On the tautological ring of ℳ g

On the tautological ring of M g , n

In this section, we briefly describe the objects under consideration for the sake of non-experts. A more detailed informal exposition of these well-known ideas is given in [PV]. When studying Riemann surfaces of some given genus g, one is naturally led to study the moduli space Mg of such objects. This space has dimension 3g−3, and has a natural compactification due to Deligne and Mumford, [DM]...

Arising from the Tautological Ring

M ar 2 00 3 COMBINATORIAL CLASSES ON M g , n ARE TAUTOLOGICAL

The combinatorial description via ribbon graphs of the moduli space of Riemann surfaces allows to define combinatorial cycles in a natural way. Witten and Kontsevich first conjectured that these classes are polynomials in the tautological classes. We answer affirmatively to this conjecture and find recursively all the polynomials. Moreover we lift the combinatorial classes to the Deligne-Mumfor...

Appendix: Polynomials Arising from the Tautological Ring

Tautological Ring of Strata of Differentials

Strata of k-differentials on smooth curves parameterize sections of the k-th power of the canonical bundle with prescribed orders of zeros and poles. Define the tautological ring of the projectivized strata using the κ and ψ classes of moduli spaces of pointed smooth curves along with the class η = O(−1) of the Hodge bundle. We show that if there is no pole of order k, then the tautological rin...