Reduction of Hyperelliptic Curves in Characteristic $\not=2$

TL;DR

This paper provides a method to explicitly construct the stable model of hyperelliptic curves over fields with residue characteristic not equal to 2, using the stable model of the projective line with marked points.

Contribution

It introduces a direct construction for the stable model of hyperelliptic curves in characteristic not 2, linking it to the stable model of the projective line with marked points.

Findings

The stable model of the hyperelliptic curve is determined by the stable model of the marked projective line.

The dual graph of the special fiber can be read off from the projective line's stable model.

A ramified extension of degree 2 may be necessary for the construction.

Abstract

Let $K$ be the quotient field of a discrete valuation ring $R$ with residue characteristic $\not=2$, and let $C$ be a hyperelliptic curve over $K$. We assume that all geometric branch points of the double covering $C\twoheadrightarrow{\mathbb P}^1_K$ are rational and mark both $C$ and ${\mathbb P}^1_K$ 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 ${\mathbb P}^1_K$. In particular, the dual graph and other information for the former can be read off directly from the corresponding information for the latter.