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

Abstract

In this note we extend the dimension theory for the spaces \widetilde{\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)=\widetilde{\mathcal{H}}_g^k(\mu) \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 [Zbl 06868654] for a weighted fundamental class of \widetilde{\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\geq 1,\mu arbitrary.