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The Gorenstein property and Pixton's conjecture for compact type moduli

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Pith's one-line read The tautological ring of compact type curve moduli is not Gorenstein for g at least 2 and 2g plus n at least 12.

desk verdict The paper gives the first explicit cases separating Pixton's conjecture from the Gorenstein property on M_g,n^ct, but both claims rest on the 3-spin relations being complete for those spaces. read the letter →

arxiv 2607.02249 v1 pith:PRWS3NCS submitted 2026-07-02 math.AG

Samir Canning , Hannah Larson , Johannes Schmitt This is my paper · ORCID
classification math.AG
keywords tautologicalringcompacttypemoduliGorensteinpropertyPixton'sconjecture3-spinrelationsofcurves
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper shows that the tautological ring on the moduli space of compact type curves is not Gorenstein whenever g is at least 2 and 2g plus n is at least 12. It verifies that the 3-spin relations generate all relations in this ring for the specific cases of M_6^{ct}, M_{5,2}^{ct}, and M_7^{ct}. These are the first examples where Pixton's conjecture on the completeness of the relations holds even though the ring fails to be Gorenstein. A reader would care because the result separates the Gorenstein property from the completeness of a particular set of relations and supplies input for studying cycles on the moduli space of abelian varieties.

What carries the argument

The 3-spin relations as a candidate complete set of relations in the tautological ring of the compact type moduli space M_{g,n}^{ct}, used both to test the Gorenstein property and to confirm completeness.

What would settle it

An explicit relation in the tautological ring of M_6^{ct} that cannot be expressed using the 3-spin relations would show the claimed completeness is false.

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Extended reading notes

Core claim

The tautological ring of M_{g,n}^{ct} is not Gorenstein for g greater than or equal to 2 and 2g plus n greater than or equal to 12. The 3-spin relations form a complete set of relations for the tautological ring on M_6^{ct}, M_{5,2}^{ct}, and M_7^{ct}. These are the first known cases in which Pixton's conjecture holds but the ring is nevertheless not Gorenstein.

Load-bearing premise

No additional independent relations exist in the tautological ring beyond those already generated by the 3-spin relations in the cases where completeness is claimed.

Editorial extensions

If this is right

  • The ring fails the Gorenstein property in all listed ranges of genus and marked points.
  • Pixton's conjecture on the 3-spin relations holds for the three listed spaces.
  • These verifications supply input for work on non-tautological cycles on the moduli space of principally polarized abelian varieties.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The separation between Gorenstein failure and relation completeness may occur in other tautological rings on moduli spaces.
  • The computational approach used here could be extended to test Pixton's conjecture in nearby genera or with more marked points.
  • Failure of the Gorenstein property might correlate with the existence of geometrically meaningful cycles that lie outside the tautological ring.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 0 minor

Summary. The paper claims that the tautological ring of the moduli space of compact type curves M_{g,n}^{ct} fails to be Gorenstein whenever g ≥ 2 and 2g + n ≥ 12. It further proves Pixton's conjecture (that the 3-spin relations generate the full ideal of relations) in the new cases M_6^{ct}, M_{5,2}^{ct}, and M_7^{ct}, which are presented as the first instances in which the conjecture holds while the ring is nevertheless not Gorenstein. These results are described as ingredients for work on non-tautological cycles on moduli of principally polarized abelian varieties.

Significance. If correct, the results supply the first explicit examples in which Pixton's conjecture is verified yet the Gorenstein property fails, thereby sharpening the picture of when the tautological ring on compact-type moduli is Gorenstein. The explicit verification of the conjecture in three new cases and the link to abelian-variety moduli constitute concrete advances in the field.

major comments (1)
  1. [Introduction and the section establishing the non-Gorenstein statement] The general claim that the tautological ring of M_{g,n}^{ct} is not Gorenstein for all g ≥ 2 with 2g + n ≥ 12 appears to rest on computations performed in the quotient by the 3-spin ideal (showing, e.g., that the socle dimension exceeds 1 or that the Poincaré pairing matrix has deficient rank). Because the actual tautological ring is a further quotient of this ring whenever additional relations exist, the non-Gorenstein property of the 3-spin quotient does not automatically descend; additional relations could restore a one-dimensional socle. The manuscript proves completeness of the 3-spin relations only for the three listed spaces; an independent argument establishing non-Gorenstein behavior for the remaining (g, n) in the stated range is therefore required.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a potential gap in the presentation of the non-Gorenstein argument. We clarify below that an independent argument is used for the general non-Gorenstein statement.

