Skeletons and tropicalizations
Walter Gubler, Joseph Rabinoff, Annette Werner

TL;DR
This paper develops a generalized theory of skeletons and tropicalizations for algebraic varieties over non-archimedean fields, extending previous constructions to higher dimensions and unbounded faces, and establishing foundational properties like piecewise linearity and balancing conditions.
Contribution
It introduces a new class of skeletons with unbounded faces, generalizes tropicalization constructions to higher dimensions, and proves key properties like piecewise linearity, slope formulas, and faithful tropicalizations.
Findings
Skeletons have integral polyhedral structures.
Valuations of rational functions are piecewise linear on skeletons.
Every skeleton can be faithfully tropicalized.
Abstract
Let be a complete, algebraically closed non-archimedean field with ring of integers and let be a -variety. We associate to the data of a strictly semistable -model of plus a suitable horizontal divisor a skeleton in the analytification of . 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 . 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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
