The kappa ring of the moduli of curves of compact type: I

TL;DR

This paper investigates the structure of the kappa ring within the tautological ring of the moduli space of curves of compact type, providing generators, bases, Betti numbers, and a universality property across genera.

Contribution

It introduces new relations via stable quotient geometry, computes bases and Betti numbers, and establishes a universality property linking higher genus kappa rings to genus 0.

Findings

Derived minimal generators for the kappa ring.

Computed bases and Betti numbers of the kappa rings.

Established a universality property relating higher genus to genus 0 kappa rings.

Abstract

The subalgebra of the tautological ring of the moduli of curves of compact type generated by the kappa classes is studied in all genera. Relations, constructed via the virtual geometry of the moduli of stable quotients, are used to obtain minimal sets of generators. Bases and Betti numbers of the kappa rings are computed. A universality property relating the higher genus kappa rings to the genus 0 rings is stated and proved in a sequel. The lambda_g formula for Hodge integrals arises as the simplest consequence.