Is there a direct translation between Tropical and Algebraic geometry?

Question

I am thinking of doing research in Tropical Geometry for my senior thesis, and I was wondering to what degree we can translate problems in Algebraic Geometry to Tropical Geometry.

We know that there are many analogous theorems from Algebraic Geometry in the Tropical realm, and that tropical curves arise naturally when we degenerate algebraic curves through the tropical log limit map $\log_q$, but I was wondering how well defined the translation layer between them is.

For instance, if we have a statement in the Algebraic Geometry world, is there

1. a way to directly find an analogous statement in the Tropical world (or say there is no analogous statement) and
2. a way to test if the statement should hold (like if the statement is compatible with the log map)?

I know these are very vague questions, but I was just wondering if anyone had any input. Also, what is this log limit map actually doing, or telling us? Are there any types of degeneration that can tell us about other properties? Is there a sense in which the log degeneration is the most natural one?

Answer 1

Andreas Gathmann has explored this connection:

Ideally, every construction in algebraic geometry should have a combinatorial counterpart in tropical geometry. One may thus hope to obtain results in algebraic geometry by looking at the tropical (i.e. combinatorial) picture first and then trying to transfer the results back to the original algebro-geometric setting. In this expository article we will for simplicity restrict ourselves mainly to the well-established theory of tropical plane curves. As tropical curves are simply images of classical curves we can hope to find tropical — and thus combinatorial — versions of many results known from classical geometry.