pith. sign in

arxiv: 2606.29656 · v1 · pith:SOSBHF7Znew · submitted 2026-06-28 · 🧮 math.AG

Chow rings, cohomology rings, and point counts of moduli spaces of curves

Hannah Larson This is my paper

Pith reviewed 2026-06-30 07:12 UTC · model grok-4.3

classification 🧮 math.AG
keywords moduli spaces of curvesChow ringscohomology ringstautological ringspoint countsfinite fieldsstable curves
0
0 comments X

The pith

Chow rings of moduli spaces of curves are tautological when generated by kappa and psi classes.

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

The paper explains the relationship between Chow rings, cohomology rings, and point counts over finite fields for the moduli spaces M_{g,n} and their compactifications by stable curves. It then surveys the known cases in which the Chow ring coincides with its tautological subring, the cohomology is generated by tautological classes, and the number of points over a finite field of size q is a polynomial in q. A sympathetic reader would care because these properties show that the geometry of curve moduli spaces is governed by a small set of explicit classes and link algebraic geometry to arithmetic questions over finite fields.

Core claim

After relating the three invariants, the paper surveys the state of knowledge on when the Chow rings of M_{g,n} and bar M_{g,n} are tautological, when their cohomology groups are tautological, and when their point counts over fields of size q are given by a polynomial in q.

What carries the argument

The moduli spaces M_{g,n} and bar M_{g,n} together with the tautological subrings of their Chow and cohomology rings and the polynomial character of their point counts.

If this is right

  • In all known cases the tautological Chow ring is generated by the kappa and psi classes.
  • Tautological cohomology implies that all cohomology classes arise from the tautological ring.
  • Polynomial point counts imply the space behaves arithmetically like a variety with mixed Tate motive.
  • The three properties hold simultaneously for many small values of g and n.

Where Pith is reading between the lines

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

  • The same generation questions could be asked for moduli spaces of higher-dimensional varieties if similar tautological rings exist.
  • Polynomial point-count formulas might be used to predict the Betti numbers or Hodge numbers of the spaces.
  • New compactifications or degeneration techniques could settle the questions for larger g.

Load-bearing premise

The state-of-the-art results surveyed on tautological rings and polynomial point counts accurately reflect the current literature.

What would settle it

An explicit computation showing that the Chow ring of some M_{g,n} with small g and n is not generated by the tautological classes, or that a point count over F_q fails to be a polynomial in q when the survey claims it is.

read the original abstract

In this expository article, we present on state-of-the art results regarding three closely related invariants of moduli spaces of curves: their Chow rings, cohomology rings, and point counts over finite fields. We study the moduli space $\mathcal{M}_{g,n}$, parameterizing smooth genus $g$ curves with $n$ marked points, as well as its compactification by stable curves $\overline{\mathcal{M}}_{g,n}$. After explaining the relationship between these different invariants, we survey what is know regarding the following related questions: When are the Chow rings tautological? When are the cohomology groups tautological? And when are the point counts over fields of size $q$ given by a polynomial in $q$?

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.

Referee Report

0 major / 1 minor

Summary. This expository survey presents state-of-the-art results on three invariants of the moduli spaces of curves ℓ_{g,n} and its Deligne-Mumford compactification: Chow rings, cohomology rings, and point counts over finite fields. After relating these invariants, the paper surveys known results on the questions of when the Chow rings are tautological, when the cohomology groups are tautological, and when the point counts over ℓ_q are given by a polynomial in q.

Significance. As a survey organizing existing results on tautological rings and polynomial point counts for moduli spaces of curves, the manuscript can serve as a useful reference for algebraic geometers working in this area if the cited results are represented accurately. The paper advances no new theorems or computations.

minor comments (1)
  1. [Abstract] Abstract: 'what is know' is a typographical error and should read 'what is known'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We are pleased that the survey is viewed as potentially serving as a useful reference, provided the cited results are represented accurately.

Circularity Check

0 steps flagged

Expository survey with no original derivations or predictions

full rationale

This manuscript is explicitly an expository survey of existing literature on tautological Chow rings, cohomology, and polynomial point counts for moduli spaces of curves. It advances no new theorems, equations, fitted parameters, or first-principles derivations. All presented results are attributed to external references, with no self-citation chains or ansatzes that reduce claims to the paper's own inputs. The central questions surveyed are standard in the field and the presentation introduces no internal reductions or load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

As an expository survey the paper introduces no new free parameters, axioms, or invented entities. It relies on standard background in algebraic geometry including the definition of moduli spaces of curves and the notion of tautological rings.

