  In this text, we merge ideas around the tropical hyperfield with the theory
of ordered blueprints to give a new formulation of tropical scheme theory. The
key insight is that a nonarchimedean absolute value can be considered as a
morphism into the tropical hyperfield. In turn, ordered blueprints make it
possible to consider the base change of a classical variety to the tropical
hyperfield. We call this base change the scheme theoretic tropicalization of
the classical variety.
  Our first main result describes the Berkovich analytification and the
tropicalization of a classical variety as sets of rational points of scheme
theoretic tropicalizations, including a characterization of the respective
topologies. Our second main result shows that the Giansiracusa bend relations
can be derived by a natural construction from the scheme theoretic
tropicalization.

Algebraic Geometry