Relations in the tautological ring

TL;DR

This paper derives explicit relations among kappa classes in the tautological ring of moduli spaces of genus g curves, confirming a conjecture by Faber-Zagier and advancing understanding of the structure of these relations.

Contribution

It provides a proof of the Faber-Zagier conjectural relations among kappa classes using stable quotient geometry, transforming complex relations into simpler forms.

Findings

Established the Faber-Zagier relations as valid tautological relations.

Connected the relations to the virtual geometry of moduli spaces of stable quotients.

Simplified the relations to a form involving a single series with factorial coefficients.

Abstract

These notes cover our series of three lectures at Humboldt University in Berlin for the October 2010 conference "Intersection theory on moduli space" (organized by G. Farkas). The topic concerns relations among the kappa classes in the tautological ring of the moduli space of genus g curves. After a discussion of classical constructions in Wick form, we derive an explicit set of relations obtained from the virtual geometry of the moduli space of stable quotients. In a series of steps, the stable quotient relations are transformed to simpler and simpler forms. Our final result establishes a previously conjectural set of tautological relations proposed a decade ago by Faber-Zagier. The Faber-Zagier relations are defined using g and a single series in one variable with coefficients (6i)!/(3i)!(2i)!. Whether these relations span the complete set of relations among the kappa classes on the moduli space of genus g curves is an interesting question.