I am reading this Brief Introduction to Tropical Geometry and I am trying to understand section 5.4.
In particular I want to understand the following construction given at the beginning of the chapter:
Let us start with an affine linear space $\tilde{\mathcal{L}} \subset \mathbb{C}^n$ of dimension $k$, and assume $\mathcal{L}=$ $\tilde{\mathcal{L}} \cap\left(\mathbb{C}^{\times}\right)^n \subset\left(\mathbb{C}^{\times}\right)^n$ is non-empty. We obtain a natural hyperplane arrangement $\mathcal{A}$ in $\mathbb{C} P^k$ by compactifying $\mathcal{L}$ to $\overline{\mathcal{L}} \cong \mathbb{C} P^k \subset \mathbb{C} P^n$. This arrangement consists of the intersection of $\overline{\mathcal{L}}$ with all coordinates hyperplanes in $\mathbb{C} P^n$
Furthermore, there is a stratification of this space:
Let us explain what do we mean by "intersection properties". A hyperplane arrangement $\mathcal{A}=\{ H_0, \dots, H_n \}$ in $\mathbb{C} P^k$ is a stratified space, here we refer to the strata as flats. Each flat can be indexed by the maximum subset $I \subset \mathcal{A}$ of hyperplanes which contains it; label such a flat $F_I$. The flats form a partially ordered set, the order being given by inclusion and is known as the lattice of flats of the arrangement $\mathcal{A}$.
I was wondering if I could get an example of this compactification and stratification in 2 or 3 dimensions with an affine linear space of dimension 1. This is most relevant to my research, and I feel like if I really focus on an example like this, I can understand enough of what's needed to get to what I really want - the local models for tropical curves (1d tropical manifolds, defined in Chapter 7 of the brief introduction).