Dimension Theory of the Moduli Space of Twisted $K$-Differentials

Abstract

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 [G. Farkas and R. Pandharipande, J. Inst. Math. Jussieu 17, No. 3, 615–672 (2018; Zbl 06868654)] to the case $k>1$. In particular, we show that the intersection ${\mathcal{H}}_g^k(\mu )={\tilde{\mathcal{H}}}_g^k(\mu )\cap {\mathcal{M}}_{g,n}$ is a union of smooth components of the expected dimensions for all $k\ge 0$. We also extend a conjectural formula from [Zbl 06868654] for a weighted fundamental class of ${\tilde{\mathcal{H}}}_g^k(\mu )$ and provide evidence in low genus. If true, this conjecture gives a recursive way to compute the cycle class $[{\overline{\mathcal{H}}}_g^k(\mu )]$ of the closure of ${\mathcal{H}}_g^k(\mu )$ for $k\ge 1,\mu$ arbitrary.