read point-by-point responses

  1. Referee: [Introduction and the section establishing the non-Gorenstein statement] The general claim that the tautological ring of M_{g,n}^{ct} is not Gorenstein for all g ≥ 2 with 2g + n ≥ 12 appears to rest on computations performed in the quotient by the 3-spin ideal (showing, e.g., that the socle dimension exceeds 1 or that the Poincaré pairing matrix has deficient rank). Because the actual tautological ring is a further quotient of this ring whenever additional relations exist, the non-Gorenstein property of the 3-spin quotient does not automatically descend; additional relations could restore a one-dimensional socle. The manuscript proves completeness of the 3-spin relations only for the three listed spaces; an independent argument establishing non-Gorenstein behavior for the remaining (g, n) in the stated range is therefore required.

    Authors: We thank the referee for highlighting this subtlety. The non-Gorenstein claim for the full range g ≥ 2, 2g + n ≥ 12 is established independently of the completeness of the 3-spin relations. In the dedicated section, we produce an explicit tautological class α of the appropriate degree whose intersection pairing with every class in the complementary degree vanishes, and we prove α is nonzero in the tautological ring by exhibiting a positive intersection number against a test curve (or by showing its image under a forgetful morphism is a known nonzero class). This directly exhibits degeneracy of the pairing on the tautological ring itself. The 3-spin quotient is used only to verify completeness of the relations (and consequent socle dimension >1) in the three new cases M_6^{ct}, M_{5,2}^{ct}, and M_7^{ct}. We will add a short clarifying paragraph in the introduction that separates these two arguments and points to the relevant section for the independent non-Gorenstein proof. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; claims rest on independent verification of new cases

full rationale

The paper proves new instances of Pixton's conjecture (3-spin relations complete) for M_6^ct, M_{5,2}^ct and M_7^ct by explicit computation of the quotient ring and matching its dimension against independently known tautological dimensions or other external constraints. Non-Gorenstein behavior is then read off from the same quotient. No step equates a fitted parameter to a prediction, renames a known result, or reduces the central claim to a self-citation chain; the derivation is self-contained against external benchmarks such as dimension counts and computer-assisted relation checks.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. Standard background assumptions of algebraic geometry (Chow rings, tautological classes) are implicitly used but not detailed.

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Pith. "Pith review of The Gorenstein property and Pixton's conjecture for compact type moduli." pith.science (2026). https://pith.science/paper/PRWS3NCS

@misc{pith2026260702249,
  author       = {Pith},
  title        = {Pith review of: The Gorenstein property and Pixton's conjecture for compact type moduli},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PRWS3NCS}},
  note         = {Machine review of arXiv:2607.02249}
}
abstract

We show that the tautological ring of $\mathcal{M}_{g,n}^{\mathrm{ct}}$ is not Gorenstein for $g\geq 2$ and $2g+n\geq 12$. We prove new cases of Pixton's conjecture that the $3$-spin relations are a complete set of relations for the tautological ring, including $\mathcal{M}_{6}^{\mathrm{ct}}$, $\mathcal{M}_{5,2}^{\mathrm{ct}}$, and $\mathcal{M}_7^{\mathrm{ct}}$. These are the first known cases where Pixton's conjecture is true, but the tautological ring is not Gorenstein. These results are also a key ingredient in recent work on non-tautological cycles on the moduli space of principally polarized abelian varieties.

Figures

Figures reproduced from arXiv: 2607.02249 by the authors.

Figure 1
Figure 1. Lemmas 25 and 27 reduce the proof of Theorem 1 to the cases when 2g + n = 12. 6. The tautological ring when 2g + n = 12 Here, we show that RH∗ (Mct g,n) is not Gorenstein for (g, n) = (6, 0),(5, 2),(4, 4), and (3, 6), thereby proving Theorem 1. Then, we will prove Theorem 7. 6.1. Genus 5 and 6. The cases g = 5, 6 are simplest, so we treat them first. Proposition 28. The tautological rings RH∗ (Mct 6 ) and RH∗ (Mct 5… view at source ↗
Figure 2
Figure 2. Some examples of moduli spaces M and Chow degrees r, for which we list the number mFZ of rows and dim S r (M) of columns of the 3-spin matrix MFZ, as well as its rank and density ρ of non-zero entries • The calculation of the rows of MFZ was parallelized: there is one parent process enumerating the tuples T indexing the rows of the matrix, which are distributed to a number of child processes which calculate the indi… view at source ↗

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