The Moduli Space of Cyclic Covers in Positive Characteristic

We study the $p$-rank stratification of the moduli space $\operatorname{\mathcal{A}\mathcal{S}\mathcal{W}}_{(d_{1},d_{2},\ldots ,d_{n})}$, which represents $\mathbb{Z}/p^{n}$-covers in characteristic $p>0$ whose $\mathbb{Z}/p^{i}$-subcovers have conductor $d_{i}$. In particular, we identify the irreducible components of the moduli space and determine their dimensions. To achieve this, we analyze the ramification data of the represented curves and use it to classify all the irreducible components of the space. In addition, we provide a comprehensive list of pairs $(p,(d_{1},d_{2},\ldots ,d_{n}))$ for which $\operatorname{\mathcal{A}\mathcal{S}\mathcal{W}}_{(d_{1},d_{2},\ldots ,d_{n})}$ in characteristic $p$ is irreducible. Finally, we investigate the geometry of $\operatorname{\mathcal{A}\mathcal{S}\mathcal{W}}_{(d_{1},d_{2},\ldots ,d_{n})}$ by studying the deformations of cyclic covers that vary the $p$-rank and the number of branch points.