Modular interpretation of a non-reductive Chow quotient

Abstract

The space of n distinct points and a disjoint parameterized hyperplane in projective d-space up to projectivity---equivalently, configurations of n distinct points in affine d-space up to translation and homothety---has a beautiful compactification introduced by Chen-Gibney-Krashen. This variety, constructed inductively using the apparatus of Fulton-MacPherson configuration spaces, is a parameter space of certain pointed rational varieties whose dual intersection complex is a rooted tree. This generalizes $\overline{M}_{0,n}$ and shares many properties with it. In this paper, we prove that the normalization of the Chow quotient of $(\mathbb{P}^d)^n$ by the diagonal action of the subgroup of projectivities fixing a hyperplane, pointwise, is isomorphic to this Chen-Gibney-Krashen space $T_{d,n}$. This is a non-reductive analogue of Kapranov's famous quotient construction of $\overline{M}_{0,n}$, and indeed as a special case we show that $\overline{M}_{0,n}$ is the Chow quotient of $(\mathbb{P}^1)^{n-1}$ by an action of a semidirect product of the additive and multiplicative group.