A compactification of the moduli space of marked vertical lines in ${\mathbb{C}}^2$

For $r\ge 1$ and $$\mathbf{n}$ \in $\mathbb{Z}${\geq0}^r\setminus{$\mathbf{0}$}$, we construct a proper complex variety ${\overline{2M}}_{\mathbf{n}}$. ${\overline{2M}}_{\mathbf{n}}$ is locally toric, and it is equipped with a forgetful map $\overline{2M}{$\mathbf{n}$} \to \overline M_{0,r+1}$\mathrm{.}Thisspaceisacompactificationof$2M_{$\mathbf{n}$}$, the configuration space of marked vertical lines in ${\mathbb{C}}^2$ up to translations and dilations. In the appendices, we give several examples and show how the stratification of ${\overline{2M}}_{\mathbf{n}}$ can be used to recursively compute its virtual Poincar'{e} polynomial.