Reduction of Hyperelliptic Curves in Characteristic $\not=2$

Tim Gehrunger; Richard Pink

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arXiv · 2112.05550

Reduction of Hyperelliptic Curves in Characteristic $\not=2$

Tim Gehrunger·Richard Pink

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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 $C\twoheadrightarrow{\mathbb P}^1_K$ are rational and mark both $C$ and ${\mathbb P}^1_K$ 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 ${\mathbb P}^1_K$. In particular, the dual graph and other information for the former can be read off directly from the corresponding information for the latter.

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BibTeX RIS

Tim Gehrunger, Richard Pink. 2021-12-10. Reduction of Hyperelliptic Curves in Characteristic $\not=2$. https://arxiv.org/abs/2112.05550

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