Pointed Trees Of Projective Spaces

Document Type


Publication Date


Published In

Journal Of Algebraic Geometry


We introduce a smooth projective variety T(d,n) which compactifies the space of configurations of it distinct points oil 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 ($) over bar (0,n+1), the moduli space of such curves. The variety T(d,n) shares many properties with (M) over bar (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 (0,n+1). This call be used to describe its Chow groups and Chow motive generalizing [Trans. Airier. Math. Soc. 330 (1992), 545-574]. It also allows us to compute its Poincare polynomials, giving all alternative to the description implicit in [Progr. Math., vol. 129, Birkhauser, 1995, pp. 401-417]. We give a presentation of the Chow rings of T(d,n), exhibit explicit dual bases for the dimension I 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 (M) over bar (0,n+1), solving a problem posed by Fulton and MacPherson.


This work is a preprint available from arXiv.org at arXiv:math/0505296.