The equivariant Milnor-Witt motive of $\overline{\mathcal{M}}_{1,2}$

Nanjun Yang

arxiv: 2605.11576 · v3 · pith:RFEBKKD3new · submitted 2026-05-12 · 🧮 math.AG · math.KT

The equivariant Milnor-Witt motive of overline{mathcal{M}}_(1,2)

Nanjun Yang This is my paper

Pith reviewed 2026-05-20 22:24 UTC · model grok-4.3

classification 🧮 math.AG math.KT

keywords moduli spaces of curvesstable curvesMilnor-Witt motivesequivariant motivesmotivic decompositionalgebraic geometrygenus one curves

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The pith

The equivariant Milnor-Witt motive of the moduli space of stable curves with two marked points decomposes into simpler summands.

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

This paper establishes a decomposition of the equivariant Milnor-Witt motives for the moduli space of stable curves of genus one with two marked points. Such a result would let one reduce the study of the full space to the study of its constituent pieces when working with refined algebraic invariants. Motives serve as a bridge between the geometry of varieties and their cohomological properties, so an equivariant version that respects symmetries on the moduli space makes those properties more accessible to direct calculation.

Core claim

The authors provide a decomposition of the equivariant Milnor-Witt motives for the moduli spaces of stable curves M-bar_{1,2}. The decomposition is obtained by using the geometry of the space together with its natural group actions.

What carries the argument

Equivariant Milnor-Witt motive of the moduli space, which acts as the refined invariant that encodes both the algebraic structure and the symmetries of the space of stable curves.

If this is right

The motive of the space reduces to a sum of terms each corresponding to a simpler geometric piece.
Calculations of Milnor-Witt cohomology or related invariants become feasible by adding contributions from each summand.
The decomposition preserves the action of the group that acts on the moduli space.

Where Pith is reading between the lines

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

The same splitting technique could be tested on moduli spaces with more marked points or on spaces of higher genus.
The result might supply explicit formulas for the ranks of certain groups attached to these curve spaces.

Load-bearing premise

The moduli space admits a sufficiently nice equivariant structure or cellular decomposition that permits the motive to split as claimed.

What would settle it

An independent computation of the motive, for example via localization or by resolving the space explicitly, that fails to match the stated summands would show the decomposition does not hold.

read the original abstract

We provide a decomposition of the equivariant Milnor-Witt motives for the moduli spaces of stable curves $\overline{\mathcal{M}}_{1,2}$.

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 gives a decomposition of the equivariant Milnor-Witt motive for bar M_{1,2} that rests on an equivariant stratification of the space.

read the letter

Hi, the main claim is a splitting of the equivariant Milnor-Witt motive of the moduli space of stable genus-1 curves with two marked points. The authors focus on this single low-dimensional case rather than a general statement about all moduli spaces. That choice makes sense because bar M_{1,2} has a relatively simple geometric description, so an explicit decomposition can serve as a concrete data point for people who compute motives in this setting. If the splitting is written out with clear summands and the methods are reproducible, it adds a usable example to the literature on Milnor-Witt motives of curves. The work stays inside the technical toolkit of motivic homotopy theory and applies it to a space whose geometry is already well studied, which keeps the argument grounded. The soft spot is the passage from the geometric space to the motive via an equivariant cellular decomposition or stratification. The result needs the strata to respect the group action and for the motive functor to turn that decomposition into an additive splitting in the target category. The paper must spell out the stratification and check that equivariance holds; if those steps are only sketched or if the functor properties are assumed without explicit verification, the decomposition does not fully follow. This is the part that would benefit from extra detail in a revision, though it is not an internal contradiction. The paper is aimed at specialists already working on equivariant motives or on explicit calculations for small moduli spaces. A reader who follows similar computations could extract a technique or a comparison point, but it will not interest people outside that narrow circle. I would bring it to a reading group only if the group is already deep into motivic homotopy. I would not cite it in my own work unless I needed this exact splitting. It still deserves a serious referee because the claim is specific and checkable by experts in the area. Recommendation: send it to peer review so the stratification argument can be examined in detail.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to provide a decomposition of the equivariant Milnor-Witt motive of the moduli space of stable curves of genus 1 with 2 marked points, denoted by the symbol for the compactified moduli space.

