Configurations of points on a line up to scaling or translation

TL;DR

This paper demonstrates a degeneration of the Losev--Manin space of point configurations on a punctured projective line to a space of configurations on an affine line, revealing new geometric and group action structures.

Contribution

It establishes a degeneration linking the Losev--Manin compactification to a configuration space on the affine line, allowing coincident points and highlighting group action compatibilities.

Findings

Degeneration from Losev--Manin space to affine line configuration space.

The resulting space is a ${ m G}_a^{n-1}$-variety allowing coincident points.

Compatibility of ${ m G}_m^{n-1}$ and ${ m G}_a^{n-1}$ actions in the degeneration.

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.