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For a variety X (over some proper fields), if Trop(X) is a tropicalization of X, then we know that Trop(X) is a polyhedral complex. If we consider the interior of the support of that polyhedral complex, then does it have some geometric meaning (in the sense of tropicalization) ?

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First of all, $\mathrm{Trop}(X)$ is a set which can be equipped with the structure of a polyhedral complex; there is not a canonical way to do it. However, there are better and worse choices. In particular, one can choose the polyhedral structure so that, if $w$ and $w'$ are in the relative interiors of the same face, then the initial ideals $\mathrm{In}_w(X)$ and $\mathrm{In}_{w'}(X)$ are equal.

Let $X$ be pure of dimension $d$ and choose such a polyhedral structure. For such a choice, if $w$ is in the relative interior of a face then the automorphisms of $\mathrm{In}_w(X)$ contain a $d$-dimensional torus. I wrote this up in Section 2.2 of my thesis (apologies for the horrible writing). You can also extract this from the discussion in Section 9.2 of Sturmfels Solving Systems of Polynomial Equations. Remark 3.4.4 in Sturmfels and Maclagan's textbook-in-progress states this result as well, and the surrounding discussion is probably the clearest of any of these sources.

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David Speyer/ I have actually read (more likely browsed through) those three references and my question was that for the (relative) interior of Trop(X) do we have counterpart in X? –  user42154 Jan 27 at 15:28

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