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

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