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 P 1 \{0, ∞} modulo scaling degenerates (isotrivially) to a compactification of the space of configurations of n points on 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 G n−1 a -variety, which mirrors the fact that the Losev-Manin space is toric. The degeneration is compatible with the actions of G n−1 m and G n−1 a in the sense that these actions fit together globally in the total space of the degeneration.