Journal of Algebraic Geometry

Author(s): L. Chen; A. Gibney; D. Krashen
Journal: J. Algebraic Geom. 18 (2009), 477-509.
Posted: November 19, 2008
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Abstract: We introduce a smooth projective variety $T_{d,n}$ which compactifies the space of configurations of

distinct points on affine

-space modulo translation and homothety. The points in the boundary correspond to

-pointed stable rooted trees of

-dimensional projective spaces, which for

, are

-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

, 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

and codimension

cycles. The variety $T_{d,n}$ is embedded in the Fulton-MacPherson spaces

for any smooth variety

, 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

. This also gives an inductive presentation of the Chow groups of

analogous to Keel's presentation for $\overline{M}_{0,n+1}$ , solving a problem posed by Fulton and MacPherson.

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L. Chen
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
Address at time of publication: Department of Mathematics and Statistics, Swarthmore College, 500 College Avenue, Swarthmore, Pennsylvania 19081
Email: lchen@math.ohio-state.edu, lchen@swarthmore.edu

A. Gibney
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
Address at time of publication: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: agibney@math.upenn.edu

D. Krashen
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut
Address at time of publication: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: daniel.krashen@yale.edu, dkrashen@math.uga.edu

PII: S 1056-3911(08)00494-3
Received by editor(s): March 1, 2007
Received by editor(s) in revised form: June 12, 2007
Posted: November 19, 2008
Additional Notes: The authors were supported during this work by the National Science Foundation under agreements DMS-0432701, DMS-0509319, and DMS-0111298, respectively.

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