Pointed trees of projective spaces

Abstract

We introduce a smooth projective variety T d , n T_{d,n} which compactifies the space of configurations of n n distinct points on affine d d -space modulo translation and homothety. The points in the boundary correspond to n n -pointed stable rooted trees of d d -dimensional projective spaces, which for d = 1 d = 1 , are ( n + 1 ) (n+1) -pointed stable rational curves. In particular, T 1 , n T_{1,n} is isomorphic to M ¯ 0 , n + 1 \overline {M}_{0,n+1} , the moduli space of such curves. The variety T d , n T_{d,n} shares many properties with M ¯ 0 , n + 1 \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 T_{d,i} for i > n i > n , and it has an inductive construction analogous to but differing from Keel’s for M ¯ 0 , n + 1 \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 T_{d,n} , exhibit explicit dual bases for the dimension 1 1 and codimension 1 1 cycles. The variety T d , n T_{d,n} is embedded in the Fulton-MacPherson spaces X [ n ] X[n] for any smooth variety X X , and we use this connection in a number of ways. In particular we give a family of ample divisors on T d , n T_{d,n} , and an inductive presentation of the Chow motive of X [ n ] X[n] . This also gives an inductive presentation of the Chow groups of X [ n ] X[n] analogous to Keel’s presentation for M ¯ 0 , n + 1 \overline {M}_{0,n+1} , solving a problem posed by Fulton and MacPherson.