Minimal regular normal crossings models of superelliptic curves

TL;DR

This paper develops an explicit method to construct minimal regular normal crossings models of superelliptic curves over discretely valued fields, using normalization of a carefully chosen model of the projective line.

Contribution

It provides a novel explicit construction of minimal models for superelliptic curves via normalization, leveraging Mac Lane's valuation theory.

Findings

Explicit minimal models are obtained through normalization.

The approach applies to covers with degree not divisible by the residue characteristic.

The method simplifies the process of understanding the models' structure.

Abstract

Let $K$ be a complete discretely valued field with perfect residue field $k$. If $X \to \mathbb{P}^1_K$ is a $\mathbb{Z}/d$-cover with $\text{char } k \nmid d$, we compute the minimal regular normal crossings model $\mathcal{X}$ of $X$ as the normalization of an explicit normal model $\mathcal{Y}$ of $\mathbb{P}^1_K$ in $K(X)$. The model $\mathcal{Y}$ is given using Mac Lane's description of discrete valuations on the rational function field $K(\mathbb{P}^1)$.