the same cellular data.
The method applies directly to other cellular spaces such as projective spaces and their products.

Where Pith is reading between the lines

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

This suggests the possibility of applying similar stratifications to compute invariants for other moduli spaces.
Comparisons between different base fields become more systematic through the cellular chains.
One could verify the computations explicitly for small n against known values.

Load-bearing premise

The space of stable genus zero curves with n marked points admits a stratification by spaces that are cohomologically trivial with respect to the strictly A1-invariant sheaves.

What would settle it

A mismatch between the cellular chain homology and the actual (co)homology groups for some known case of the moduli space with a specific sheaf coefficient.

read the original abstract

In this article we set up and showcase cellular computations for (co)homology with values in strictly $\mathbb{A}^1$-invariant sheaves. These computations encapsulate many classical invariants like Chow groups and singular cohomology of the real points. They also extend enumerative arguments from algebraically closed fields to more general fields. The spaces considered here have to admit a cellular structure. Instead of using the classical notion of cellularity, i.e. having a stratification by affine spaces, more general stratifications by cohomologically trivial spaces are used, following Morel--Sawant. Examples of cellular spaces include projective spaces and their products, but also spaces such as $\overline{M_{0,n}}$, the moduli space of stable genus $0$ curves with $n$ marked points. For these examples, we showcase the computations and show how to derive the classical results. Hopefully, the following text provides enough evidence to be convincing that such computations are doable and is encouraging to start computing the cohomology for more cellular spaces. This is part of the author's PhD thesis.

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 sets up Morel-Sawant cellular computations for ĂM_{0,n} that recover Chow groups and real cohomology, but asserts the needed stratification without exhibiting it or checking the vanishing condition.

read the letter

The core point is that this work applies the Morel-Sawant version of cellularity to ĂM_{0,n} so that (co)homology with strictly A^1-invariant sheaves can be read off a filtration whose quotients are cohomologically trivial. It recovers the usual Chow ring and the singular cohomology of the real points, and it frames the same method for projective spaces first as a check.

It does a clean job of showing how the spectral sequence collapses in these cases and how the approach extends enumerative counts from algebraically closed fields to arbitrary base fields. The examples are chosen sensibly and the recovery of known answers serves as a sanity test.

The weak point is exactly the one the stress-test flags: the paper states that ĂM_{0,n} carries a stratification by cohomologically trivial strata but does not display the stratification (for instance via dual graphs) or verify that each stratum satisfies H^i(-,F)=0 for i>0 when F is strictly A^1-invariant. Without that step the spectral sequence cannot be invoked, so the encapsulation claim rests on an unshown assertion.

The manuscript is written for people already working in A^1-homotopy theory who want a template for moduli-space calculations. A reader who needs the explicit cellular data for ĂM_{0,n} will still have to supply it. The thinking is coherent on its own terms and the citations to Morel-Sawant are appropriate.

I would send it to peer review; the framework is worth referee attention even if the central verification step needs to be written out.

Referee Report

1 major / 0 minor

Summary. The manuscript develops a framework for (co)homology computations with coefficients in strictly A^1-invariant sheaves on cellular spaces in the sense of Morel-Sawant, i.e., spaces admitting a stratification whose successive quotients are cohomologically trivial. It applies the resulting spectral sequence to recover Chow groups and singular cohomology of real points for projective spaces and their products, and claims to do the same for the moduli space ar M_{0,n} of stable genus-0 curves with n marked points, thereby extending enumerative results from algebraically closed fields to arbitrary fields. The text presents these as examples and states that the computations are feasible for other cellular spaces.

Significance. If the cellular stratification of ar M_{0,n} is established with the required vanishing property, the approach would supply a uniform method for computing A^1-homology invariants of moduli spaces that simultaneously captures algebraic cycle classes and topological information over general base fields. This could streamline extensions of classical enumerative geometry and provide concrete test cases for A^1-homotopy theory beyond affine spaces.

major comments (1)

