Log smooth curves over discrete valuation rings


We give necessary and sufficient conditions for log smoothness of a proper regular arithmetic surface with smooth geometrically connected generic fibre over a discrete valuation ring with perfect residue field. As an application, we recover known criteria for log smooth reduction of minimal normal crossings models of curves.

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    It is likely that Y/G is log smooth over V but we won’t need this.


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I am indebted to Arne Smeets for raising the question answered by Corollary 4, for useful correspondence and comments on my early results, and for sending me [15], which motivated me to look for Theorem 1. I also thank Lorenzo Ramero and Takeshi Saito for valuable comments. Finally, I am very grateful to the referee, whose comments led to considerable improvements.

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Correspondence to Rémi Lodh.

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Lodh, R. Log smooth curves over discrete valuation rings. manuscripta math. (2021). https://doi.org/10.1007/s00229-020-01268-1

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Mathematics Subject Classification

  • 14A21
  • 14H20
  • 11G20
  • 14F20
  • 11S15
  • 19L10