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Constructing smooth and fully faithful tropicalizations for Mumford curves.(English)Zbl 1440.14283

Summary: 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 \(X^{\text{an}}\) 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.

MSC:

14T25 Arithmetic aspects of tropical varieties
14G22 Rigid analytic geometry
32P05 Non-Archimedean analysis
14T15 Combinatorial aspects of tropical varieties
14T20 Geometric aspects of tropical varieties

Cite

References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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