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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 of off its tropicalization $\mbox{Trop(X)}=\bigcap_{f\in I}\mbox{trop}(f)$?

As a very basic example, it is known that $\dim(X)=\dim_{\mathbb{R}}\mbox{Trop}(X)$.

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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 of it's its tropicalization $\mbox{Trop(X)}=\bigcap_{f\in I}\mbox{trop}(f)$?

As a very basic example, it is known that $\dim(X)=\dim_{\mathbb{R}}\mbox{Trop}(X)$.

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# What can we learn from the tropicalization of an algebraic variety?

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 of it's tropicalization $\mbox{Trop(X)}=\bigcap_{f\in I}\mbox{trop}(f)$?

As a very basic example, it is known that $\dim(X)=\dim_{\mathbb{R}}\mbox{Trop}(X)$.