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.

中文翻译：

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记录离散估值环上的平滑曲线

我们给出了一个适当的必要算术条件，该条件用于在具有理想残差场的离散评估环上，通过光滑的几何连接的通用光纤在适当的规则算术曲面上进行对数平滑。作为一种应用，我们恢复了已知标准，用于对数最小法线交叉模型的对数平滑减少。