Pointed trees of projective spaces

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 $\overline {M}_{0,n+1}$, the moduli space of such curves. The variety $T_{d,n}$ shares many properties with $\overline {M}_{0,n+1}$. For example, as we prove, the boundary is a smooth normal crossings divisor whose components are products of $T_{d,i}$ for $i < n$, and it has an inductive construction analogous to but differing from Keel’s for $\overline {M}_{0,n+1}$. This can be used to describe its Chow groups and Chow motive generalizing [Trans. Amer. Math. Soc. 330 (1992), 545–574]. It also allows us to compute its Poincaré polynomials, giving an alternative to the description implicit in [Progr. Math., vol. 129, Birkhäuser, 1995, pp. 401–417]. 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 any smooth variety $X$, and we use this connection in a number of ways. In particular we give a family of ample divisors on $T_{d,n}$, and an inductive presentation of the Chow motive of $X[n]$. This also gives an inductive presentation of the Chow groups of $X[n]$ analogous to Keel’s presentation for $\overline {M}_{0,n+1}$, solving a problem posed by Fulton and MacPherson.