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

Abstract

For $r\ge 1$ and $\mathbf{n}\in {\mathbb{Z}}_{\ge 0}^r\setminus \{0\}$, 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 ${\overline{2M}}_{\mathbf{n}}\to {\overline{M}}_{0,r+1}$. This space is a compactification of $2M_{\mathbf{n}}$, 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 Poincaré polynomial.

37 pages, 20 figures