Pointed trees of projective spaces

Abstract

We introduce a smooth projective variety $T_{d,n}$ which compactifies the space of configurations of $n$ distinct points on affine $d$-space modulo translation and homothety. The points in the boundary correspond to $n$-pointed stable rooted trees of $d$-dimensional projective spaces, which for $d = 1$, are $(n+1)$-pointed stable rational curves. In particular, $T_{1,n}$ is isomorphic to $\bar{M}_{0,n+1}$, the moduli space of such curves. The variety $T_{d,n}$ shares many properties with $\bar{M}_{0,n}$. For example, as we prove, the boundary is a smooth normal crossings divisor whose components are products of $T_{d,i}$ for $i < n$, it has an inductive construction analogous to but differing from Keel's for $\bar{M}_{0,n}$ which can be used to describe its Chow groups, Chow motive and Poincar\'e polynomials, generalizing \cite{Keel,Man:GF}. We give a presentation of the Chow rings of $T_{d,n}$, exhibit explicit dual bases for the dimension 1 and codimension 1 cycles. The variety $T_{d,n}$ is embedded in the Fulton-MacPherson spaces $X[n]$ for \textit{any} smooth variety $X$ and we use this connection in a number of ways. For example, to give a family of ample divisors on $T_{d,n}$ and to give an inductive presentation of the Chow groups and the Chow motive of $X[n]$ analogous to Keel's presentation for $\bar{M}_{0,n}$, solving a problem posed by Fulton and MacPherson.