Spaces of multiscaled lines with collision

Abstract

We study varieties ${\mathcal{A}}_n$ arising as equivariant compactifications of the space of $n$ points in $\mathbb{C}$ up to overall translation. We define ${\mathcal{A}}_n$ and examine its basic geometric properties before constructing an isomorphism to an augmented wonderful variety. We show that ${\mathcal{A}}_n$ is in a canonical way a resolution of the space ${\overline{P}}_n$ considered by Zahariuc, proving along the way that the resolution constructed by Zahariuc is equivalent to ours.

46 pages, comments welcome!