Tautological ring of strata of differentials

TL;DR

This paper studies the structure of the tautological ring of strata of differentials on smooth curves, revealing generators depending on the presence of poles of order $k$, thus advancing understanding of their algebraic properties.

Contribution

It introduces a description of the tautological ring of strata of $k$-differentials, identifying generators based on pole conditions, which was previously unexplored.

Findings

Tautological ring generated by $ ext{ exteta}$ when no pole of order $k$ exists.

When poles of order $k$ are present, the ring is generated by corresponding $ ext{ extpsi}$ classes.

Provides a criterion for the generators of the tautological ring based on pole orders.

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 tautological class $\eta$ 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 $k$.