Regular models of hyperelliptic curves

TL;DR

This paper constructs explicit regular models with normal crossings for hyperelliptic curves over discretely valued fields, introducing the novel concept of MacLane cluster pictures to connect clusters and valuations.

Contribution

It introduces the new notion of MacLane cluster pictures and provides explicit constructions of regular models for hyperelliptic curves.

Findings

Explicit regular models with normal crossings are constructed.

MacLane cluster pictures effectively link clusters and valuations.

The method applies to hyperelliptic curves over fields with residue characteristic not 2.

Abstract

Let $K$ be a complete discretely valued field of residue characteristic not $2$ and $O_K$ its ring of integers. We explicitly construct a regular model over $O_K$ with strict normal crossings of any hyperelliptic curve $C/K:y^2=f(x)$. For this purpose, we introduce the new notion of ''MacLane cluster picture'', that aims to be a link between clusters and MacLane valuations.