Math @ Duke

Publications [#345767] of Joseph D Rabinoff
Papers Published
 Gubler, W; Rabinoff, J; Werner, A, Skeletons and tropicalizations,
Advances in Mathematics, vol. 294
(May, 2016),
pp. 150215 [doi]
(last updated on 2021/05/13)
Abstract: Let K be a complete, algebraically closed nonarchimedean field with ring of integers K and let X be a Kvariety. We associate to the data of a strictly semistable K model X of X plus a suitable horizontal divisor H a skeleton S(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 BakerPayneRabinoff from curves to higher dimensions. Every such skeleton has an integral polyhedral structure. We show that the valuation of a nonzero rational function is piecewise linear on S(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 wellknown result by SturmfelsTevelev. 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. ∙ ∙


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