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.

References
- David Eisenbud,
*Commutative algebra*, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR **1322960**
- W. Fulton and R. Pandharipande,
*Notes on stable maps and quantum cohomology*.
- William Fulton,
*Intersection theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR **99d:14003**
- William Fulton and Robert MacPherson,
*A compactification of configuration spaces*, Ann. of Math. (2) **139** (1994), no. 1, 183–225. MR **1259368**, DOI https://doi.org/10.2307/2946631
- Angela Gibney, Sean Keel, and Ian Morrison,
*Towards the ample cone of $\overline M_{g,n}$*, J. Amer. Math. Soc. **15** (2002), no. 2, 273–294. MR **1887636**, DOI https://doi.org/10.1090/S0894-0347-01-00384-8
- Paul Hacking, Sean Keel, and Jenia Tevelev,
*Compactification of the moduli space of hyperplane arrangements*, J. Algebraic Geom. **15** (2006), no. 4, 657–680. MR **2237265**, DOI https://doi.org/10.1090/S1056-3911-06-00445-0
- Robin Hartshorne,
*Algebraic geometry*, Springer-Verlag, New York, 1977, Graduate Texts in Mathematics, No. 52. MR **57:3116**
- M. M. Kapranov,
*Chow quotients of Grassmannians. I*, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 29–110. MR **1237834**
- M. M. Kapranov,
*Veronese curves and Grothendieck-Knudsen moduli space $\overline M_{0,n}$*, J. Algebraic Geom. **2** (1993), no. 2, 239–262. MR **1203685**
- Ralph Kaufmann,
*The intersection form in ${\scr H}^*(\overline {\scr M}_{0n})$ and the explicit Künneth formula in quantum cohomology*, Internat. Math. Res. Notices **19** (1996), 929–952. MR **1422369**, DOI https://doi.org/10.1155/S107379289600058X
- Sean Keel,
*Intersection theory of moduli space of stable $n$-pointed curves of genus zero*, Trans. Amer. Math. Soc. **330** (1992), no. 2, 545–574. MR **1034665**, DOI https://doi.org/10.1090/S0002-9947-1992-1034665-0
- Sean Keel and Jenia Tevelev,
*Geometry of Chow quotients of Grassmannians*, Duke Math. J. **134** (2006), no. 2, 259–311. MR **2248832**, DOI https://doi.org/10.1215/S0012-7094-06-13422-1
- Li Li,
*Chow Motive of Fulton-MacPherson configuration spaces and wonderful compactifications*, arXiv:math.AG/0611459.
- Ju. I. Manin,
*Correspondences, motifs and monoidal transformations*, Mat. Sb. (N.S.) **77 (119)** (1968), 475–507 (Russian). MR **0258836**
- Yu. I. Manin,
*Generating functions in algebraic geometry and sums over trees*, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 401–417. MR **1363064**, DOI https://doi.org/10.1007/978-1-4612-4264-2_14
- Andrei Mustaţă and Magdalena Anca Mustaţă,
*Intermediate moduli spaces of stable maps*, Invent. Math. **167** (2007), no. 1, 47–90. MR **2264804**, DOI https://doi.org/10.1007/s00222-006-0006-1
- Dragos Oprea,
*Divisors on the moduli spaces of stable maps to flag varieties and reconstruction*, J. Reine Angew. Math. **586** (2005), 169–205. MR **2180604**, DOI https://doi.org/10.1515/crll.2005.2005.586.169
- Dragos Oprea,
*Tautological classes on the moduli spaces of stable maps to $\Bbb P^r$ via torus actions*, Adv. Math. **207** (2006), no. 2, 661–690. MR **2271022**, DOI https://doi.org/10.1016/j.aim.2006.01.002
- Dragos Oprea,
*The tautological rings of the moduli spaces of stable maps to flag varieties*, J. Algebraic Geom. **15** (2006), no. 4, 623–655. MR **2237264**, DOI https://doi.org/10.1090/S1056-3911-06-00452-8
- Rahul Pandharipande,
*Intersections of $\mathbf Q$-divisors on Kontsevich’s moduli space $\overline M_{0,n}(\mathbf P^r,d)$ and enumerative geometry*, Trans. Amer. Math. Soc. **351** (1999), no. 4, 1481–1505. MR **1407707**, DOI https://doi.org/10.1090/S0002-9947-99-01909-1

