A cluster criterion for potential degeneracy of superelliptic curves

TL;DR

This paper establishes a criterion based on cluster data for the potential degeneracy of superelliptic curves over discretely valued fields, providing explicit methods to construct minimal regular models when conditions are met.

Contribution

It introduces a new cluster criterion for potential degeneracy of superelliptic curves and links it to convex hulls in the Berkovich projective line, with explicit model construction.

Findings

Cluster data criterion characterizes potential degeneracy.

Explicit construction of minimal regular models under the criterion.

Interpretation of cluster data via convex hulls in Berkovich space.

Abstract

Let $K$ be a field with a discrete valuation; let $p$ be a prime; and let $C$ be the curve defined by an equation of the form $y^p = f(x)$. We show that the curve $C$ has a model over an algebraic extension of $K$ whose special fiber consists of genus-$0$ components and has at worst nodal singularities if and only if the cluster data of the roots of $f$ satisfies a certain criterion, and when these hold, we show explicitly how to build the minimal regular model of $C$. We develop an interpretation of cluster data in terms of a convex hull in the Berkovich projective line and express the above results directly in terms of this convex hull.