Reduction of Hyperelliptic Curves in Characteristic $\ne 2$

Abstract

Let $K$ be the quotient field of a discrete valuation ring $R$ with residue characteristic $\ne 2$, and let $C$ be a hyperelliptic curve over $K$. We assume that all geometric branch points of the double covering $C\twoheadrightarrow {\mathbb{P}}_K^1$ are rational and mark both $C$ and ${\mathbb{P}}_K^1$ with these branch points. After possibly replacing $R$ by a ramified extension of degree $2$, we give a direct construction for the stable model of $C$ as a marked curve over $R$. We deduce that the closed fiber of this stable model is determined completely by the closed fiber of the stable model of the marked ${\mathbb{P}}_K^1$. In particular, the dual graph and other information for the former can be read off directly from the corresponding information for the latter.