[Abstract (paragraph on cellular spaces) and the section presenting the ar M_{0,n} example] The central claim that ar M_{0,n} admits a Morel-Sawant cellular structure (and therefore that its (co)homology with strictly A^1-invariant coefficients can be computed via the cellular spectral sequence) rests on the assertion that it possesses a stratification by cohomologically trivial strata. No explicit stratification (for instance by dual graphs or boundary divisors) is exhibited, nor is the required vanishing H^i(X_j, F)=0 for i>0 and F strictly A^1-invariant verified for the successive quotients X_j. Without this step the spectral sequence cannot be invoked and the encapsulation of Chow groups or real singular cohomology does not follow.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comment on the treatment of ar M_{0,n}. We respond to the major comment below.

read point-by-point responses

Referee: [Abstract (paragraph on cellular spaces) and the section presenting the ar M_{0,n} example] The central claim that ar M_{0,n} admits a Morel-Sawant cellular structure (and therefore that its (co)homology with strictly A^1-invariant coefficients can be computed via the cellular spectral sequence) rests on the assertion that it possesses a stratification by cohomologically trivial strata. No explicit stratification (for instance by dual graphs or boundary divisors) is exhibited, nor is the required vanishing H^i(X_j, F)=0 for i>0 and F strictly A^1-invariant verified for the successive quotients X_j. Without this step the spectral sequence cannot be invoked and the encapsulation of Chow groups or real singular cohomology does not follow.

Authors: We agree that the manuscript asserts the cellularity of ar M_{0,n} (and states that the spectral sequence applies) without exhibiting an explicit stratification or verifying the vanishing conditions for the successive quotients. The text presents ar M_{0,n} as an example alongside projective spaces but does not carry out the full construction for it. We will revise the abstract and the relevant section to remove the implication that the computations are fully showcased for ar M_{0,n} in the present work, or alternatively to add a concise description of the dual-graph stratification together with an inductive argument that the quotients are cohomologically trivial (using that the strata are themselves products of lower ar M_{0,k} and that the differences satisfy the required vanishing by known properties of the moduli spaces). Either change will make the claims accurate. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external Morel-Sawant framework

full rationale

The paper assumes ar M_{0,n} admits a Morel-Sawant cellular stratification (as one of several listed examples) and then performs the (co)homology computation under that hypothesis. No equation or claim reduces a derived quantity to a fitted parameter or self-defined input by construction. The central premise is justified by citation to Morel-Sawant rather than any self-citation chain, and the work contains no fitted inputs, ansatzes smuggled via prior self-work, or renamings of known results presented as new derivations. The derivation is therefore self-contained against the stated external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities can be extracted.

pith-pipeline@v0.9.1-grok · 5713 in / 1039 out tokens · 23055 ms · 2026-06-26T13:13:33.937596+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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

[1]

Tom Bachmann, Baptiste Calmès, Frédéric Déglise, Jean Fasel, and Paul Arne Østvær,Milnor-Witt Motives, (2022),https://arxiv.org/abs/2004.06634

work page arXiv 2022
[2]

Daniel DuggerNotes on the Milnor conjectures, (2004),https://arxiv.org/ abs/math/0408436

work page internal anchor Pith review Pith/arXiv arXiv 2004
[3]

Rains,The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points

Pavel Etingof, André Henriques, Joel Kamnitzer, and Eric M. Rains,The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points. Ann. of Math. 171 (2010), no. 2, 731–777

2010
[4]

K-Theory 11 (2013), no

Jean Fasel,The projective bundle theorem forIj-cohomology, J. K-Theory 11 (2013), no. 2, 413–464

2013
[5]

Jean Fasel,Lectures on Chow-Witt groups, Motivic homotopy theory and re- fined enumerative geometry, Contemp. Math. 745 (2020), 83–121

2020
[6]

Jan Hennig,A cellular Milnor-Witt (co)homology computation for the moduli space of stable, genus 0 curves with marked points, (2025), PhD Thesis

2025
[7]

Jan Hennig,The real cycle class isomorphism for linear schemes, (2025), arXiv 2511.02549

work page arXiv 2025
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K-Theory 6 (2021), no

Jens Hornbostel, Matthias Wendt, Heng Xie, and Marcus Zibrowius,The real cycle class map, Ann. K-Theory 6 (2021), no. 2, 239–317

2021
[9]

Maxim Kontsevich, and Yuri Manin,Gromov-Witten classes, quantum coho- mology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562

1994
[10]

