Dimension theory of the moduli space of twisted $k$-differentials

In this note we extend the dimension theory for the spaces ${\tilde{\mathcal{H}}}_g^k(\mu )$ of twisted $k$-differentials defined by Farkas and Pandharipande in [FP15] to the case $k>1$. In particular, we show that the intersection ${\tilde{\mathcal{H}}}_g^k(\mu )\cap {\mathcal{M}}_{g,n}$ is a union of smooth components of the expected dimensions for all $k\ge 0$. We also extend a conjectural formula from [FP15] for a weighted fundamental class of ${\tilde{\mathcal{H}}}_g^k(\mu )$ and provide evidence in low genus.