Limits of tropicalizations
This paper is a follow-up to the third author's now quite well-known paper ``Analytification is the limit of all tropicalizations'', [\textit{S. Payne}, Math. Res. Lett. 16, No. 2--3, 543--556 (2009; Zbl 1193.14077)]. In the latter, Payne establishes a fundamental link between tropicalization and non-archimedean analytification (in the sense of Berkovich) which has served as a focal point in many applications of tropical geometry and subsequent developments in the subject. It is difficult to overstate the importance of this paper of Payne, even though he seems to imply in it that all the results were known in some form in much earlier literature. Regardless of that point, this paper cleaned up the story, elucidated the connection, and established a fundamentally important bridge between two rapidly developing fields. The main result from that paper is the following. The tropicalization of a variety depends on the choice of an embedding in a toric variety defined over a valued field. Since tropicalization is functorial with respect to toric morphisms, as one varies the embedding (while keeping the valued field fixed), one obtains an inverse system of tropicalizations, and the limit of this system (when one topologizes the tropicalizations appropriately using the Euclidean topology) is homeomorphic to the Berkovich analytification of the original variety if the latter is quasi-projective. If the variety is affine then one can use a much smaller system: instead of embeddings in arbitrary toric varieties, it suffices to consider embeddings in affine space. The present paper looks closer at this result and derives more precise conditions on the inverse system in order to guarantee that the limit remains the Berkovich analytification. See the introduction for the main statement. The idea is, in essence, to see exactly what properties were really used in Payne's argument and to adapt a more general argument based on an abstraction of these properties. This leads to two improvements of Payne's original result: (1) one can very often use a much smaller system of toric embeddings and still get the analytification, and (2) this limit result holds for any closed subscheme of a toric variety, one need not assume quasi-projective. Reviewer's remark: The reviewer would like to mention in passing that my brother and I have also re-examined Payne's original result and adapted it to a scheme-theoretic setting, where we found that there is an embedding in a slight generalization of a toric variety (allowing, for instance, non-finite type) such that the tropicalization of this single embedding is the Berkovich analytification, and moreover applying our scheme-theoretic tropicalization we obtained the moduli space of valuations on the original scheme, so that the \(\mathbb{T}\)-points are the Berkovich analytification but all higher rank valuations appears as other points as well [\textit{J. Giansiracusa} and \textit{N. Giansiracusa}, ``The universal tropicalization and the Berkovich analytification'', \url{arxiv:1410.4348}]. There seems ample opportunity to explore this topic further, for instance to try combining the ideas of this present paper with this scheme-theoretic setting to obtain smaller inverse systems that still manifest this moduli space of valuations.
scientific article; zbMATH DE number 6361319
scientific article