Ulirsch [@CFPU]. However, these results are a bit different in spirit, as the authors start with a given skeleton and then make a construction that works for *some* algebraic curve with that skeleton. We also only care about the skeleton of the curve in our construction, but the map we construct works for *every* curve with that skeleton.

While the main body of our text deals with general Mumford curves, i.e. we do not use any additional properties, our main technique of lifting tropical meromorphic functions can also be used to construct nice tropicalizations for all curves with a given explicit skeleton. We exhibit this in Section \[sec:genus2\] for a special skeleton of genus $2$.

Smooth tropicalizations
-----------------------

We consider another property of tropicalizations: smoothness. Roughly speaking, an embedded tropical curve is smooth if locally, at every vertex, the tropical curve looks like the 1-dimensional fan in ${\mathbf{R}}^k$ whose rays are $e_1,\dots,e_k, -\sum e_i$.

We define in Definition \[def:local-degree-of-non-smoothness\] an invariant of an embedded tropical curve that measures how singular that tropical curve is. We prove the following resolution of singularities result (Corollary \[Cor resolution of singularities\]) by showing the we can inductively lower this invariant via re-embedding.

\[ThmB\] Let 