Galois Covers of Degree $p$ and Semi-Stable Reduction of Curves in Mixed Characteristics

Abstract

In this paper we study the degeneration of a Galois cover $f:\mathcal{Y}\to \mathcal{X}$ of degree $p$ above a proper and smooth curve $\mathcal{X}$ over a complete discrete valuation ring of mixed characteristics $(0,p)$. We exhibit geometric and combinatorial degeneration data which describe the geometry of the special fiber of a minimal semi-stable model of $\mathcal{Y}$.