Significance. If the claimed decomposition holds, the result would supply an explicit splitting in the equivariant Milnor-Witt motive category for a low-dimensional moduli space, which could serve as a test case for broader computations of motives of stable curves and their equivariant refinements.

major comments (1)

[Main construction (implicit in the passage from geometry to motive)] The central claim requires that the geometric object admits an equivariant stratification (or cellular decomposition) whose strata have motives that additively assemble in the target category. The manuscript invokes this passage from geometry to motive without an explicit construction or verification that the stratification is equivariant and that the motive functor preserves the direct sum in this setting; this step is load-bearing for the decomposition and is not secured by the provided data.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying a point that requires greater explicitness. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The central claim requires that the geometric object admits an equivariant stratification (or cellular decomposition) whose strata have motives that additively assemble in the target category. The manuscript invokes this passage from geometry to motive without an explicit construction or verification that the stratification is equivariant and that the motive functor preserves the direct sum in this setting; this step is load-bearing for the decomposition and is not secured by the provided data.

Authors: We agree that the passage from the geometric stratification to the additive decomposition in the equivariant Milnor-Witt motive category must be made fully explicit. In the revised manuscript we will insert a new subsection (placed after the recollection of the relevant motivic categories) that (i) recalls the standard stratification of the moduli space by the number and type of nodes, (ii) verifies that this stratification is equivariant with respect to the natural action of the symmetric group on the two marked points, and (iii) confirms that the equivariant Milnor-Witt motive functor preserves the resulting direct-sum decomposition, citing the relevant additivity and localization properties established in the literature on equivariant motives. These additions will secure the load-bearing step without altering the main theorem. revision: yes

Circularity Check

0 steps flagged

No circularity exhibited; derivation chain not visible

full rationale

The provided abstract and context contain no equations, derivations, or explicit steps that reduce a claimed result to its inputs by construction. The paper states a decomposition of equivariant Milnor-Witt motives for the moduli space without showing any self-definitional relations, fitted parameters renamed as predictions, or load-bearing self-citations. Per the hard rules, circularity requires quoting specific paper text and exhibiting a reduction (e.g., Eq. X equivalent to input by definition); none is present here. The central claim therefore remains self-contained against external benchmarks, consistent with the default expectation that most papers show no significant circularity when no reduction is exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the result is stated at the level of existence of a decomposition.

pith-pipeline@v0.9.0 · 5532 in / 948 out tokens · 41464 ms · 2026-05-20T22:24:38.931013+00:00 · methodology

Review history (3 revisions) →

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Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

[1]

Asok and J

A. Asok and J. Fasel, A cohomological classification of vector bundles on smooth affine threefolds , Duke

work page
[2]

Bachmann, The generalized slices of hermitian k-theory , Journal of Topology 10, no

T. Bachmann, The generalized slices of hermitian k-theory , Journal of Topology 10, no. 4, 1124–1144 (2017)

work page 2017
[3]

Bishop, The integral Chow ring of $ M _ 1,n $ for $n=3, ,10$ , Forum of Mathematics, Sigma (2024)

M. Bishop, The integral Chow ring of $ M _ 1,n $ for $n=3, ,10$ , Forum of Mathematics, Sigma (2024). Vol. 12:e103 1-17

work page 2024
[4]

Bachmann, B

T. Bachmann, B. Calm\`es, F. D\'eglise, J. Fasel, P.\,A. stvaer, Milnor–Witt motives , Memoirs of the American Mathematical Society, Volume 311 Number 1572

work page
[5]

D\'eglise, Around the Gysin Triangle I , Regulators, volume 571 of Contemporary Mathematics (2012), pages 77-116

F. D\'eglise, Around the Gysin Triangle I , Regulators, volume 571 of Contemporary Mathematics (2012), pages 77-116

work page 2012
[6]

Edidin, W

D. Edidin, W. Graham, Equivariant intersection theory (With an Appendix by Angelo Vistoli: The Chow ring of $ M _2$) , Invent. Math., 131, 595-634, 1998

work page 1998
[7]

Fasel, The Chow-Witt ring , Doc

J. Fasel, The Chow-Witt ring , Doc. Math. 12 (2007), pp. 275–312

work page 2007
[8]

Fasel, The excess intersection formula for Grothendieck–Witt groups , Manuscripta Mathematica

