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Torelli loci, product cycles, and the homomorphism conjecture for $\mathcal{A}_g$

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Pith's one-line read This paper establishes new cases of the conjecture that tautological projection on the Chow ring of the moduli space of abelian varieties is multiplicative, and uses two independent calculations—excess intersection and Gromov-Witten wall-cr

desk verdict Genuinely new homomorphism-property computations, but Theorem 7 currently leans on two unpublished co-authored preprints. read the letter →

arxiv 2601.04353 v3 pith:ZBEJCBF6 submitted 2026-01-07 math.AG

Samir Canning , Lycka Drakengren , Jeremy Feusi , Daniel Holmes , Aitor Iribar López , Denis Nesterov , Dragos Oprea , Rahul Pandharipande
show 2 more authors
Johannes Schmitt Zheming Sun
This is my paper · ORCID
classification math.AG MSC 14K1014C2514H1014N35
keywords tautologicalringmoduliofabelianvarietiesTorellilocushomomorphismpropertyproductcyclesGorensteinkernelwall-crossingAbel-Jacobimaps
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's central claim is that the canonical tautological projection taut on the Chow ring of the moduli space A_g of principally polarized abelian varieties commutes with multiplication for the Jacobian locus times product loci in new cases: for ([J_g],[A_2×A_{g-2}]) for 2≤g≤8 and ([J_6],[A_3×A_3]) on A_6. The proof proceeds by computing the pullback of product cycles along the Torelli map, which is done twice independently: once via excess intersection on the fiber product (stratified by colored extremal trees) and once via families Gromov-Witten theory and a wall-crossing formula; both computations agree. The same framework produces explicit nonzero classes in the Gorenstein kernels of the tautological rings of moduli spaces of compact-type curves, namely a generator of the kernel in R^5(M^ct_{5,2}) and a nonzero class in R^5(M^ct_{4,4}). If correct, these results extend the homomorphism property to Torelli-type cycles and support the broader conjecture that taut is a Q-algebra homomorphism.

What carries the argument

The central object is the Torelli fiber product Z = M_g^ct ×_{A_g}(A_r × A_{g-r}), the moduli of compact-type curves whose Jacobian decomposes as a product. The excess-intersection computation uses a stratification of Z indexed by g-colored extremal trees (stable trees with genus and color data recording which curve components contribute to which product factor), together with local monomial equations for Z and a residual-intersection recursion expressing the Torelli pullback as a sum of tautological classes. The wall-crossing computation instead identifies Z with a moduli space of stable maps from compact-type curves to fibers of the universal abelian variety in the minimal curve class, and

What would settle it

Compute the Torelli pullback for g=9 using either method and test the predicted equality Tor*([A_2×A_{g-2}]) = Tor*(taut([A_2×A_{g-2}])); a nonzero difference would disprove the expected infinite-family extension. A second direct check: locate any nonreduced stratum in the fiber product for some other partition, since the genus-4 Torelli square is already known to be nonreduced.

Watch

Extended reading notes

Core claim

Theorem 7 states that for each genus g=2,...,8, multiplying the Jacobian cycle [J_g] by the product locus [A_2×A_{g-2}] and then applying tautological projection gives the same class as projecting each factor and multiplying; the same holds for [J_6] times [A_3×A_3] on A_6. Concretely, the paper computes the Torelli pullback Tor*([A_2×A_{g-2}]) as a tautological class on the moduli space of compact-type curves and shows it equals Tor*(taut([A_2×A_{g-2}])). Along the way, the paper defines 'generalized product loci' PR_{g,s} in the universal fiber product X_g^s and computes their tautological projections as a determinant in universal theta divisors and Poincaré classes. Pulling these projecti

Load-bearing premise

The load-bearing premise is that the Torelli fiber product Z is reduced, as asserted by a separate preprint the paper relies on; the wall-crossing branch additionally depends on an unpublished comparison identifying the reduced virtual class with the Torelli pullback. If either premise fails, the computed Torelli pullbacks are not justified.

