A guide to tropicalizations

By Walter Gubler. 2011

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Tropicalizations form a bridge between algebraic and convex geometry. We generalize basic results from tropical geometry which are well-known for special ground elds to arbitrary non-archimedean valued elds. To achieve this, we develop a theory of toric schemes over valuation rings of rank 1. As a basic tool, we use techniques from non-archimedean analysis. MSC2010: 14T05, 14M25, 32P05 X) of polyhedra in R n . This process is called tropicaliza- tion and it can be used to transform a problem from algebraic geometry into a corresponding problem in convex geometry which is usually easier. If the toric co- ordinates are well suited to the problem, it is sometimes possible to use a solution of the convex problem to solve the original algebraic problem. Another strategy is to vary the ambient torus to compensate the loss of information due to the tropi- calization process. Tropicalization originates from a paper of Bergman (Berg) on logarithmic limit sets. The convex structure of the tropical variety Trop(X) was worked out by Bieri{Groves (BG) with applications to geometric group theory in mind. Sturmfels (Stu) pointed out that Trop(X) is a subcomplex of the Grobner complex. In fact, the polyhedral complex Trop(X) has some natural weights satisfying a balancing condition which appears rst in Speyer's thesis (Spe). This relies on the description

Subfields: Computational Theory and Mathematics, Geometry and Topology

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