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Integration on Artin toric stacks and Euler characteristics


Authors: Dan Edidin and Yogesh More
Journal: Proc. Amer. Math. Soc. 141 (2013), 3689-3699
MSC (2010): Primary 14M25, 14C15, 14D23
DOI: https://doi.org/10.1090/S0002-9939-2013-11849-6
Published electronically: July 12, 2013
MathSciNet review: 3091761
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Abstract | References | Similar Articles | Additional Information

Abstract: There is a well-developed intersection theory on smooth Artin stacks with quasi-affine diagonal. However, for Artin stacks whose diagonal is not quasi-finite, the notion of the degree of a Chow cycle is not defined. In this paper we propose a definition for the degree of a cycle on Artin toric stacks whose underlying toric varieties are complete. As an application we define the Euler characteristic of an Artin toric stack with complete good moduli space, extending the definition of the orbifold Euler characteristic. An explicit combinatorial formula is given for 3-dimensional Artin toric stacks.


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

  • [Al] Jarod Alper, Good moduli spaces for Artin stacks, math.AG/08042242 (2008).
  • [BCS] Lev A. Borisov, Linda Chen, and Gregory G. Smith, The orbifold Chow ring of toric Deligne-Mumford stacks, J. Amer. Math. Soc. 18 (2005), no. 1, 193-215 (electronic). MR 2114820 (2006a:14091), https://doi.org/10.1090/S0894-0347-04-00471-0
  • [CLS] David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322 (2012g:14094)
  • [EG98] Dan Edidin and William Graham, Equivariant intersection theory, Invent. Math. 131 (1998), no. 3, 595-634. MR 1614555 (99j:14003a), https://doi.org/10.1007/s002220050214
  • [EG03] Dan Edidin and William Graham, Riemann-Roch for quotients and Todd classes of simplicial toric varieties, Comm. Algebra 31 (2003), no. 8, 3735-3752. Special issue in honor of Steven L. Kleiman. MR 2007382 (2004h:14015), https://doi.org/10.1081/AGB-120022440
  • [EM] Dan Edidin and Yogesh More, Partial desingularizations of good moduli spaces of Artin toric stacks, Michigan Math. J. 61 (2012), no. 3, 451-474. MR 2975255, https://doi.org/10.1307/mmj/1347040252
  • [Gil] Henri Gillet, Intersection theory on algebraic stacks and $ Q$-varieties, Proceedings of the Luminy conference on algebraic $ K$-theory (Luminy, 1983), 1984, pp. 193-240. MR 772058 (86b:14006), https://doi.org/10.1016/0022-4049(84)90036-7
  • [Kre] Andrew Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), no. 3, 495-536. MR 1719823 (2001a:14003), https://doi.org/10.1007/s002220050351
  • [Iwa] Isamu Iwanari, Integral chow rings of toric stacks, Int. Math. Res. Not. IMRN 24 (2009), 4709-4725. MR 2564373 (2010m:14003), https://doi.org/10.1093/imrn/rnp110
  • [MS] Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098 (2006d:13001)
  • [Vis] Angelo Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613-670. MR 1005008 (90k:14004), https://doi.org/10.1007/BF01388892

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Additional Information

Dan Edidin
Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211
Email: edidind@missouri.edu

Yogesh More
Affiliation: Department of Mathematics, SUNY College at Old Westbury, Old Westbury, New York 11568
Email: yogeshmore80@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-2013-11849-6
Received by editor(s): January 10, 2012
Published electronically: July 12, 2013
Additional Notes: The first author was partially supported by NSA grant H98230-08-1-0059 while preparing this article.
Communicated by: Lev Borisov
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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