Editorial extensions

If this is right

  • For 2≤g≤8 the equality Tor*([A_2×A_{g-2}]) = Tor*(taut([A_2×A_{g-2}])) holds in the tautological ring of M^ct_g; pushing forward under the Torelli map gives the homomorphism property for ([J_g],[A_2×A_{g-2}]).
  • For g≥7, Tor*([A_3×A_{g-3}]) = 0, so [J_g]·[A_3×A_{g-3}] = 0 in the Chow ring of A_g; the same vanishing mechanism yields [J_g]·[A_4×A_{g-4}]=0 for 8≤g≤13 and [J_g]·[A_5×A_{g-5}]=0 for 10≤g≤11.
  • The determinant formula for taut_s([PR_{g,s}]) gives a closed expression for the tautological projections of generalized product cycles on X_g^s for all s.
  • The classes aj*(Δ_{5,1}) and aj*(Δ_{4,2}) are explicit nonzero elements of the Gorenstein kernels, with aj*(Δ_{5,1}) generating the kernel K_{5,2}; Theorem 48 verifies kernel membership for s=1 for all g and for g≤4, s=2.
  • The two independent computations agree for all verified cases, and the paper expects the equality Tor*([A_2×A_{g-2}]) = Tor*(taut([A_2×A_{g-2}])) to hold for all g≥2, reducing the infinite family to a single Hodge-integral identity.

Reading between the lines

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

  • If the determinant structure observed for PR_{g,s} persists for other special cycles, one could predict tautological projections of more general product cycles without running the full tree sum; a testable check is to compute the analogous projection for r=3 and higher self-products and compare with the closed determinant form.
  • Speculation 50 implies a strong generation statement: all Gorenstein kernels of compact-type moduli spaces would be generated from Abel-Jacobi pullbacks of these generalized products using only the formal operations listed in the paper. A concrete first test is to verify membership property (P1) in the already-understood cases g≤7 before attacking the proposed harder case in codimension 7.
  • The success of the excess-intersection method depends on reducedness of Z, a property that is not automatic (the genus-4 Torelli square is nonreduced). The wall-crossing route may offer a way to bypass reducedness over partial compactifications, where the excess-intersection equations would need a more delicate deformation-theoretic analysis.
  • The equivalence of the homomorphism property with the single κ_1-paired Hodge integral identity suggests that the infinite genus family could be proven by analytic or Hodge-integral methods rather than by enumerating the exponentially many colored extremal trees.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the canonical tautological projection taut: CH^*(A_g) → R^*(A_g) and tests the conjecture that taut is a Q-algebra homomorphism on special cycles. The main new result is Theorem 7: the homomorphism property for the pairs ([J_g],[A_2 × A_{g-2}]) for 2 ≤ g ≤ 8 and for ([J_6],[A_3 × A_3]) on A_6. The verification is based on two independent computations of the Torelli pullback Tor^*([A_r × A_{g-r}]): a direct excess-intersection analysis of the Torelli fiber product Z (Sections 2–3) and a wall-crossing formula for families Gromov-Witten classes (Section 4). The paper also defines tautological projections on the universal fiber products X_g^s, computes the projection of generalized product cycles PR_{g,s} (Theorem 12), and uses Abel-Jacobi pullbacks to construct elements in Gorenstein kernels, including nonzero classes in R^5(M_{5,2}^{ct}) and R^5(M_{4,4}^{ct}) (Theorem 48).

Significance. If correct, Theorem 7 provides the first evidence for the homomorphism property for the Torelli locus against non-elliptic product loci, going substantially beyond the cases covered by [10] and [27]. The agreement of two independently implemented computational methods for g=4,5 is a genuine strength, as is the release of the code [26,37]. The framework connecting Torelli pullbacks to families Gromov-Witten theory and wall-crossing is original and likely to be influential. However, the central theorem is explicitly conditional on several external, unpublished or co-authored inputs: the reducedness of the Torelli fiber product [17], the wall-crossing comparisons and blowup descriptions [27], and the Gorenstein kernel calculations [8]. The manuscript is transparent about these dependencies, but as it stands the proofs of Theorems 5, 6, and 48 are not self-contained.

