Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Pointed trees of projective spaces


Authors: L. Chen, A. Gibney and D. Krashen
Journal: J. Algebraic Geom. 18 (2009), 477-509
DOI: https://doi.org/10.1090/S1056-3911-08-00494-3
Published electronically: November 19, 2008
MathSciNet review: 2496455
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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 $\overline {M}_{0,n+1}$, 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.


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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
MR Author ID: 689485
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
MR Author ID: 728218
ORCID: 0000-0001-6826-9901
Email: daniel.krashen@yale.edu, dkrashen@math.uga.edu

Received by editor(s): March 1, 2007
Received by editor(s) in revised form: June 12, 2007
Published electronically: 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.