We study whether a given tropical curve Γ in R n can be realized as the tropicalization of an algebraic curve whose non-archimedean skeleton is faithfully represented by Γ. We give an affirmative answer to this question for a large class of tropical curves that includes all trivalent tropical curves, but also many tropical curves of higher valence. We then deduce that for every metric graph G with rational edge lengths there exists a smooth algebraic curve in a toric variety whose analytification has skeleton G, and the corresponding tropicalization is faithful. Our approach is based on a combination of the theory of toric schemes over discrete valuation rings and logarithmically smooth deformation theory, expanding on a framework introduced by Nishinou and Siebert.Date: November 5, 2018.

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Abstract. We define normalized versions of Berkovich spaces over a trivially valued field k, obtained as quotients by the action of R>0 defined by rescaling semivaluations. We associate such a normalized space to any special formal k-scheme and prove an analogue of Raynaud's theorem, characterizing categorically the spaces obtained in this way. This construction yields a locally ringed G-topological space, which we prove to be G-locally isomorphic to a Berkovich space over the field k((t)) with a t-adic valuation. These spaces can be interpreted as nonarchimedean models for the links of the singularities of k-varieties, and allow to study the birational geometry of k-varieties using techniques of non-archimedean geometry available only when working over a field with non-trivial valuation. In particular, we prove that the structure of the normalized non-archimedean links of surface singularities over an algebraically closed field k is analogous to the structure of non-archimedean analytic curves over k((t)), and deduce characterizations of the essential and of the log essential valuations, i.e. those valuations whose center on every resolution (respectively log resolution) of the given surface is a divisor.