Socle pairings on tautological rings

TL;DR

This paper investigates the properties of the $ ext{lambda}_g$ pairing on the tautological ring of the moduli space of genus $g$ stable curves of compact type, revealing explicit rank formulas and relationships between different pairings.

Contribution

It provides explicit formulas for the rank of the $ ext{lambda}_g$ pairing on various subspaces of the tautological ring, and relates these to known pairings on $M_g$.

Findings

Rank of pairing on boundary classes equals that on pure boundary strata.

Explicit formula for pairing rank in terms of partitions.

Rank increases by the rank of the $ ext{lambda}_g ext{lambda}_{g-1}$ pairing on $M_g$.

Abstract

We study some aspects of the $\lambda_g$ pairing on the tautological ring of $M_g^c$, the moduli space of genus $g$ stable curves of compact type. We consider pairing kappa classes with pure boundary strata, all tautological classes supported on the boundary, or the full tautological ring. We prove that the rank of this restricted pairing is equal in the first two cases and has an explicit formula in terms of partitions, while in the last case the rank increases by precisely the rank of the $\lambda_g\lambda_{g - 1}$ pairing on the tautological ring of $M_g$.