pith-pipeline@v0.9.1-grok · 5638 in / 1047 out tokens · 54355 ms · 2026-06-30T07:12:45.953923+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

55 extracted references · 13 canonical work pages · 2 internal anchors

  1. [1]

    Arbarello and M

    E. Arbarello and M. Cornalba,Calculating cohomology groups of moduli spaces of curves via algebraic geometry, Inst. Hautes ´Etudes Sci. Publ. Math. (1998), no. 88, 97–127

    1998

  2. [2]

    Arbarello, M

    E. Arbarello, M. Cornalba, P. Griffiths,Geometry of algebraic curves. Volume II.With a contribution by Joseph Daniel Harris, Grundlehren Math. Wiss.,268, Springer, Heidelberg, 2011. 963 pp

    2011

  3. [3]

    Arena, S

    V. Arena, S. Canning, E. Clader, R. Haburcak, A. Li, S. Mok, C. Tamborini,Holomorphic forms and non-tautological cycles on moduli spaces of curves, 2024, to appear in Selecta Math., arXiv:2402.03874

    work page arXiv 2024

  4. [4]

    Bae and J

    Y. Bae and J. Schmitt,Chow rings of stacks of prestable curves II, J. Reine Angew. Math.800(2023), 55–106

    2023

  5. [5]

    K. A. Behrend,The Lefschetz trace formula for algebraic stacks, Invent. Math.112(1993), no. 1, 127–149

    1993

  6. [6]

    Belorousski,Chow rings of moduli spaces of pointed elliptic curves, ProQuest LLC, Ann Arbor, MI, 1998, Thesis (Ph.D.)–The University of Chicago

    P. Belorousski,Chow rings of moduli spaces of pointed elliptic curves, ProQuest LLC, Ann Arbor, MI, 1998, Thesis (Ph.D.)–The University of Chicago

    1998

  7. [7]

    Bergstr¨ om, Cohomology of moduli spaces of curves, Github,https://github.com/ jonasbergstroem/Cohomology-of-moduli-spaces-of-curves/blob/main/README.md

    J. Bergstr¨ om, Cohomology of moduli spaces of curves, Github,https://github.com/ jonasbergstroem/Cohomology-of-moduli-spaces-of-curves/blob/main/README.md. 24

  8. [8]

    Bergstr¨ om, S

    J. Bergstr¨ om, S. Canning, D. Petersen, and J. Schmitt,Motivic Euler characteristics of moduli spaces of curves, to appear

  9. [9]

    Bergstr¨ om and C

    J. Bergstr¨ om and C. F. Faber,Cohomology of moduli spaces via a result of Chenevier and Lannes, ´Epijournal G´ eom. Alg´ ebrique7(2023), Art. 20, 14 pp

    2023

  10. [10]

    Bergstr¨ om, C

    J. Bergstr¨ om, C. Faber and S. Payne,Polynomial point counts and odd cohomology vanishing on moduli spaces of stable curves, 2022, arXiv:2206.07759

    work page arXiv 2022

  11. [11]

    Burkhardt,Weight 11 Compactly Supported Cohomology of Moduli Spaces of Curves in excess four, 2024, arXiv:2407.16372

    L. Burkhardt,Weight 11 Compactly Supported Cohomology of Moduli Spaces of Curves in excess four, 2024, arXiv:2407.16372

    work page arXiv 2024

  12. [12]

    van den Bogaart and B

    T. van den Bogaart and B. Edixhoven, Algebraic stacks whose number of points over finite fields is a polynomial, inNumber fields and function fields—two parallel worlds, 39–49, Progr. Math., 239, Birkh¨ auser Boston, Boston, MA

  13. [13]

    Canning,The tautological ring of Mg,n is rarely Gorenstein, to appear in Geometry and Topology

    S. Canning,The tautological ring of Mg,n is rarely Gorenstein, to appear in Geometry and Topology. arXiv:2406.10516

    work page arXiv

  14. [14]

    Canning and H

    S. Canning and H. Larson,The Chow rings of the moduli spaces of curves of genus7,8, and9, Journal of Algebraic Geometry,33, (2024), no. 1, 55–116, arXiv:2104.05820

    work page arXiv 2024

  15. [15]

    Canning and H

    S. Canning and H. Larson,On the Chow and cohomology rings of moduli spaces of stable curves, to appear in Journal of the European Math Society,arXiv:2208.02357

    work page arXiv

  16. [16]

    Canning, H

    S. Canning, H. Larson and S. Payne,The eleventh cohomology group of Mg,n, Forum of Math., Sigma, 11(2023), Paper No. e62, 18pp

    2023

  17. [17]

