Relations in the tautological ring of the moduli space of curves

TL;DR

This paper proves the Faber-Zagier relations among kappa classes on the moduli space of nonsingular genus g curves using stable quotient geometry, confirming a conjecture from 2000.

Contribution

It introduces a novel approach using stable quotient geometry to derive and prove the Faber-Zagier relations in the tautological ring.

Findings

Proof of the Faber-Zagier relations

Simplification of stable quotient relations

Advancement in understanding the tautological ring

Abstract

The virtual geometry of the moduli space of stable quotients is used to obtain Chow relations among the kappa classes on the moduli space of nonsingular genus g curves. In a series of steps, the stable quotient relations are rewritten in successively simpler forms. The final result is the proof of the Faber-Zagier relations (conjectured in 2000).