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

Technical Summary: The Gorenstein Property and Pixton's Conjecture for Compact Type Moduli

Problem Statement
The paper addresses two fundamental open problems in the intersection theory of moduli spaces of curves: the structure of the tautological rings $R^*(\mathcal{M}^{\text{ct}}_{g,n})$ for compact type curves and the validity of Pixton's conjecture regarding the completeness of 3-spin relations. Specifically, the authors investigate whether these rings satisfy the Gorenstein property (where the intersection pairing is perfect) and whether the ideal of tautological relations is generated entirely by the 3-spin relations (Pixton's conjecture). While the Gorenstein property was conjectured for low genera, recent work suggested its failure in higher genera. The relationship between the validity of Pixton's conjecture and the Gorenstein property remained unclear, particularly in cases where the ring might fail to be Gorenstein.

Methodology
The authors employ a combination of theoretical reductions, structural analysis of tautological classes, and extensive computational verification using the admcycles software package.

1. Theoretical Framework:

   • Strata Algebra and Relations: The study utilizes the strata algebra $S^*(\mathcal{M}_{g,n})$ and its quotient by the 3-spin relations, denoted $R^*_{\text{FZ}}$. The authors analyze the map from the strata algebra to the tautological ring and the cohomological tautological ring.
   • Inductive Reductions: The proofs rely on gluing maps (e.g., attaching an elliptic tail) and forgetting maps to relate the properties of $\mathcal{M}^{\text{ct}}_{g,n}$ to lower genus or fewer marked points. Key lemmas establish that the existence of "invisible" classes (classes that pair to zero with all complementary classes) is preserved under these operations.
   • Theorem $\star$: The authors utilize the result that any class in the 3-spin quotient $R^d_{\text{FZ}}(\mathcal{M}_{g,n})$ can be represented by decorated strata with a specific number of genus 0 vertices. This allows for precise control over the generators of the ring in high codimensions.
   • Representation Theory: For specific cases (e.g., $(g,n)=(3,6)$), the authors decompose cohomology groups into irreducible representations of the symmetric group $S_n$ to compute dimensions of invariant subspaces, bypassing direct computation of the full ring.

2. Computational Strategy:

   • Matrix Rank Calculations: To determine the dimension of the tautological ring, the authors construct matrices representing the 3-spin relations. They compute the rank of these matrices modulo a prime $p$ to obtain upper bounds on the dimension of the ring.
   • Pairing Verification: Lower bounds are established by computing the rank of intersection pairings between complementary degrees. When the upper and lower bounds coincide, the 3-spin relations are proven complete, and the pairing is perfect.
   • Optimizations: Due to the combinatorial explosion in higher genera, the authors implemented parallelization, Least Recently Used (LRU) caching, and disk-based storage for matrix rows. They also utilized sparse linear algebra libraries (LinBox) and modular arithmetic to handle large matrices efficiently.

Key Contributions and Results

1. Failure of the Gorenstein Property:
   The authors prove that the tautological rings $R^*(\mathcal{M}^{\text{ct}}_{g,n})$ and $RH^*(\mathcal{M}^{\text{ct}}_{g,n})$ are not Gorenstein for all $g \ge 2$ and $2g + n \ge 12$.

   • This extends previous results by Petersen (who showed failure for $g=2, n \ge 8$) to higher genera.
   • The proof reduces the general case to the boundary cases where $2g + n = 12$ (specifically $(6,0), (5,2), (4,4), (3,6)$).
   • For these specific cases, the authors demonstrate the existence of non-zero classes in the kernel of the intersection pairing, thereby proving the ring is not Gorenstein.

2. Completeness of Pixton's Conjecture in Non-Gorenstein Cases:
   The paper proves new cases of Pixton's conjecture, showing that the 3-spin relations are a complete set of relations for the tautological ring in Chow and cohomology for:

   • $\mathcal{M}^{\text{ct}}_6$
   • $\mathcal{M}^{\text{ct}}_{5,2}$
   • $\mathcal{M}^{\text{ct}}_7$
   • Crucially, for these cases, the authors show that while Pixton's conjecture holds (the relations are complete), the ring is not Gorenstein. This establishes the first known instances where the conjecture is true despite the failure of the Gorenstein property.

3. Partial Converse and Low Genus Results:

   • The authors prove that for $g=0, 1$ and $g \ge 2$ with $2g + n < 12$ (excluding $(2,7)$), the tautological rings are isomorphic to the cohomological tautological rings and are Gorenstein.
   • They prove that the pairing between $R^1$ and its complement is always perfect (Theorem 4), confirming the Gorenstein property in codimension 1.

4. Connection to Abelian Varieties:
   The results on $\mathcal{M}^{\text{ct}}_6$ are identified as a key ingredient in recent work proving the existence of non-tautological cycles on the moduli space of principally polarized abelian varieties ($\mathcal{A}_6$). Specifically, the failure of the Gorenstein property in $\mathcal{M}^{\text{ct}}_6$ provides a geometric explanation for the non-tautological nature of certain classes in $\mathcal{A}_6$.

Significance
The paper's primary significance lies in decoupling the Gorenstein property from Pixton's conjecture. Prior to this work, it was unclear whether the failure of the Gorenstein property implied the incompleteness of the 3-spin relations. By proving that Pixton's conjecture holds for $\mathcal{M}^{\text{ct}}_6, \mathcal{M}^{\text{ct}}_{5,2},$ and $\mathcal{M}^{\text{ct}}_7$ while simultaneously proving these rings are not Gorenstein, the authors demonstrate that the 3-spin relations fully capture the tautological ring even when the intersection pairing is degenerate.

Furthermore, the paper provides a comprehensive classification of the Gorenstein property for compact type moduli spaces, identifying the precise threshold ($2g + n = 12$) where the property fails. The computational techniques developed to handle high-codimension relations in these moduli spaces serve as a practical foundation for further investigations into the structure of tautological rings and non-tautological cycles.