Seminar in Real and Complex Geometry

Thursday, May 16, 2019, 16:00-18:00, Schreiber building, room 209

Oliver Lorscheid (IMPA, Brazil)

Tropical geometry over the tropical hyperfield


In 2013, Giansiracusa and Giansiracusa have employed concepts from F1-geometry to endow the set-theoretic tropicalization of a classical variety with the structure of a semiring scheme over the tropical semifields T. The set-theoretic tropicalization can be retrieved from the scheme-theoretic tropicalization by taking T-rational points. This discovery was the starting point of tropical scheme theory, which is a recent new branch of tropical geometry. Intuitively, the scheme-theoretic tropicalization can be seen as the base change of the classical variety to T along a non-archimedean valuation. To make this heuristics rigorous, we need to pass from (semi)fields to hyperfields and from semiring schemes to blue schemes. While these latter terminologies might appear exotic on a first sight, we will see that they provide in fact a very simple and natural language for tropical scheme theory. In particular, we can define the tropicalization literally as a base change along a valuation, which recovers both, the set-theoretic tropicalization and the scheme-theoretic tropicalization in the sense of Giansiracusa-Giansiracusa.