Résume | In 2013, Giansiracusa and Giansiracusa have found a way to use F1-geometry for tropical geometry. More precisely, they define the scheme-theoretic tropicalization of a classical variety and show that the set-theoretic tropicalization can be retrieved as the set of T-rational points.
The scheme-theoretic tropicalization carries more information than the set-theoretic tropicalization. For example, it knows about the Hilbert polynomial of the classical variety and the weights of the (maximal cells of the) set-theoretic tropicalization.
There are hopes that this will be useful for future developments, such as tropical sheaf cohomology, a cohomological approach to intersection theory, flat tropical families, and more.
However, some fundamental problems remain unsolved so far. For example, it is not clear how to approach dimension theory or decompositions into irreducible components. It is not even clear what a good notion of a tropical scheme should be since the class of semiring schemes contains too many and pathological objects.
In this talk we give an introduction to tropical scheme theory and an overview of this circle of ideas. |