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arXiv · 2607.16159

Defining reduction types of curves via minimal regular and minimal normal crossings models

Jakab Schrettner
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Abstract

We propose a definition of the reduction type of a curve over a discretely valued field in terms of the special fibre of an arbitrary regular model. We show that under this definition, the reduction type in terms of the minimal regular model determines and reduction type in terms of the minimal regular normal crossings model and vice versa. We also show that our definition is compatible with earlier classification results in low genus, namely the Kodaira-N\'eron classification for elliptic curves and the classification result of Namikawa-Ueno in genus 2. The essential new element in our definition is a suitable invariant of the singularities on the special fibre, building on the notion of equisingularity introduced by Zariski.

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Jakab Schrettner. 2026-07-17. Defining reduction types of curves via minimal regular and minimal normal crossings models. https://arxiv.org/abs/2607.16159

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