The moduli space of cyclic covers in positive characteristic

TL;DR

This paper investigates the structure and irreducibility of the moduli space of cyclic covers in positive characteristic, analyzing ramification data and deformations to classify components and their dimensions.

Contribution

It identifies irreducible components of the moduli space of cyclic covers and provides a classification based on ramification data and deformation analysis.

Findings

Classified all irreducible components of the moduli space.

Determined the dimensions of each component.

Listed pairs where the moduli space is irreducible.

Abstract

We study the $p$-rank stratification of the moduli space $\mathcal{ASW}_{(d_1,d_2,\ldots,d_n)}$, 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 $\mathcal{ASW}_{(d_1,d_2,\ldots,d_n)}$ in characteristic $p$ is irreducible. Finally, we investigate the geometry of $\mathcal{ASW}_{(d_1,d_2,\ldots,d_n)}$ by studying the deformations of cyclic covers which vary the $p$-rank and the number of branch points.