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Semistable reduction of covers of degree $p$
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abstract
Let $K$ be a local field of residue characteristic $p>0$. We explain how to compute the semistable reduction of $K$-curves $Y$ equipped with a degree-$p$ morphism from $Y$ to the projective line. This includes the reduction at $p$ of superelliptic curves of degree $p$, but our approach is not limited to Galois covers. We give particular attention to the reduction of plane quartics at $p=3$, which case is implemented in SageMath. We use the language of non-archimedean analytic geometry in the sense of Berkovich. A key tool is the different function of Cohen, Temkin, and Trushin.
Forward citations
Cited by 2 Pith papers
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Semistable Reduction of Plane Quartics
A plane quartic admits a GIT-stable plane model exactly when its stable reduction is non-hyperelliptic, and then the stable model is the unique minimal semistable model arising by resolving the cusps of the GIT model.
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Semistable reduction of smooth quartics
For smooth plane quartics, non-hyperelliptic stable reduction is equivalent to the existence of a unique GIT-stable plane model, and the stable model is obtained by cusp resolution.
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