Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

The orbifold Chow ring of toric Deligne-Mumford stacks


Authors: Lev A. Borisov, Linda Chen and Gregory G. Smith
Journal: J. Amer. Math. Soc. 18 (2005), 193-215
MSC (2000): Primary 14N35; Secondary 14C15, 14M25
DOI: https://doi.org/10.1090/S0894-0347-04-00471-0
Published electronically: November 3, 2004
MathSciNet review: 2114820
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Generalizing toric varieties, we introduce toric Deligne-Mumford stacks. The main result in this paper is an explicit calculation of the orbifold Chow ring of a toric Deligne-Mumford stack. As an application, we prove that the orbifold Chow ring of the toric Deligne-Mumford stack associated to a simplicial toric variety is a flat deformation of (but is not necessarily isomorphic to) the Chow ring of a crepant resolution.


References [Enhancements On Off] (What's this?)

  • [1] D. Abramovich, A. Corti, and A. Vistoli, Twisted bundles and admissible covers, Comm. Algebra 31 (2003), 3547-3618. MR 2007376
  • [2] D. Abramovich, T. Graber, and A. Vistoli, ``Algebraic orbifold quantum products'' in Orbifolds in mathematics and physics, pp. 1-24, Contemp. Math. 310, Amer. Math. Soc., Providence, RI, 2002.MR 1950940 (2004c:14104)
  • [3] V. V. Batyrev, Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs, J. Eur. Math.Soc. 1 (1999), 5-33.MR 1677693 (2001j:14018)
  • [4] L. A. Borisov, String cohomology of a toroidal singularity, J. Algebraic Geom. 9 (2000), 289-300.MR 1735773 (2001b:14079)
  • [5] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Math. 39, Cambridge University Press, Cambridge, 1993.MR 1251956 (95h:13020)
  • [6] D. A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), 17-50. MR 1299003 (95i:14046)
  • [7] W. Chen and Y. Ruan, A New Cohomology Theory for Orbifold, arXiv:math.AG/0004129.
  • [8] W. Chen and Y. Ruan, ``Orbifold Gromov-Witten theory'' in Orbifolds in mathematics and physics, pp. 25-85, Contemp. Math. 310, Amer. Math. Soc., Providence, RI, 2002. MR 1950941 (2004k:53145)
  • [9] P. Deligne and M. Rapoport, ``Les schémas de modules de courbes elliptiques'' in Modular functions of one variable, II, pp. 143-316, Lecture Notes in Math. 349, Springer, Berlin, 1973. MR 0337993 (49:2762)
  • [10] D. Edidin, ``Notes on the construction of the moduli space of curves'' in Recent progress in intersection theory, pp. 85-113, Trends Math., Birkhäuser Boston, Boston, MA, 2000. MR 1849292 (2002f:14039)
  • [11] D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Math. 150, Springer-Verlag, New York, 1995.MR 1322960 (97a:13001)
  • [12] D. Eisenbud and J. Harris, The geometry of schemes, Graduate Texts in Math. 197, Springer-Verlag, New York, 2000. MR 1730819 (2001d:14002)
  • [13] D. Eisenbud and B. Sturmfels, Binomial ideals, Duke Math. J. 84 (1996), 1-45.MR 1394747 (97d:13031)
  • [14] B. Fantechi and L. Göttsche, Orbifold cohomology for global quotients, Duke Math. J. 117 (2003), 197-227.MR 1971293 (2004h:14062)
  • [15] W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies 131, Princeton University Press, Princeton, NJ, 1993.MR 1234037 (94g:14028)
  • [16] Y. Jiang, The Chen-Ruan cohomology of weighted projective spaces, arXiv:math.AG/0304140.
  • [17] J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 32, Springer-Verlag, Berlin, 1996. MR 1440180 (98c:14001)
  • [18] L. Lafforgue, Pavages des simplexes, schémas de graphes recollés et compactification des $\text{PGL} \sp {n+1} \sb {r}/\text{PGL} \sb {r}$, Invent. Math. 136 (1999), 233-271. MR 1681089 (2000i:14071)
  • [19] G. Laumon and L. Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 39, Springer-Verlag, Berlin, 2000. MR 1771927 (2001f:14006)
  • [20] I. Moerdijk, ``Orbifolds as groupoids: an introduction'' in Orbifolds in mathematics and physics, pp. 205-222, Contemp. Math. 310, Amer. Math. Soc., Providence, RI, 2002. MR 1950948 (2004c:22003)
  • [21] T. Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 15, Springer-Verlag, Berlin, 1988.MR 0922894 (88m:14038)
  • [22] M. Poddar, ``Orbifold cohomology group of toric varieties'' in Orbifolds in mathematics and physics, pp. 223-231, Contemp. Math. 310, Amer. Math. Soc., Providence, RI, 2002. MR 1950949 (2003j:14068)
  • [23] Y. Ruan,Cohomology Ring of Crepant Resolutions of Orbifolds, arXiv:math.AG/0108195.
  • [24] R. Stanley, ``Generalized $H$-vectors, intersection cohomology of toric varieties, and related results'' in Commutative algebra and combinatorics, pp. 187-213, Adv. Stud. Pure Math. 11, North-Holland, Amsterdam, 1987. MR 0951205 (89f:52016)
  • [25] B. Uribe, Orbifold cohomology of the symmetric product, Comm. Anal. Geom. (to appear), arXiv:math.AT/0109125.
  • [26] A. Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), 613-670. MR 1005008 (90k:14004)
  • [27] C. A. Weibel, An introduction to homological algebra. Cambridge Studies in Advanced Math. 38, Cambridge University Press, Cambridge, 1994.MR 1269324 (95f:18001)
  • [28] T. Yasuda, Twisted jets, motivic measure and orbifold cohomology, Compos. Math. 140 (2004), 396-422. MR 2027195
  • [29] G. M. Ziegler, Lectures on polytopes, Graduate Texts in Math. 152, Springer-Verlag, New York, 1995. MR 1311028 (96a:52011)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 14N35, 14C15, 14M25

Retrieve articles in all journals with MSC (2000): 14N35, 14C15, 14M25


Additional Information

Lev A. Borisov
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: borisov@math.wisc.edu

Linda Chen
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
Address at time of publication: Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, Ohio 43210
Email: lchen@math.ohio-state.edu

Gregory G. Smith
Affiliation: Department of Mathematics, Barnard College, Columbia University, New York, New York 10027
Address at time of publication: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6 Canada
Email: ggsmith@mast.queensu.ca

DOI: https://doi.org/10.1090/S0894-0347-04-00471-0
Keywords: Deligne-Mumford stack, Chow ring, toric variety, crepant resolution
Received by editor(s): September 30, 2003
Published electronically: November 3, 2004
Additional Notes: The first author was partially supported in part by NSF grant DMS-0140172.
The second author was partially supported in part by NSF VIGRE grant DMS-9810750.
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society