Configurations of points on a line up to scaling or translation

Abstract

We prove that the Losev--Manin compactification of the space of configurations of $n$ points on ${\mathbb{P}}^1\backslash \{0,\infty \}$ modulo scaling degenerates (isotrivially) to a compactification of the space of configurations of $n$ points on ${\mathbb{A}}^1$ modulo translation. The latter resembles the compactification constructed by Ziltener and Mau--Woodward, but allows the marked points to coincide, making it a ${\mathbb{G}}_a^{n-1}$-variety, which mirrors the fact that the Losev--Manin space is toric. The degeneration is compatible with the actions of ${\mathbb{G}}_m^{n-1}$ and ${\mathbb{G}}_a^{n-1}$ in the sense that these actions fit together globally in the total space of the degeneration.

arXiv commentThe paper is superseded by arXiv:2111.13743 (which improves the methods, results, and style of the current paper) and no longer intended for publication