Stable Reduction of Curves and an Analog to a Theorem of Deuring
Abstract. Let $R$ be a complete mixed characteristic discrete valuation
ring with
algebraically closed residue field $k$ and field of fractions $K$.
For a $p$-cyclic cover $X \rightarrow {\mathbb P}_K^1$ with branch locus $B$
we are interested in the stable reduction $X_R$ of $X$.
In the case that $B$ consists of four rational points the
cover can be given birationally by the equation
$y^p=x(x-1)^\alpha (x-\lambda)^\beta$ and we determine $X_R$ in terms
of data associated to this equation.
In particular we obtain conditions for $X_R$ to be a smooth $R$-curve.
Those conditions can be thought
of as an analog to Deuring's theorem on good reduction of elliptic curves in
terms of the $j$-invariant and they look quite similar.
Finally for general $B$ we report on work in progress aiming at
qualitative results on the special fiber $X_k=X_R \otimes_R k$.