Tautological ring of strata of differentials

Abstract

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 $$\kappa $$ and $$\psi $$ classes of moduli spaces of pointed smooth curves along with the class $$\eta = \mathcal O(-1)$$ of the Hodge bundle. We show that if there is no pole of order k, then the tautological ring is generated by $$\eta $$ only, and otherwise it is generated by the $$\psi $$ classes corresponding to the poles of order k.