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 $\hat{I}º$ and $\ddot{I}ˆ$ classes of moduli spaces of pointed smooth curves along with the tautological class $\hat{I}\cdot$ of the Hodge bundle. We show that if there is no pole of order $k$, then the tautological ring is generated by $\hat{I}\cdot$ only, and otherwise it is generated by the $\ddot{I}ˆ$ classes corresponding to the poles of order $k$.