Dimension Theory of The Moduli Space of Twisted k-Differentials

Abstract

In this note we extend the dimension theory for the spaces ̃Hkg(μ) of twistedk-differentials defined by Farkas and Pandharipande in [FP18] to the case k >1. In particular, we show that the intersection Hkg(μ) = ̃Hkg(μ)∩ Mg,n is a union of smooth components of the expected dimensions for all k≥0. We also extend a conjectural formula from [FP18] for a weighted fundamental class of ̃Hkg(μ) and provide evidence in low genus. If true, this conjecture gives a recursive way to compute the cycle class [Hkg(μ)] of the closure of Hkg(μ)fork≥1,μ arbitrary.