Point Configurations and Translations

The spaces of point configurations on the projective line up to the action of $\mathrm{SL}(2,\mathbb K)$ and its maximal torus are canonically compactified by the Grothdieck-Knudsen and Losev-Manin moduli spaces $\overline M_{0,n}$ and $\overline L_n$ respectively. We examine the configuration space up to the action of the maximal unipotent group $\mathbb G_a\subseteq \mathrm{SL}(2,\mathbb K)$ and define an analogous compactification. For this we first assign a canonical quotient to the action of a unipotent group on a projective variety. Moreover, we show that similar to $\overline M_{0,n}$ and $\overline L_n$ this quotient arises in a sequence of blow-ups from a product of projective spaces.