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Mathematics > Number Theory

Title: Reduction type of non-hyperelliptic genus 3 curves

Abstract: Let $C/K: F=0$ be a smooth plane quartic over a complete discrete valuation field $K$ whose residue field is of characteristic different from $2,3, 5$ and $7$. We give various characterizations of the reduction (i.e. non-perelliptic genus 3 curve, hyperelliptic genus 3 curve or bad) of the stable model of $C$: in terms of the existence of a particular plane quartic model; in terms of the valuations of the theta constants of $C$; in terms of the valuations of the Dixmier-Ohno invariants of $C$. The last one leads to an easy computable criterion.
Comments: 26 pages
Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
MSC classes: 11G20, 14Q05, 14D10, 14D20, 14H25
Cite as: arXiv:1803.05816 [math.NT]
  (or arXiv:1803.05816v1 [math.NT] for this version)

Submission history

From: Christophe Ritzenthaler [view email]
[v1] Thu, 15 Mar 2018 15:45:22 GMT (37kb)