major comments (3)
  1. [§1.5, §2.3, §3.3 (Theorem 21, Eq. (23))] The excess-intersection proof of Theorem 5 depends crucially on the reducedness of the fiber product Z = M_g^{ct} ×_{A_g}(A_r × A_{g-r}). Reducedness is the basis for writing local equations as squarefree monomials (Theorem 21) and for the residual-intersection recursion (23) that computes the contributions Cont_T on all strata. This reducedness is not proved here but quoted from the separate co-authored preprint [17]. The risk is not hypothetical: [18] proves that the analogous Torelli square in genus 4 is nonreduced. If Z had embedded or nilpotent components, the normal-cone computation in (22)–(23) could fail to determine the pullback class, and equality (26) would be unsupported. Since (26) is the core of Theorem 7, this is a load-bearing gap. The authors should either include a proof of reducedness in this paper or make the argument of [17] available and explicitly state the exact s
  2. [§1.6, §4 (Eq. (12), Theorem 6, Theorem 27)] The wall-crossing method, presented as an independent verification of Theorem 7, depends on several statements from the unpublished preprint [27]: the equality (12) identifying Tor^![A_r × A_{g-r}] with the reduced virtual fundamental class of the moduli space of stable maps to the fibers of π_r, the wall-crossing formula of Theorem 6, and the blowup descriptions of the relevant moduli spaces of unramified maps (Theorem 27). None of these is proved in the present manuscript. Thus both routes to Theorem 7 rest on foundations not established here. The agreement of the two methods is meaningful only if both external inputs are correct. Please clarify the status of [27] and either include the needed statements with proofs or provide enough detail that the verifier can check them.
  3. [§6.4–§6.5 (Theorem 48)] The Gorenstein kernel claims, especially Theorem 10′ and the nonvanishing of Δ_{4,2}, use the results of [8] on the 1-dimensionality of K_{5,2} and on the completeness of 3-spin relations. The paper cites [8] as a preprint. While this is less central to the homomorphism property of Theorem 7, it is the main evidence for the conjectural framework on Gorenstein kernels. As with [17] and [27], the current manuscript cannot be independently verified without access to [8]. If [8] is not intended to be part of this submission, the claim should be phrased as conditional or the needed statements should be proved here.
minor comments (5)
  1. [§3.4, Example 23] The long expression for Cont_T contains several typographical ambiguities in monomials, e.g. "35z_1z_2z_2^3" and "7z_1z_2^2z_3" appear in a way that is difficult to parse. Please double-check and display the polynomial with clearly separated monomials.
  2. [§4.9–§4.11] The detailed wall-crossing computations are shown for g=4 and g=5 only. For g=6,7,8 the paper says the homomorphism property was verified using [26], but no intermediate data or summary tables are given. A table of the computed Tor^*([A_2 × A_{g-2}]) classes for all g=2,...,8 would make the two-method agreement more transparent and would help reproducibility.
  3. [§1.3, formulas for taut([J_g])] The formulas for g=7 and g=8 are long and their correctness is not evident. Since these are quoted from [61] or earlier work, it would be helpful to state explicitly whether they are inputs or outputs of the current verification.
  4. [§5.7, Proposition 43] The Capelli identity is used in a compact way. A short reminder of the exact normalization of the Capelli determinant would help the reader who is not an expert in invariant theory.
  5. [§6.3, notation ζ_T] The notation for stable graphs T in the two proofs of Proposition 47 is somewhat overloaded: T sometimes denotes a tree, sometimes a rooted tree with leaves. Please introduce separate notation or clarify.

Circularity Check

3 steps flagged · score 4.0 of 10

Self-citations are load-bearing in both routes to Theorem 7, but the central homomorphism verification is computed, not assumed.

  1. self citation load bearing [Section 1.5 (Theorem 5) and Section 2.3 (Theorem 21)]
    "A key step in the proof of Theorem 5 is the reducedness of the fiber product Z proven by Drakengren [17]. Reducedness is crucial here: we can then write local equations for Z and compute the pullback class via excess intersection theory [28] on the strata of M^ct_g."

    The excess-intersection derivation of Tor^*([A_h × A_{g-h}]) — the main input to the homomorphism verification (26) — depends on the local monomial equations of Theorem 21 and the residual recursion (23). Both are only valid if the Torelli fiber product Z is reduced, and that reducedness is imported from [17], a separate preprint by co-author Drakengren. The central computation therefore rests on a same-author result that is not proved in the present paper. This is load-bearing self-citation, though not an equation-level tautology: the theorem in [17] is a general statement about Z, not the target homomorphism identity.

  2. self citation load bearing [Section 1.6 and Section 4 (equations (12), Theorem 6, Theorem 27)]
    "By a result of [27] extending [31, Theorem 1.2], there is an equality of cycles (12) Tor^![A_r × A_{g-r}] = [M^ct_g(π_r)]^{red} under the isomorphism (11). ... The following result is established in [27]. Theorem 6 (Wall-crossing)."

