Moduli of Galois $p$-covers in mixed characteristics

Abstract

We define a proper moduli stack classifying covers of curves of prime degree $p$. The objects of this stack are torsors $Y\to \mathcal{X}$ under a finite flat $\mathcal{X}$-group scheme, with $\mathcal{X}$ a twisted curve and $Y$ a stable curve. We also discuss embeddings of finite flat group schemes of order $p$ into affine smooth $1$-dimensional group schemes.