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

Full-text PDF

Abstract | References | Additional Information

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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Additional Information

**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.