Tautological ring of moduli space of K3 surfaces

In 1983 Mumford defined the tautological ring for moduli space M_g of smooth projective curves of genus g>=2. Since then, understanding the tautological ring has become an interesting subject in moduli theory. As a generalization of the story to surfaces, Marian-Oprea-Pandharipande in 2016 introduced the tautological ring on the moduli spaces F_g of quasi-polarised K3 surfaces of genus g. It is the subring of Chow ring generated by pushforward of kappa classes from all special sub-families. In this talk, I will explain a strategy to show that the tautological of F_g for g<=5 is the whole Chow ring. Our strategy is based on the tools of the equivariant Chow theory due to Totaro and Edidin-Graham, together with a geometric stratification for these moduli spaces. This is a similar result to the moduli space of curve M_g in which the Chow ring of M_g is tautological for g<=6 due to many people.