The Moduli Space of Cyclic Covers in Positive Characteristic

Abstract We study the p-rank stratification of the moduli space ASW (₃_₁, d₂, , d₍), which represents Z/p^n-covers in characteristic p0 whose Z/p^i-subcovers have conductor d₈. 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, (d₁, d₂, , d₍) ) for which ASW (₃_₁, d₂, , d₍) in characteristic p is irreducible. Finally, we investigate the geometry of ASW (₃_₁, d₂, , d₍) by studying the deformations of cyclic covers that vary the p-rank and the number of branch points.