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arXiv · 2508.15974

A cluster criterion for potential degeneracy of superelliptic curves

Jeffrey Yelton
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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.

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Jeffrey Yelton. 2025-08-21. A cluster criterion for potential degeneracy of superelliptic curves. https://arxiv.org/abs/2508.15974

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