References
- David Eisenbud,
*Commutative algebra*, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. MR **1322960 (97a:13001)**
- W. Fulton and R. Pandharipande,
*Notes on stable maps and quantum cohomology*.
- William Fulton,
*Intersection theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR **99d:14003**
- William Fulton and Robert MacPherson,
*A compactification of configuration spaces*, Ann. of Math. (2) **139** (1994), no. 1, 183–225. MR **1259368 (95j:14002)**
- Angela Gibney, Sean Keel, and Ian Morrison,
*Towards the ample cone of $\overline M_ {g,n}$*, J. Amer. Math. Soc. **15** (2002), no. 2, 273–294 (electronic). MR **1887636 (2003c:14029)**
- Paul Hacking, Sean Keel, and Jenia Tevelev,
*Compactification of the moduli space of hyperplane arrangements*, J. Algebraic Geom. **15** (2006), no. 4, 657–680. MR **2237265**
- Robin Hartshorne,
*Algebraic geometry*, Springer-Verlag, New York, 1977, Graduate Texts in Mathematics, No. 52. MR **57:3116**
- M. M. Kapranov,
*Chow quotients of Grassmannians. I*, I. M. Gel$’$fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 29–110. MR **1237834 (95g:14053)**
- ---,
*Veronese curves and Grothendieck-Knudsen moduli space $\overline M_ {0,n}$*, J. Algebraic Geom. **2** (1993), no. 2, 239–262. MR **1203685 (94a:14024)**
- Ralph Kaufmann,
*The intersection form in ${H}^*(\overline {M}_{0n})$ and the explicit Künneth formula in quantum cohomology*, Internat. Math. Res. Notices (1996), no. 19, 929–952. MR **1422369 (98c:14006)**
- Sean Keel,
*Intersection theory of moduli space of stable $n$-pointed curves of genus zero*, Trans. Amer. Math. Soc. **330** (1992), no. 2, 545–574. MR **1034665 (92f:14003)**
- Sean Keel and Jenia Tevelev,
*Geometry of Chow quotients of Grassmannians*, Duke Math. J. **134** (2006), no. 2, 259–311. MR **2248832**
- Li Li,
*Chow Motive of Fulton-MacPherson configuration spaces and wonderful compactifications*, arXiv:math.AG/0611459.
- Ju. I. Manin,
*Correspondences, motifs and monoidal transformations*, Mat. Sb. (N.S.) **77 (119)** (1968), 475–507. MR **0258836 (41:3482)**
- Yu. I. Manin,
*Generating functions in algebraic geometry and sums over trees*, The moduli space of curves (Texel Island, 1994), Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 401–417. MR **1363064 (97e:14065)**
- Andrei Mustaţă and Magdalena Anca Mustaţă,
*Intermediate moduli spaces of stable maps*, Invent. Math. **167** (2007), no. 1, 47–90. MR **2264804**
- Dragos Oprea,
*Divisors on the moduli spaces of stable maps to flag varieties and reconstruction*, J. Reine Angew. Math. **586** (2005), 169–205. MR **2180604 (2006k:14062)**
- ---,
*Tautological classes on the moduli spaces of stable maps to $\mathbb {P}^ r$ via torus actions*, Adv. Math. **207** (2006), no. 2, 661–690. MR **2271022**
- ---,
*The tautological rings of the moduli spaces of stable maps to flag varieties*, J. Algebraic Geom. **15** (2006), no. 4, 623–655. MR **2237264**
- Rahul Pandharipande,
*Intersections of $\mathbf {Q}$-divisors on Kontsevich’s moduli space $\overline M_ {0,n}(\mathbf {P}^ r,d)$ and enumerative geometry*, Trans. Amer. Math. Soc. **351** (1999), no. 4, 1481–1505. MR **1407707 (99f:14068)**

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.