[Paper Review] The kappa ring of the moduli of curves of compact type: II

Rahul Pandharipande|ArXiv.org|Jun 15, 2009

Algebraic Geometry and Number Theory16 references13 citations

TL;DR

This paper establishes a universality theorem for the $κ$ ring of the moduli space of curves of compact type, proving that relations among $κ$ classes in genus 0 extend universally to all higher genera via virtual geometry of stable maps. The key result confirms that the $κ$ ring in higher genus is a canonical quotient of the genus 0 $κ$ ring, and provides strong evidence for the Gorenstein conjecture by verifying the non-degeneracy of the socle pairing.

ABSTRACT

The subalgebra of the tautological ring of the moduli of curves of compact type generated by the kappa classes is studied. Relations, constructed via the virtual geometry of the moduli of stable maps, are used to prove universality results relating the kappa rings in genus 0 to higher genus. Predictions for kappa classes of the Gorenstein conjecture are proven.

Motivation & Objective

• To establish a universality property for the $κ$ ring of the moduli space of curves of compact type across all genera.
• To prove that relations among $κ$ classes in genus 0 imply corresponding relations in all higher genera.
• To provide evidence for the Gorenstein conjecture on the tautological ring of $M_{g,n}^c$ by verifying the non-degeneracy of the socle evaluation pairing.
• To derive a basis for the $κ$ ring in higher genus using the genus 0 ring as a universal source.

Proposed method

• Uses virtual geometry of the moduli space of stable maps to construct universal relations among $κ$ classes.
• Applies intersection theory on strata classes in the tautological ring $R^*(M_{g,n}^c)$ to derive relations.
• Employs generating functions and matrix identities involving $κ$ and $ψ$ classes to analyze the structure of the $κ$ ring.
• Leverages the pushforward of $ψ$ monomials via forgetful maps to relate $ψ$ and $κ$ classes.
• Uses the identity $x\frac{d}{dx}\mathsf{F}_{\emptyset} = -\mathsf{F}_0$ and generating series to prove triangularity of key matrices.
• Proves lower-triangularity of matrix $\mathsf{L}_0(d)$ with $\pm1$ diagonal entries to establish determinant formulas and rank bounds.

Experimental results

Research questions

• RQ1 Does every relation among $κ$ classes in genus 0 extend to all higher genera in the $κ$ ring of $M_{g,n}^c$?
• RQ2 Can the $κ$ ring of $M_{g,n}^c$ for $g>0$ be canonically realized as a quotient of the genus 0 $κ$ ring?
• RQ3 Is the socle evaluation pairing on the $κ$ ring non-degenerate, providing evidence for the Gorenstein conjecture?
• RQ4 What is the precise basis for the $κ^d(M_{g,n}^c)$ space in terms of $κ$ monomials?

Key findings

• The $κ$ ring in genus $g$ with $n$ marked points is canonically a quotient of the genus 0 $κ$ ring on $2g+n$ marked points, via the map $\iota_{g,n}$.
• A $\mathbb{Q}$-basis for $\kappa^d(M_{g,n}^c)$ is given by $\{\kappa_{\mathbf{p}} \mid \mathbf{p} \in P(d, 2g-2+n-d)\}$ for $n>0$.
• The dimension of $\kappa^d(M_{0,n}^c)$ is $|P(d, n-2-d)|$, confirming the rank of the genus 0 $κ$ ring.
• The space of relations among $κ$ monomials of degree $d$ valid across all $M_{g,n}^c$ with fixed $\zeta = 2g-2+n$ has rank at least $|P(d)| - |P(d, \zeta - d)|$.
• The matrix $\mathsf{L}_0(d)$ is lower triangular with diagonal entries $\pm1$, proving the determinant formula and supporting the universality result.
• Theorem 2 confirms that the socle evaluation $L_\xi(\gamma) = \int_{\overline{M}_{g,n}} \overline{\gamma} \cdot \lambda_g \cdot \xi$ is non-trivial for non-zero $\xi$, providing strong evidence for the Gorenstein conjecture.