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

Schmitt, Johannes

Documenta mathematica, Tome 23 (2018), pp. 871-894.

Résumé

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.

Détail

Classification : 14H10, 30F30
Keywords: strata of $k$-differentials, deformation theory, tautological classes, double ramification cycles