Wild monodromy and automorphisms of curves

Abstract

Let $R$ be a complete discrete valuation ring (DVR) of mixed characteristic $(0,p)$ with field of fractions $K$ containing the $p$th roots of unity. This article is concerned with semistable models of $p$-cyclic covers of the projective line $C\to {\mathbb{P}}_K^1$. We start by providing a new construction of a semistable model of $C$ in the case of an equidistant branch locus. If the cover is given by the Kummer equation $Z^p=f(X_0)$, we define what we call the monodromy polynomial $L(Y)$ of $f(X_0)$, a polynomial with coefficients in $K$. Its zeros are key to obtaining a semistable model of $C$. As a corollary, we obtain an upper bound for the minimal extension $K\prime /K$, over which a stable model of the curve $C$ exists. Consider the polynomial $L(Y)\Pi (Y^p-f(y_i))$, where the $y_i$ range over the zeros of $L(Y)$. We show that the splitting field of this polynomial always contains $K\prime$ and that, in some instances, the two fields are equal