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

Nathaniel Bottman; Alexei Oblomkov

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arXiv · 1910.02037

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

Nathaniel Bottman·Alexei Oblomkov

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Abstract

For $r \geq 1$ 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 $\overline{2M}_{\mathbf{n}} \to \overline M_{0,r+1}$. 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 Poincar\'{e} polynomial.

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Nathaniel Bottman, Alexei Oblomkov. 2019-10-04. A compactification of the moduli space of marked vertical lines in $\mathbb{C}^2$. https://arxiv.org/abs/1910.02037

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