I often hear people speaking of the many connections between algebraic varieties and tropical geometry and how geometric information about a variety can be read off from the associated tropical variety. Although I have seen some concrete examples of this, I am curious about how much we can get out of this correspondence in general. More precisely, my question is the following:
Which information of $X=V(I)$ can be
read off its tropicalization
$\mbox{Trop(X)}=\bigcap_{f\in I}\mbox{trop}(f)$?
As a very basic example, it is known that $\dim(X)=\dim_{\mathbb{R}}\mbox{Trop}(X)$.
Best Answer
By work of Matt Baker, the dimension of a linear system on a curve is bounded above by the dimension of the corresponding tropical linear system on the corresponding tropical curve -- see Lemma 2.8 of his paper, "Specialization of linear systems from curves to graphs."