Bend relations, valuations, and universal tropicalization

Jeffrey Giansiracusa (University of Swansea)

Wednesday 21st October, 2015 16:00-17:00 Maths 522

Abstract

Let X be a locally integral scheme over a non-archimedean valued field k.  An embedding of X into a scheme equipped with a model over the field with one element F_1 determines a tropicalization of X, which is described by certain canonical equations called the 'bend relations'.  The base-change adjunction between F_1 and k yields a universal embedding of X into an ambient space with F_1 model, and thus there is a universal tropicalization.  Moreover, in this case solutions to the bend relations are exactly non-archimedean valuations on the structure sheaf of X.  Thus the Berkovich analytification can be realized as an algebraic moduli space of valuations.

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