    The wall-crossing route to Theorem 7 identifies the desired Torelli pullback with a reduced virtual class via (12), then evaluates it using Theorem 6 and the blowup descriptions of Theorem 27. All three statements are attributed to [27], an in-preparation preprint by three co-authors of the present paper (Feusi, Iribar López, Nesterov). Thus the 'second independent method' is not independent of the author group: its foundational comparison is imported from the same team's unpublished work. This is a self-citation chain, although the wall-crossing formula itself is a general theorem rather than a restatement of the homomorphism property.

1 more flagged steps
  1. uniqueness imported from authors [Section 6.4, proof of Theorem 10′]
    "By [8, Theorem 7], K_{5,2} is 1-dimensional. Therefore, Theorem 10′ is a consequence of parts (i) and (iii) of Theorem 48 for Δ_{5,1} ∈ K_{5,2}."

    The claim that aj^*(Δ_{5,1}) is the generator of the Gorenstein kernel of R^5(M^ct_{5,2}) is forced by the one-dimensionality of K_{5,2}, quoted from [8]. That paper is a co-authored preprint by Canning, Larson, and Schmitt; Canning and Schmitt are also co-authors here. The construction and non-vanishing of Δ_{5,1} are carried out in the present paper, so the central algebraic content is independent, but the uniqueness/dimension statement that upgrades the class to 'the generator' is imported from the same author group's prior work.

full rationale

No step in the paper is self-definitional in the sense that a fitted parameter is later called a prediction: the equality (26), Tor^*([A_h × A_{g-h}]) = Tor^*(taut([A_h × A_{g-h}])), is computed by two concrete algorithms (excess intersection and wall-crossing) and then compared with an independently known formula for taut([A_h × A_{g-h}]). The homomorphism property then follows by pushing forward and applying the tautological projection; this is a genuine derivation, not a tautology. However, both computational routes rest on same-group unpublished foundations: reducedness of the Torelli fiber product is taken from [17] by co-author Drakengren, and the wall-crossing comparison (12), Theorem 6, and the blowup descriptions are taken from [27] by three co-authors. Additionally, the statement that Δ_{5,1} generates K_{5,2} uses the one-dimensionality result from co-authored [8]. These are load-bearing self-citations, and the 'independent methods' claim is weaker than it appears because both methods depend on the same research group's unpublished results. The core verification still has independent mathematical content — the computations are explicit, reproducible, and compared with known Hodge-integral formulas — so the paper is only partially circular rather than reducible by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numeric free parameters are fitted; the only constants are Bernoulli numbers from the known evaluation (2). The central claims rest on several theorems imported from co-authored, partly unpublished sources ([17], [27], [8]) and from the authors' earlier work [9,10], which are listed here as axioms because the reader pays for them upstream.

assumptions (5)
  • domain assumption The fiber product Z = M_g^ct ×_{A_g}(A_r × A_{g-r}) is reduced (Drakengren [17]).
    Theorem 5's excess-intersection recursion (23) requires reducedness to write local equations; cited to a co-author's separate preprint arXiv:2601.02592.
  • domain assumption The wall-crossing formula (Theorem 6) and the blowup descriptions (Theorem 27) of moduli of unramified maps hold as stated.
    Taken from [27], which is 'in preparation'; no proof appears in this paper.
  • domain assumption The Gorenstein kernel K_{5,2} is 1-dimensional and the 3-spin relations are complete for the relevant cases.
    Used in Theorem 48 and Theorem 10' via [8], a non-archival PDF by overlapping authors.
  • standard math The vanishing R^{>2g-3}(M_g^ct)=0 from [23,30] and the formula for taut([A_2×A_{g-2}]) from [9].
    Imported results used in Sections 1.7 and 3.6 to reduce homomorphism questions to computable equalities.
  • standard math The identity θ^{g+1}=0 in CH^*(X_g) and the Sp-invariant theory presentation of R^*(X_g^s).
    Used in Theorem 39 and Proposition 43; standard results from [34], [63].

