The tropicalization of an algebraic variety X is a combinatorial shadow of X, which is sensitive to a closed embedding of X into a toric variety. Given a good embedding, the tropicalization can provide a lot of information about X. We construct two types of these good embeddings for Mumford curves: fully faithful tropicalizations, which are embeddings such that the tropicalization admits a continuous section to the associated Berkovich space \(X^{{{\,\mathrm{an}\,}}}\) of X, and smooth tropicalizations. We also show that a smooth curve that admits a smooth tropicalization is necessarily a Mumford curve. Our key tool is a variant of a lifting theorem for rational functions on metric graphs.

中文翻译：

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为Mumford曲线构建平滑且完全忠实的热带化

代数品种的热带化X是的组合阴影X，这是一个封闭的嵌入敏感X成复曲面品种。有了良好的嵌入，热带化可以提供有关X的许多信息。我们为Mumford曲线构造了两种类型的良好嵌入：完全忠实的热带化，其嵌入使得热带化使相关联的Berkovich空间\（X ^ {{{\\ mathrm {an} \，}} } \）的X，并使热带化顺利进行。我们还表明，允许平滑热带化的平滑曲线必然是Mumford曲线。我们的关键工具是度量图上有理函数的提升定理的一种变体。