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• W310036830 abstract "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." @default.
• W310036830 created "2016-06-24" @default.
• W310036830 creator A5022968687 @default.
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• W310036830 date "2014-04-28" @default.
• W310036830 modified "2023-09-27" @default.
• W310036830 title "Skeletons and tropicalizations" @default.
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• W310036830 hasPublicationYear "2014" @default.
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