Constructing smooth and fully faithful tropicalizations for Mumford curves

Philipp Jell ^[1]
1. [1] University of Regensburg
  
  University of Regensburg
  
  Kreisfreie Stadt Regensburg, Alemania
Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 26, Nº. 4, 2020
Idioma: inglés
DOI: 10.1007/s00029-020-00586-2
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Resumen
- 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 Xan 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.