Jesse Leo Kass, and Kirsten Wickelgren,An arithmetic count of the lines on a smooth cubic surface, Compos. Math. 157 (2021), no. 4, 677–709

2021
[11]

Sean Keel,Intersection theory of moduli space of stablen-pointed curves of genus zero. Trans. Amer. Math. Soc. 330 (1992), no. 2, 545–574

1992
[12]

Martin Markl, Steve Shnider, and Jim Stasheff,Operads in algebra, topol- ogy and physics, American Mathematical Society, Mathematical Surveys and Monographs 96 (2002), x+349

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[13]

Fabien Morel,A 1-algebraic topology over a field, Springer, Lecture Notes in Mathematics 2052 (2012), x+259

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[14]

Fabien Morel, and Anand Sawant,CellularA1-homology and the motivic ver- sion of Matsumoto’s theorem, Adv. Math. 434 (2023) 14 JAN HENNIG

2023
[15]

Dmitri Orlov, Alexander Vishik, and Vladimir Voevodsky,An exact sequence forK M ∗ /2with applications to quadratic forms, Ann. of Math. (2) 165 (2007), no. 1, 1–13 Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany Email address:jan.hennig@uni-duesseldorf.de

2007

[1] [1]

Tom Bachmann, Baptiste Calmès, Frédéric Déglise, Jean Fasel, and Paul Arne Østvær,Milnor-Witt Motives, (2022),https://arxiv.org/abs/2004.06634

work page arXiv 2022

[2] [2]

Daniel DuggerNotes on the Milnor conjectures, (2004),https://arxiv.org/ abs/math/0408436

work page internal anchor Pith review Pith/arXiv arXiv 2004

[3] [3]

Rains,The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points

Pavel Etingof, André Henriques, Joel Kamnitzer, and Eric M. Rains,The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points. Ann. of Math. 171 (2010), no. 2, 731–777

2010

[4] [4]

K-Theory 11 (2013), no

Jean Fasel,The projective bundle theorem forIj-cohomology, J. K-Theory 11 (2013), no. 2, 413–464

2013

[5] [5]

Jean Fasel,Lectures on Chow-Witt groups, Motivic homotopy theory and re- fined enumerative geometry, Contemp. Math. 745 (2020), 83–121

2020

[6] [6]

Jan Hennig,A cellular Milnor-Witt (co)homology computation for the moduli space of stable, genus 0 curves with marked points, (2025), PhD Thesis

2025

[7] [7]

Jan Hennig,The real cycle class isomorphism for linear schemes, (2025), arXiv 2511.02549

work page arXiv 2025

[8] [8]

K-Theory 6 (2021), no

Jens Hornbostel, Matthias Wendt, Heng Xie, and Marcus Zibrowius,The real cycle class map, Ann. K-Theory 6 (2021), no. 2, 239–317

2021

[9] [9]

Maxim Kontsevich, and Yuri Manin,Gromov-Witten classes, quantum coho- mology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562

1994

[10] [10]

Jesse Leo Kass, and Kirsten Wickelgren,An arithmetic count of the lines on a smooth cubic surface, Compos. Math. 157 (2021), no. 4, 677–709

2021

[11] [11]

Sean Keel,Intersection theory of moduli space of stablen-pointed curves of genus zero. Trans. Amer. Math. Soc. 330 (1992), no. 2, 545–574

1992

[12] [12]

Martin Markl, Steve Shnider, and Jim Stasheff,Operads in algebra, topol- ogy and physics, American Mathematical Society, Mathematical Surveys and Monographs 96 (2002), x+349

2002

[13] [13]

Fabien Morel,A 1-algebraic topology over a field, Springer, Lecture Notes in Mathematics 2052 (2012), x+259

2052

[14] [14]

Fabien Morel, and Anand Sawant,CellularA1-homology and the motivic ver- sion of Matsumoto’s theorem, Adv. Math. 434 (2023) 14 JAN HENNIG

2023

[15] [15]

Dmitri Orlov, Alexander Vishik, and Vladimir Voevodsky,An exact sequence forK M ∗ /2with applications to quadratic forms, Ann. of Math. (2) 165 (2007), no. 1, 1–13 Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany Email address:jan.hennig@uni-duesseldorf.de

2007