[Paper Review] On the tautological rings of M_{g, 1} and its universal Jacobian

Qizheng Yin|arXiv (Cornell University)|Jun 17, 2012

Advanced Differential Equations and Dynamical Systems18 references8 citations

TL;DR

This paper introduces a novel method to generate tautological relations in the moduli space of curves $\mathcal{M}_{g,1}$ using the $\mathfrak{sl}_2$-action on the Chow ring of the universal Jacobian. By applying differential operators derived from this action, the authors prove that the tautological ring $\mathcal{R}(\mathcal{M}_{g,1})$ is generated by $\kappa_1,\ldots,\kappa_{\lfloor g/3\rfloor}$ and $\psi$, confirm Faber’s conjectures for $g \leq 19$, and reprove them for $\mathcal{M}_g$ up to $g \leq 23$, recovering the Faber-Zagier relations at $g=24$. The method provides a systematic, combinatorial approach to tautological relations with broad computational and theoretical implications.

ABSTRACT

We give a new method of producing relations in the tautological ring R(M_{g, 1}), using the sl_2-action on the Chow ring of the universal Jacobian. With these relations, we prove that R(M_{g, 1}) is generated by κ_i for i no greater than g/3, together with ψ. Our computation shows that Faber's conjectures for M_{g, 1} are true for g up to 19. Further, by pushing relations forward to M_g, we obtain a new proof of Faber's conjectures (for M_g) for g up to 23. For g = 24, our method recovers all the Faber-Zagier relations. We also give an algebraic proof of an identity of Morita.

Motivation & Objective

• To develop a systematic method for generating tautological relations in the tautological ring $\mathcal{R}(\mathcal{M}_{g,1})$ using geometric and algebraic structures.
• To prove that $\mathcal{R}(\mathcal{M}_{g,1})$ is generated by $\kappa_1,\ldots,\kappa_{\lfloor g/3\rfloor}$ and $\psi$, resolving a key part of Faber’s conjectures for $\mathcal{M}_{g,1}$.
• To extend these results to $\mathcal{M}_g$ via pushforward, re-proving Faber’s conjectures for $g \leq 23$ and recovering the Faber-Zagier relations at $g=24$.
• To provide an algebraic proof of Morita’s identity using the $\mathfrak{sl}_2$-action framework.

Proposed method

• Leverage the $\mathfrak{sl}_2$-action on the Chow ring of the universal Jacobian $J \to \mathcal{M}_{g,1}$ to define a differential operator $\mathcal{D}$ acting on the tautological ring $\mathcal{T}(J)$.
• Construct relations in $\mathcal{T}(J)$ by applying $\mathcal{D}$ to polynomials in the classes $p_{i,j}$ and $\psi$ that vanish for geometric reasons.
• Restrict the resulting relations to the subalgebra $\mathcal{R}(\mathcal{M}_{g,1})$, which is isomorphic to a quotient of $\mathcal{T}(J)$, to obtain tautological relations in $\mathcal{M}_{g,1}$.
• Use computer algebra to compute dimensions of tautological cohomology groups modulo a large prime, ensuring numerical stability and accuracy despite large coefficients.
• Apply the pushforward map from $\mathcal{M}_{g,1}$ to $\mathcal{M}_g$ to transfer relations and verify consistency with known Faber-Zagier relations.
• Use the 'Dutch house' combinatorial framework to organize and simplify complex relations, making them computationally tractable.

Experimental results

Research questions

• RQ1 Can the $\mathfrak{sl}_2$-action on the universal Jacobian be used to systematically generate tautological relations in $\mathcal{M}_{g,1}$?
• RQ2 Is the tautological ring $\mathcal{R}(\mathcal{M}_{g,1})$ generated by $\kappa_1,\ldots,\kappa_{\lfloor g/3\rfloor}$ and $\psi$?
• RQ3 Do the relations derived from the $\mathfrak{sl}_2$-action confirm Faber’s conjectures for $\mathcal{M}_{g,1}$ up to genus 19?
• RQ4 Can these relations be pushed forward to $\mathcal{M}_g$ to reprove Faber’s conjectures for $g \leq 23$?
• RQ5 Does the method recover the Faber-Zagier relations at $g=24$?
• RQ6 Can an algebraic proof of Morita’s identity be constructed within this framework?

Key findings

• The tautological ring $\mathcal{R}(\mathcal{M}_{g,1})$ is generated by $\kappa_1,\ldots,\kappa_{\lfloor g/3\rfloor}$ and $\psi$, as proven via the $\mathfrak{sl}_2$-action method.
• Faber’s conjectures for $\mathcal{M}_{g,1}$ are confirmed to be true for all $g \leq 19$ using computer-assisted verification of the relation space.
• By pushing forward relations to $\mathcal{M}_g$, the authors reprove Faber’s conjectures for $\mathcal{M}_g$ up to genus $g=23$.
• At $g=24$, the method recovers exactly the set of Faber-Zagier relations, indicating the completeness of the relation space at this genus.
• The computation reveals that for $g=24$ and codimension 13, one relation is missing in $\mathcal{M}_{g,1}$, and for $\mathcal{M}_{24}$, one relation is missing in codimension 12, suggesting potential gaps near the middle codimension.
• An algebraic proof of Morita’s identity is established using the $\mathfrak{sl}_2$-action and the differential operator $\mathcal{D}$, providing a new conceptual derivation.