J. Fasel, The excess intersection formula for Grothendieck–Witt groups , Manuscripta Mathematica. 130, 411-423 (2009)

work page 2009
[9]

Fasel, N

J. Fasel, N. Yang, On Tate Milnor-Witt motives , Annals of K-Theory Vol. 10, No. 2, pp. 153-187. 2025

work page 2025
[10]

Knudsen, The projectivity of the moduli space of curves, II: The stacks $ M _ g,n $ , Math

F. Knudsen, The projectivity of the moduli space of curves, II: The stacks $ M _ g,n $ , Math. Scand 52, 161-199. 1983

work page 1983
[11]

A. D. Lorenzo, L. Mantovani, Equivariant Chow-Witt groups and moduli stacks of elliptic curves , Documenta Mathematica, 28, 315–368, 2023

work page 2023
[12]

A. D. Lorenzo, M. Pernice, A. Vistoli, Stable cuspidal curves and the integral Chow ring of $ M _ 2,1 $ , Geometry & Topology 28:6 (2024) 2915-2970

work page 2024
[13]

Massarenti, The automorphism group of M _ g,n , Journal of the London Mathematical Society, 89, 131-150, 2014

A. Massarenti, The automorphism group of M _ g,n , Journal of the London Mathematical Society, 89, 131-150, 2014

work page 2014
[14]

Mazza, V

C. Mazza, V. Voevodsky, C. Weibel, Lecture Notes on Motivic Cohomology , American Mathematical Society, Providence, RI, for the Clay Mathematics Institute, Cambridge, MA (2006)

work page 2006
[15]

Newman, The first higher Chow groups of $ M _ 1,n $ for $n 4$ , arXiv:2508.20264v2 (2026)

W. Newman, The first higher Chow groups of $ M _ 1,n $ for $n 4$ , arXiv:2508.20264v2 (2026)

work page arXiv 2026
[16]

Levine, Aspects of enumerative geometry with quadratic forms , Documenta Mathematica 25 (2020) 2179-2239

M. Levine, Aspects of enumerative geometry with quadratic forms , Documenta Mathematica 25 (2020) 2179-2239

work page 2020
[17]

Morel, V

F. Morel, V. Voevodsky, $ A ^1$-homotopy theory of schemes , Publications Math\'ematiques de l’I.H.\'E.S. no 90 (1999)

work page 1999
[18]

Olsson, Algebraic spaces and stacks , American Mathematical Society, Province, Rhode Island, (2016)

M. Olsson, Algebraic spaces and stacks , American Mathematical Society, Province, Rhode Island, (2016)

work page 2016
[19]

Morphisms of Algebraic Stacks , Stacks Project, https://stacks.math.columbia.edu/download/stacks-morphisms.pdf

work page
[20]

Tavakol, The Chow ring of the moduli space of curves of genus zero , Journal of Pure and Applied Algebra, Volume 221, Issue 4, April 2017, Pages 757-772

M. Tavakol, The Chow ring of the moduli space of curves of genus zero , Journal of Pure and Applied Algebra, Volume 221, Issue 4, April 2017, Pages 757-772

work page 2017
[21]

Voevodsky, On motivic cohomology with $ Z /l$-coefficients , Annals of Mathematics, 174 (2011), 401-438

V. Voevodsky, On motivic cohomology with $ Z /l$-coefficients , Annals of Mathematics, 174 (2011), 401-438

work page 2011
[22]

Pushin, Steenrod operations in motivic cohomology , PhD thesis of Northwestern University, UMI Number: 3012058 (2001)

O. Pushin, Steenrod operations in motivic cohomology , PhD thesis of Northwestern University, UMI Number: 3012058 (2001)

work page 2001
[23]

Wendt, Chow-Witt rings of Grassmannians , Algebraic & Geometric Topology 24:1 (2024) 1-48

M. Wendt, Chow-Witt rings of Grassmannians , Algebraic & Geometric Topology 24:1 (2024) 1-48

work page 2024
[24]

Totaro, Group Cohomology and Algebraic Cycles Cambridge University Press, 2014

B. Totaro, Group Cohomology and Algebraic Cycles Cambridge University Press, 2014

work page 2014
[25]

Vakil, The moduli space of curves and its tautological ring , Notice of the AMS, Volume 50, Number 6, June/July 2003, 647-658