    Canning, H

    S. Canning, H. Larson and S. Payne,Extensions of tautological rings and motivic structures in the cohomology of Mg,n, Forum Math. Pi12(2024), Paper No. e23, 31 pp

    2024

  18. [18]

    Moduli spaces of curves with polynomial point counts

    S. Canning, H. Larson, S. Payne and T. Willwacher,Moduli spaces of curves with polynomial point counts, arXiv:2410.19913 (2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  19. [19]

    Canning, H

    S. Canning, H. Larson, S. Payne and T. Willwacher,The motivic structuresLS 12 andS 16 in the coho- mology of moduli spaces of curves, arXiv:2411.12652 (2024)

    work page arXiv 2024

  20. [20]

    Canning, H

    S. Canning, H. Larson, S. Payne and T. Willwacher,F A-modules of holomorphic forms on Mg,n, arXiv:2509.08774 (2025)

    work page arXiv 2025

  21. [21]

    M. Chan, S. Galatius and S. Payne,Tropical curves, graph complexes, and top weight cohomology of Mg, J. Amer. Math. Soc. 34 (2021), no. 2, 565–594

    2021

  22. [22]

    M. Chan, S. Galatius and S. Payne,Topology of moduli spaces of tropical curves with marked points, inFacets of algebraic geometry. Vol. I, 77–131, London Math. Soc. Lecture Note Ser., 472, Cambridge Univ. Press, Cambridge

  23. [23]

    Deligne,La conjecture de Weil

    P. Deligne,La conjecture de Weil. I, Inst. Hautes ´Etudes Sci. Publ. Math. No. 43 (1974), 273–307

    1974

  24. [24]

    Deligne,Formes modulaires et repr´ esentationsl-adiques, inS´ eminaire Bourbaki

    P. Deligne,Formes modulaires et repr´ esentationsl-adiques, inS´ eminaire Bourbaki. Vol. 1968/69: Ex- pos´ es 347–363, Exp. No. 355, 139–172, Lecture Notes in Math., 175, Springer, Berlin

    1968

  25. [25]

    Deligne and D

    P. Deligne and D. B. Mumford,The irreducibility of the space of curves of given genus, Inst. Hautes ´Etudes Sci. Publ. Math. No. 36 (1969), 75–109

    1969

  26. [26]

    Faber,Chow rings of moduli spaces of curves, PhD Thesis, Universiteit van Amsterdam (1988)

    C. Faber,Chow rings of moduli spaces of curves, PhD Thesis, Universiteit van Amsterdam (1988)

    1988

  27. [27]

    Faber,Chow rings of moduli spaces of curves

    C. Faber,Chow rings of moduli spaces of curves. I. The Chow ring of M 3, Ann. of Math. (2)132 (1990), no. 2, 331–419

    1990

  28. [28]

    Faber,Chow rings of moduli spaces of curves

    C. Faber,Chow rings of moduli spaces of curves. II. Some results on the Chow ring of M 4, Ann. of Math. (2)132(1990), no. 3, 421–449

    1990

  29. [29]

    C. F. Faber and R. V. Pandharipande,Tautological and non-tautological cohomology of the moduli space of curves, inHandbook of moduli. Vol. I, 293–330, Adv. Lect. Math. (ALM), 24, Int. Press, Somerville, MA

  30. [30]

    Faro and C

    D. Faro and C. Tamborini,Non-tautological cycles on moduli spaces of smooth pointed curves, arXiv:2410.04400, 2024

    work page arXiv 2024

  31. [31]

    Getzler and M

    E. Getzler and M. M. Kapranov,Modular operads, Compositio Math.110(1998), no. 1, 65–126

    1998

  32. [32]

    Graber and R

    T. Graber and R. Pandharipande,Constructions of nontautological classes on moduli spaces of curves. Michigan Math. J.51(2003), no. 1, 93–109

    2003

  33. [33]

    Graber and R

    T. Graber and R. D. Vakil,On the tautological ring of Mg,n, Turkish J. Math.25(2001), no. 1, 237–243

    2001

  34. [34]

    Izadi,The Chow ring of the moduli space of curves of genus5, The moduli space of curves (Texel Island, 1994), Progr

    E. Izadi,The Chow ring of the moduli space of curves of genus5, The moduli space of curves (Texel Island, 1994), Progr. Math., vol. 129, Birkh¨ auser Boston, Boston, MA, 1995, pp. 267–304. MR 1363060 25

    1994

  35. [35]

    Keel,Intersection theory of moduli space of stablen-pointed curves of genus zero, Trans

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

    1992

  36. [36]

