Topology of amoebas of linear spaces

Consider a $d$-dimensional linear subspace $V$ of ${\mathbb C}^n$, and let $M$ be the intersection of $V$ with $(\mathbb{C}^*)^n$. Then $M$ is the complement of a collection $H$ of $n$ hyperplanes in $V$, and (virtually) any arrangement of hyperplanes arises in this way. It is a classical problem to study the topology of $M$ in terms of the combinatorics (for example the matroid) of $H$.

Questions:

1. What are the fibers of the map $\text{Log}: M \to A$, where $A$ is the amoeba of $V$?

2. What conditions on $H$ will guarantee that this map is a homeomorphism?

3. What can we say in general about the topology of the amoeba of a linear space?

4. How does this relate to Federico Ardila's characterization of the tropicalization of $V$ in terms of the matroid of $H$?

When $d=1$, the answer to (1) is easy. In this case, $H$ is a collection of points on a complex line. If there exist three points that do not lie on a common real line, then $\text{Log}$ is injective. If all $n$ points lie on a real line, then the fibers of $\text{Log}$ are the orbits of the $\mathbb{Z}_2$ action given by reflection over this line.

Higher dimensional examples of hyperplane arrangements such that $\text{Log}$ is injective can be constructed by taking a product of $d$ copies of three generic points on a complex line, and then adding arbitrarily many more hyperplanes to this collection of 3d-hyperplanes in $V=\mathbb{C}^d$. But there should be many examples that are simpler than these.

(contributed by Nicholas Proudfoot)

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