Point configurations and translations

Abstract

The spaces of point configurations on the projective line up to the action of \({{\rm SL}(2,\mathbb{K})}\) 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 \({\mathbb{G}_a \subseteq {\rm SL}(2,,\mathbb{K})}\) 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.