    Laumon,Fonctions zˆ etas des vari´ et´ es de Siegel de dimension trois, Ast´ erisque No

    G. Laumon,Fonctions zˆ etas des vari´ et´ es de Siegel de dimension trois, Ast´ erisque No. 302 (2005), 1–66

    2005

  37. [37]

    Liu,On the Chow rings of the moduli spacesM 5,8 andM 5,9, arXiv:2509.02950 (2025)

    Y. Liu,On the Chow rings of the moduli spacesM 5,8 andM 5,9, arXiv:2509.02950 (2025)

    work page arXiv 2025

  38. [38]

    Lu,On O’Grady’s generalized Franchetta conjecture for genus11K3 surfaces, arXiv:2511.16875 (2025)

    Y. Lu,On O’Grady’s generalized Franchetta conjecture for genus11K3 surfaces, arXiv:2511.16875 (2025)

    work page arXiv 2025

  39. [39]

    J. S. Milne, ´Etale cohomology, Princeton Mathematical Series, No. 33, Princeton Univ. Press, Princeton, NJ, 1980

    1980

  40. [40]

    D. B. Mumford,Towards an enumerative geometry of the moduli space of curves, inArithmetic and geometry, Vol. II, 271–328, Progr. Math., 36, Birkh¨ auser Boston, Boston, MA

  41. [41]

    Pandharipande,A calculus for the moduli space of curves

    R. Pandharipande,A calculus for the moduli space of curves. Algebraic geometry: Salt Lake City 2015, 459–487. Proc. Sympos. Pure Math., 97.1

    2015

  42. [42]

    Pandharipande and A

    R. Pandharipande and A. Pixton,Relations in the tautological ring of the moduli space of curves. Pure Appl. Math. Q. 17 (2021), no. 2, 717–771

    2021

  43. [43]

    R. V. Pandharipande, A. Pixton and D. Zvonkine,Relations on Mg,n via3-spin structures, J. Amer. Math. Soc.28(2015), no. 1, 279–309

    2015

  44. [44]

    Payne and T

    S. Payne and T. Willwacher,Weight 2 compactly supported cohomology of moduli spaces of curves, Duke Math. J.173(2024), no. 16, 3107–3178

    2024

  45. [45]

    Payne and T

    S. Payne and T. Willwacher,Weight 11 compactly supported cohomology of moduli spaces of curves, Int. Math. Res. Not. IMRN2024, no. 8, 7060–7098

  46. [46]

    Penev and R

    N. Penev and R. Vakil,The Chow ring of the moduli space of curves of genus six, Algebr. Geom.2 (2015), no. 1, 123–136

    2015

  47. [47]

    Petersen,The structure of the tautological ring in genus one, Duke Math

    D. Petersen,The structure of the tautological ring in genus one, Duke Math. J.163(2014), no. 4, 777–793

    2014

  48. [48]

    Petersen,Cohomology of local systems on the moduli of principally polarized abelian surfaces, Pacific J

    D. Petersen,Cohomology of local systems on the moduli of principally polarized abelian surfaces, Pacific J. Math.275(2015), no. 1, 39–61

    2015

  49. [49]

    Petersen, A spectral sequence for stratified spaces and configuration spaces of points, Geom

    D. Petersen, A spectral sequence for stratified spaces and configuration spaces of points, Geom. Topol. 21(2017), no. 4, 2527–2555

    2017

  50. [50]

    Conjectural relations in the tautological ring of $\bar{M}_{g,n}$

    A. Pixton,Conjectural relations in the tautological ring of Mg,n, arXiv:1207.1918 (2012)

    work page internal anchor Pith review Pith/arXiv arXiv 1918

  51. [51]

    G. F. B. Riemann,Theorie der Abel’schen Functionen, J. Reine Angew. Math.54(1857), 115–155

  52. [52]

    Totaro,The motive of a classifying space, Geom

    B. Totaro,The motive of a classifying space, Geom. Topol.20(2016), no. 4, 2079–2133

    2016

  53. [53]

    Wahl,Homological stability for the mapping class groups of non-orientable surfaces, Invent

    N. Wahl,Homological stability for the mapping class groups of non-orientable surfaces, Invent. Math. 171(2008), no. 2, 389–424

    2008

  54. [54]

    Weissauer, Four dimensional Galois representations, Ast´ erisque No

    R. Weissauer, Four dimensional Galois representations, Ast´ erisque No. 302 (2005), 67–150

    2005

  55. [55]

    van Zelm,Nontautological bielliptic cycles, Pacific J

    J. van Zelm,Nontautological bielliptic cycles, Pacific J. Math.294(2018), no. 2, 495–504. 26

    2018