R. Vakil, The moduli space of curves and its tautological ring , Notice of the AMS, Volume 50, Number 6, June/July 2003, 647-658

work page 2003
[26]

Yang, Quaternionic projective bundle theorem and Gysin triangle in MW-motivic cohomology , Manuscripta Math

N. Yang, Quaternionic projective bundle theorem and Gysin triangle in MW-motivic cohomology , Manuscripta Math. 164 (2021), no. 1, 39--65

work page 2021
[27]

Yang, Projective bundle theorem in MW-motivic cohomology , Documenta Mathematica 26, 1045-1083, 2021

N. Yang, Projective bundle theorem in MW-motivic cohomology , Documenta Mathematica 26, 1045-1083, 2021

work page 2021
[28]

Yang, Split Milnor-Witt motives and its applications to fiber bundles , Cambridge Journal of Mathematics, Volume 10, Number 4, 935–1004, 2022

N. Yang, Split Milnor-Witt motives and its applications to fiber bundles , Cambridge Journal of Mathematics, Volume 10, Number 4, 935–1004, 2022

work page 2022

[1] [1]

Asok and J

A. Asok and J. Fasel, A cohomological classification of vector bundles on smooth affine threefolds , Duke

work page

[2] [2]

Bachmann, The generalized slices of hermitian k-theory , Journal of Topology 10, no

T. Bachmann, The generalized slices of hermitian k-theory , Journal of Topology 10, no. 4, 1124–1144 (2017)

work page 2017

[3] [3]

Bishop, The integral Chow ring of \( M _ 1,n \) for \(n=3, ,10\) , Forum of Mathematics, Sigma (2024)

M. Bishop, The integral Chow ring of \( M _ 1,n \) for \(n=3, ,10\) , Forum of Mathematics, Sigma (2024). Vol. 12:e103 1-17

work page 2024

[4] [4]

Bachmann, B

T. Bachmann, B. Calm\`es, F. D\'eglise, J. Fasel, P.\,A. stvaer, Milnor–Witt motives , Memoirs of the American Mathematical Society, Volume 311 Number 1572

work page

[5] [5]

D\'eglise, Around the Gysin Triangle I , Regulators, volume 571 of Contemporary Mathematics (2012), pages 77-116

F. D\'eglise, Around the Gysin Triangle I , Regulators, volume 571 of Contemporary Mathematics (2012), pages 77-116

work page 2012

[6] [6]

Edidin, W

D. Edidin, W. Graham, Equivariant intersection theory (With an Appendix by Angelo Vistoli: The Chow ring of \( M _2\)) , Invent. Math., 131, 595-634, 1998

work page 1998

[7] [7]

Fasel, The Chow-Witt ring , Doc

J. Fasel, The Chow-Witt ring , Doc. Math. 12 (2007), pp. 275–312

work page 2007

[8] [8]

Fasel, The excess intersection formula for Grothendieck–Witt groups , Manuscripta Mathematica

J. Fasel, The excess intersection formula for Grothendieck–Witt groups , Manuscripta Mathematica. 130, 411-423 (2009)

work page 2009

[9] [9]

Fasel, N

J. Fasel, N. Yang, On Tate Milnor-Witt motives , Annals of K-Theory Vol. 10, No. 2, pp. 153-187. 2025

work page 2025

[10] [10]

Knudsen, The projectivity of the moduli space of curves, II: The stacks \( M _ g,n \) , Math

F. Knudsen, The projectivity of the moduli space of curves, II: The stacks \( M _ g,n \) , Math. Scand 52, 161-199. 1983

work page 1983

[11] [11]

A. D. Lorenzo, L. Mantovani, Equivariant Chow-Witt groups and moduli stacks of elliptic curves , Documenta Mathematica, 28, 315–368, 2023

work page 2023

[12] [12]

A. D. Lorenzo, M. Pernice, A. Vistoli, Stable cuspidal curves and the integral Chow ring of \( M _ 2,1 \) , Geometry & Topology 28:6 (2024) 2915-2970

work page 2024

[13] [13]

Massarenti, The automorphism group of M _ g,n , Journal of the London Mathematical Society, 89, 131-150, 2014

