Elliptic-Cover Cycles and the Gorenstein Kernel

Definitions and mathematical statement

Let $\mathcal M_g^{\mathrm{ct}}$ denote the moduli stack of stable curves of genus $g$ of compact type, meaning that their dual graphs are trees. Their Jacobians are principally polarized abelian varieties, giving the Torelli morphism \[ \operatorname{Tor}\colon \mathcal M_g^{\mathrm{ct}}\longrightarrow \mathcal A_g, \] where $\mathcal A_g$ denotes the moduli stack of principally polarized abelian varieties of dimension $g$. The source does not specify the ground field or the range of genera. Under the customary rational Chow interpretation, write \[ R=R^*(\mathcal M_g^{\mathrm{ct}}) \subseteq A^*(\mathcal M_g^{\mathrm{ct}})\otimes{\mathbb{Q}} \] for the tautological ring. This coefficient and cycle-theoretic convention requires confirmation from the cited report.

For clarity, a graded pairing formulation of a Gorenstein kernel is as follows. Given a specified socle degree $s$ and trace functional $\varepsilon\colon R^s\to{\mathbb{Q}}$, set \[ K^k=\{\alpha\in R^k:\varepsilon(\alpha\beta)=0 \text{ for every }\beta\in R^{s-k}\}, \qquad K=\bigoplus_{k\geq0}K^k, \] with $R^j=0$ for $j<0$. Thus $K$ is the radical of the multiplication pairing followed by $\varepsilon$. The supplied record does not specify the socle degree, trace, or an equivalent definition of its intended Gorenstein kernel; these data must be matched to the source before this formulation is definitive.

The source denotes elliptic-cover Noether--Lefschetz loci by $\operatorname{NL}_d$ and asks about their Torelli pullbacks \[ c_d=\operatorname{Tor}^*[\operatorname{NL}_d]. \] It does not supply a definition of these loci, their ambient spaces, the permitted degrees $d$, their cycle multiplicities, or the pullback construction. In particular, membership of every intended $c_d$ in $R$ must be established before treating these classes as generators in that ring.

Question, with unresolved formulation data explicit. Can all of the Gorenstein kernel of $R^*(\mathcal M_g^{\mathrm{ct}})$ be explained using the classes $c_d$? If "explained using" means generation as a graded ideal, then, after specifying the preceding data, the proposed precise question is whether \[ K=(c_d:d\in D)_R, \] or, equivalently, for every $k\geq0$, \[ K^k=\sum_{d\in D}R^{k-r_d}c_d, \qquad r_d=\operatorname{codim}(c_d), \] where $D$ is the intended set of degrees. This is a conditional formulation: the record does not distinguish ideal generation from linear spanning or another construction, and it does not determine $D$, $r_d$, or the intended quantifier over $g$.

The supplied dated assessment says that particular Torelli-pullback classes are proved to lie in the kernel, while the whole kernel is not identified. It describes this question as requiring a larger leap than Questions 1 and 2. These are supplied source qualifications, not independently verified results or a resolution of the question.

Short English statement

The question asks whether elliptic-cover Noether--Lefschetz classes pulled back along the Torelli morphism account for the entire Gorenstein kernel of the tautological ring of compact-type curves. An ideal-generation interpretation is displayed conditionally. The precise loci, degree and genus ranges, pairing, and meaning of generation remain unspecified. The supplied status is open; this statement-only draft requires human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005559$, title "Elliptic-Cover Cycles and the Gorenstein Kernel," exact source problem code catalogue entry, Oberwolfach Reports Open Problems collection; proposed year: 2023. No proposer is identified in the record.
• Recent Trends in Algebraic Geometry (2023), Oberwolfach Reports, pp. 1485--1542, DOI 10.4171/owr/2023/27. Supplied report link: https://doi.org/10.4171/owr/2023/27.
• Supplied literature reference: Pandharipande, "Maps to a moving elliptic curve," https://doi.org/10.4171/owr/2023/27. This is the attribution attached to the reference; it does not by itself identify the proposer of Question 3.
• Literature checked at: 2026-08-22. This exact date records supplied dated curation, not an independently verified assessment. The supplied status is "open." The assessment states that particular Torelli-pullback classes lie in the kernel but that the entire kernel has not been identified, and characterizes Question 3 as requiring a larger leap. The record requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. Those are supplied verification claims; no independent source consultation is asserted here.