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

TL;DR

This paper extends the dimension theory of twisted k-differentials to all k≥0, showing their intersection with moduli space is smooth of expected dimension and supporting a conjectural class formula with low genus evidence.

Contribution

It generalizes the dimension theory for twisted k-differentials to all k≥0 and extends a conjectural class formula, providing new insights into their geometric structure.

Findings

Intersection with moduli space is smooth of expected dimension for all k≥0.

Extended a conjectural formula for the weighted fundamental class of these spaces.

Provided evidence supporting the conjectural formula in low genus cases.

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