Reduction of Hyperelliptic Curves in Characteristic \not=2

2021/12/10 by Tim Gehrunger, Richard Pink, Gehrunger, Tim +1 · 3 citations
Mathematics · #11G20) #14H30 (14H10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2112.05550

openalex publication_date 2021/12/10 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

Abstract

Let K be the quotient field of a discrete valuation ring R with residue characteristic \not=2, and let C be a hyperelliptic curve over K. We assume that all geometric branch points of the double covering C\twoheadrightarrow\mathbb P¹_K are rational and mark both C and \mathbb P¹_K with these branch points. After possibly replacing R by a ramified extension of degree 2, we give a direct construction for the stable model of C as a marked curve over R. We deduce that the closed fiber of this stable model is determined completely by the closed fiber of the stable model of the marked \mathbb P¹_K. In particular, the dual graph and other information for the former can be read off directly from the corresponding information for the latter.

Cited by

• A cluster criterion for potential degeneracy of superelliptic curves
• The Classification of the Stable Marked Reduction of Genus 2 Curves in Residue Characteristic 2
• Computing the Stable Reduction of Hyperelliptic Curves in Residue Characteristic 2

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