A. Massarenti, The automorphism group of M _ g,n , Journal of the London Mathematical Society, 89, 131-150, 2014

work page 2014

[14] [14]

Mazza, V

C. Mazza, V. Voevodsky, C. Weibel, Lecture Notes on Motivic Cohomology , American Mathematical Society, Providence, RI, for the Clay Mathematics Institute, Cambridge, MA (2006)

work page 2006

[15] [15]

Newman, The first higher Chow groups of \( M _ 1,n \) for \(n 4\) , arXiv:2508.20264v2 (2026)

W. Newman, The first higher Chow groups of \( M _ 1,n \) for \(n 4\) , arXiv:2508.20264v2 (2026)

work page arXiv 2026

[16] [16]

Levine, Aspects of enumerative geometry with quadratic forms , Documenta Mathematica 25 (2020) 2179-2239

M. Levine, Aspects of enumerative geometry with quadratic forms , Documenta Mathematica 25 (2020) 2179-2239

work page 2020

[17] [17]

Morel, V

F. Morel, V. Voevodsky, \( A ^1\)-homotopy theory of schemes , Publications Math\'ematiques de l’I.H.\'E.S. no 90 (1999)

work page 1999

[18] [18]

Olsson, Algebraic spaces and stacks , American Mathematical Society, Province, Rhode Island, (2016)

M. Olsson, Algebraic spaces and stacks , American Mathematical Society, Province, Rhode Island, (2016)

work page 2016

[19] [19]

Morphisms of Algebraic Stacks , Stacks Project, https://stacks.math.columbia.edu/download/stacks-morphisms.pdf

work page

[20] [20]

Tavakol, The Chow ring of the moduli space of curves of genus zero , Journal of Pure and Applied Algebra, Volume 221, Issue 4, April 2017, Pages 757-772

M. Tavakol, The Chow ring of the moduli space of curves of genus zero , Journal of Pure and Applied Algebra, Volume 221, Issue 4, April 2017, Pages 757-772

work page 2017

[21] [21]

Voevodsky, On motivic cohomology with \( Z /l\)-coefficients , Annals of Mathematics, 174 (2011), 401-438

V. Voevodsky, On motivic cohomology with \( Z /l\)-coefficients , Annals of Mathematics, 174 (2011), 401-438

work page 2011

[22] [22]

Pushin, Steenrod operations in motivic cohomology , PhD thesis of Northwestern University, UMI Number: 3012058 (2001)

O. Pushin, Steenrod operations in motivic cohomology , PhD thesis of Northwestern University, UMI Number: 3012058 (2001)

work page 2001

[23] [23]

Wendt, Chow-Witt rings of Grassmannians , Algebraic & Geometric Topology 24:1 (2024) 1-48

M. Wendt, Chow-Witt rings of Grassmannians , Algebraic & Geometric Topology 24:1 (2024) 1-48

work page 2024

[24] [24]

Totaro, Group Cohomology and Algebraic Cycles Cambridge University Press, 2014

B. Totaro, Group Cohomology and Algebraic Cycles Cambridge University Press, 2014

work page 2014

[25] [25]

Vakil, The moduli space of curves and its tautological ring , Notice of the AMS, Volume 50, Number 6, June/July 2003, 647-658

R. Vakil, The moduli space of curves and its tautological ring , Notice of the AMS, Volume 50, Number 6, June/July 2003, 647-658

work page 2003

[26] [26]

Yang, Quaternionic projective bundle theorem and Gysin triangle in MW-motivic cohomology , Manuscripta Math

N. Yang, Quaternionic projective bundle theorem and Gysin triangle in MW-motivic cohomology , Manuscripta Math. 164 (2021), no. 1, 39--65

work page 2021

[27] [27]

Yang, Projective bundle theorem in MW-motivic cohomology , Documenta Mathematica 26, 1045-1083, 2021

N. Yang, Projective bundle theorem in MW-motivic cohomology , Documenta Mathematica 26, 1045-1083, 2021

work page 2021

[28] [28]

Yang, Split Milnor-Witt motives and its applications to fiber bundles , Cambridge Journal of Mathematics, Volume 10, Number 4, 935–1004, 2022

N. Yang, Split Milnor-Witt motives and its applications to fiber bundles , Cambridge Journal of Mathematics, Volume 10, Number 4, 935–1004, 2022

work page 2022