Skeletons and tropicalizations

By Walter Gubler, Joseph Rabinoff, Annette Werner. 2014

78 citations recorded by the source corpus.

Let $K$ be a complete, algebraically closed non-archimedean field with ring of integers $K^\circ$ and let $X$ be a $K$-variety. We associate to the data of a strictly semistable $K^\circ$-model $\mathscr X$ of $X$ plus a suitable horizontal divisor $H$ a skeleton $S(\mathscr X,H)$ in the analytification of $X$. This generalizes Berkovich's original construction by admitting unbounded faces in the directions of the components of H. It also generalizes constructions by Tyomkin and Baker--Payne--Rabinoff from curves to higher dimensions. Every such skeleton has an integral polyhedral structure. We show that the valuation of a non-zero rational function is piecewise linear on $S(\mathscr X, H)$. For such functions we define slopes along codimension one faces and prove a slope formula expressing a balancing condition on the skeleton. Moreover, we obtain a multiplicity formula for skeletons and tropicalizations in the spirit of a well-known result by Sturmfels--Tevelev. We show a faithful tropicalization result saying roughly that every skeleton can be seen in a suitable tropicalization. We also prove a general result about existence and uniqueness of a continuous section to the tropicalization map on the locus of tropical multiplicity one.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c2589682f9e967dc5a3ff06c7f466db6c72ea883

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In Computational Theory and Mathematics

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Citations in the corpus

Degeneration of Linear Series from the Tropical Point of View and Applications (2015, result): “A generalization of the Slope Formula to the context of non-Archimedean varieties and tropical complexes is proved in [GRW14].”
Functorial tropicalization of logarithmic schemes: the case of constant coefficients (2013, background): “ive a detailed comparison of these two cases. These considerations togetherwith the above two theoremslead to the following incarnation of the principle of faithful tropicalization (see e.g. [BPR16], [GRW16], and [CHW14]). Corollary 1.3. Suppose that Yis a proper and scho¨n subvariety of a T-toric v”