Constructing smooth and fully faithful tropicalizations for Mumford curves
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Constructing smooth and fully faithful tropicalizations for Mumford curves Philipp Jell1
© The Author(s) 2020
Abstract 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 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. Keywords Tropical geometry · Smooth tropical curves · Mumford curves · Extended skeleta · Faithful tropicalization Mathematics Subject Classification Primary: 14T05; Secondary: 14G22 · 32P05
1 Introduction Let K be a field that is algebraically closed and complete with respect to a nonarchimedean non-trivial absolute value. Given a closed subvariety X of a toric variety Y over K , one can associate a so-called tropical variety Trop(X ) which is a polyhedral complex. Note however, that Trop(X ) is not an invariant of X , but depends on the embedding into Y . In good situations, Trop(X ) can retain a lot of information about X . Let us mention here work by Katz, Markwig and Markwig on the j-invariant of elliptic curves [21,
The author was supported by the DFG Research Fellowship JE 856/1-1 and by the Institute Mittag-Leffler and the “Vergstiftelsen”.
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Philipp Jell [email protected] Universität Regensburg, 93040 Regensburg, Germany 0123456789().: V,-vol
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22] and work by Itenberg, Mikhalkin, Katzarkov and Zharkov on recovering Hodge numbers in degenerations of complex projective varieties [17]. In the latter work, a smoothness condition for tropical varieties in arbitrary codimension appears: a tropical variety is called smooth if it is locally isomorphic to the Bergman fan of a matroid. (See Definition 2.6 for an equivalent definition for curves.) For tropical hypersurfaces, this is equivalent to the associated subdivision of the Newton polytope being a primitive triangulation, which is the definition of smoothness that is generally used for tropical hypersurfaces [17, Remark p. 24]. The definition in [17] is motivated by complex analytic geometry. A complex variety is smooth if it is locally isomorphic to open subsets of Cn in the analytic topology. Bergman fans of matroids are the local models for linear spaces in tropical geometry, thus it makes sense to call a tropical variety smooth if it is locally isomorphic to the Bergman fan of a matroid. This smoothness condition has been shown to imply many tropical analogues of classical theorems from complex and algebraic geometry, for example intersection theory, Poincaré duality and a Lefschetz (1, 1)-theorem [18,19,31
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