Relations in the tautological ring

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.

27 pages

References in corpus (1)

• The kappa ring of the moduli of curves of compact type: II

Cited by in corpus (5)

• Conjectural relations in the tautological ring of ${\bar{M}}_{g,n}$
• Tautological and non-tautological cohomology of the moduli space of curves
• On the tautological rings of M_{g, 1} and its universal Jacobian
• Canonical metric on the space of symplectic invariant tensors and its applications
• Cycles on curves and Jacobians: a tale of two tautological rings