# Models of Hyperelliptic Curves with Tame Potentially Semistable Reduction

## TL;DR

This paper demonstrates that the combinatorial cluster picture, combined with Galois action and leading coefficient, fully determines the special fiber structure of hyperelliptic curves with tame potentially semistable reduction.

## Contribution

It provides an explicit combinatorial description of the special fiber of the minimal model of hyperelliptic curves using cluster pictures and Galois data.

## Key findings

- Cluster picture determines the special fiber structure.
- Explicit description of the special fiber is given.
- Galois action influences the reduction type.

## Abstract

Let $C$ be a hyperelliptic curve $y^2 = f(x)$ over a discretely valued field $K$. The $p$-adic distances between the roots of $f(x)$ can be described by a completely combinatorial object known as the cluster picture. We show that the cluster picture of $C$, along with the leading coefficient of $f$ and the action of $\mathrm{Gal}(\bar{K}/K)$ on the roots of $f$, completely determines the combinatorics of the special fibre of the minimal strict normal crossings model of $C$. In particular, we give an explicit description of the special fibre in terms of this data.