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Cite this review

Pith. "Pith review of Torelli loci, product cycles, and the homomorphism conjecture for $\mathcal{A}_g$." pith.science (2026). https://pith.science/paper/ZBEJCBF6

@misc{pith2026260104353,
  author       = {Pith},
  title        = {Pith review of: Torelli loci, product cycles, and the homomorphism conjecture for $\mathcalA_g$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZBEJCBF6}},
  note         = {Machine review of arXiv:2601.04353}
}
abstract

The tautological $\mathbb{Q}$-subalgebra $\mathsf{R}^*(\mathcal{A}_g) \subset \mathsf{CH}^*(\mathcal{A}_g)$ of the Chow ring of the moduli space of principally polarized abelian varieties is generated by the Chern classes of the Hodge bundle. There is a canonical $\mathbb{Q}$-linear projection operator $\mathsf{taut}: \mathsf{CH}^*(\mathcal{A}_g) \rightarrow \mathsf{R}^*(\mathcal{A}_g).$ We present here new calculations of intersection products of the Torelli locus in $\mathcal{A}_g$ with the product loci $\mathcal{A}_{r}\times \mathcal{A}_{g-r} \rightarrow \mathcal{A}_g$ for $r\leq 3$. The results suggest that $\mathsf{taut}$ is a $\mathbb{Q}$-algebra homomorphism, at least for special cycles. We discuss a conjectural framework for this homomorphism property. Our calculations follow two independent approaches. The first is a direct study of the excess intersection geometry of the fiber product of the Torelli and product morphisms. The second recasts the geometry in terms of families Gromov-Witten classes, which are computed by a wall-crossing formula related to unramified maps. We define tautological projections of cycles on the fiber products $\mathcal X_g^s \to \mathcal A_g$ of the universal family. We compute these projections for a class of product cycles on $\mathcal X_g^s$ in terms of a determinant involving the universal theta divisors and Poincar\'e classes. Using Abel-Jacobi pullbacks of product cycles on $\mathcal X_g^s$ and their projections, we construct a new family of classes which we conjecture to lie in the Gorenstein kernels of the tautological rings $\mathsf{R}^*(\mathcal M^{\mathrm{ct}}_{g,n})$. In particular, we construct nontrivial elements of the Gorenstein kernels of $\mathsf{R}^5(\mathcal{M}_{5,2}^{\mathrm{ct}})$ and $\mathsf{R}^5(\mathcal{M}_{4,4}^{\mathrm{ct}})$.

Figures

Figures reproduced from arXiv: 2601.04353 by the authors.

Figure 1
Figure 1. Star-shaped graph with partition labels on edges. 4.6. Blowup description for r = 2. The wall-crossing formula in Theorem 6 is a sum of contri￾butions indexed by star-shaped graphs. For r = 2, only two types of moduli spaces appear in the wall-crossing terms: (37) Mct,un 2 (π2,(1), . . . ,(1) | {z } k ) ◦ and Mct,un 1 (π2,(1, 1),(1), . . . ,(1) | {z } k ) • . Indeed, for non-product abelian surfaces, the curves in t… view at source ↗
Figure 2
Figure 2. An example of an unramified map from a nodal genus 2 curve to a Fulton-MacPherson degeneration of an abelian surface. locus Z1. With more marked points, there can be more Fulton-MacPherson expansions, which are realized via the iterated blowups of Zℓ . 4.8. Blowup description of classes. Using Theorem 27, we can describe all the classes appearing in the wall-crossing formula of Theorem 6 in terms of tautological and… view at source ↗
Figure 3
Figure 3. Graphs for r = 2 and g = 4, where (12 ) denotes the partition (1, 1). The first graph corresponds to the blowup τ : Mf2,1 → Mct 2,1 along Z1. The normal bundle of Z1 ⊂ Mct 2,1 is trivial, therefore we have τ∗(E 2 1 ) = −Z1 , τ∗(E ℓ 1 ) = 0 for ℓ ̸= 2 . Here, E1 is the unique exceptional divisor. The second graph corresponds to the iterated blowup τ : Mf2,2 → Mct 2,2 along Z2 and Z1. In￾specting equation (13), we see… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Graphs for r = 2 and g = 5, where (12 ) denotes the partition (1, 1). The first graph corresponds to Mf2,1, and the same analysis applies as in the case of g = 4. For the second graph, we analyze the moduli space Mf2,2. Since the degrees in z of the corresponding I￾fun…

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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

    math.AG 2026-07 unverdicted novelty 7.0 of 10

    Tautological ring of M_{g,n}^{ct} is not Gorenstein for g≥2 and 2g+n≥12; Pixton's 3-spin relations are complete for M_6^{ct}, M_{5,2}^{ct}, M_7^{ct}, first such cases without Gorenstein property.

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