On the global quotient structure of the space of twisted stable maps to a quotient stack
Authors:
Dan Abramovich, Tom Graber, Martin Olsson and Hsian-Hua Tseng
Journal:
J. Algebraic Geom. 16 (2007), 731-751
DOI:
https://doi.org/10.1090/S1056-3911-07-00443-2
Published electronically:
March 7, 2007
MathSciNet review:
2357688
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Abstract |
References |
Additional Information
Abstract: Let ${\mathcal {X}}$ be a tame proper Deligne-Mumford stack of the form $[M/G]$ where $M$ is a scheme and $G$ is an algebraic group. We prove that the stack ${\mathcal {K}} _{g,n}({\mathcal {X}},d)$ of twisted stable maps is a quotient stack and can be embedded into a smooth Deligne-Mumford stack. When $G$ is finite, we give a more precise construction of ${\mathcal {K}}_{g,n}( {\mathcal {X}},d)$ using Hilbert schemes and admissible $G$-covers.
References
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References
- D. Abramovich, A. Corti, and A. Vistoli. Twisted bundles and admissible covers. Comm. Algebra 31 (2003) 3547–3618. MR 2007376 (2005b:14049)
- D. Abramovich, T. Graber, and A. Vistoli. Algebraic orbifold quantum products. Orbifolds in mathematics and physics (Madison, WI, 2001), 1–24, Amer. Math. Soc., 2002. MR 1950940 (2004c:14104)
- D. Abramovich and A. Vistoli. Compactifying the space of stable maps. J. Amer. Math. Soc. 15 (2002) 27–75. MR 1862797 (2002i:14030)
- K. Behrend. Gromov-Witten invariants in algebraic geometry. Invent. Math. 127 (1997), no. 3, 601–617. MR 1431140 (98i:14015)
- P. Deligne and D. Mumford. The irreducibility of the space of curves of given genus. Inst. Hautes Éudes Sci. Publ. Math. no. 36 (1969), 75–109. MR 0262240 (41:6850)
- D. Edidin, B. Hassett, A. Kresch, and A. Vistoli. Brauer groups and quotient stacks. Amer. J. Math. 123 (2001), no. 4, 761–777. MR 1844577 (2002f:14002)
- C. Faber and R. Pandharipande. Hodge integrals and Gromov-Witten theory. Invent. Math. 139 (2000), no. 1, 173–199. MR 1728879 (2000m:14057)
- W. Fulton and J. Harris. Representation theory. A first course. Springer-Verlag, 1991. MR 1153249 (93a:20069)
- W. Fulton and R. Pandharipande. Notes on stable maps and quantum cohomology. Algebraic geometry (Santa Cruz, 1995), 45–96, Amer. Math. Soc., 1997. MR 1492534 (98m:14025)
- A. Grothendieck. Fondements de la géométrie algébrique. [Extraits du Seminaire Bourbaki, 1957–1962.] (French). Secrétariat mathématique, 1962. MR 0146040 (26:3566)
- T. Graber and R. Pandharipande. Localization of virtual classes. Invent. Math. 135 (1999), no. 2, 487–518. MR 1666787 (2000h:14005)
- R. Hartshorne. Algebraic geometry. Springer-Verlag, 1977. MR 0463157 (57:3116)
- L. Illusie. Complexe cotangent et déformations I. Lecture Notes in Mathematics, Vol. 239, Springer-Verlag. MR 0491680 (58:10886a)
- G. Laumon and L. Moret-Bailly. Champs algébriques. Springer-Verlag, 2000. MR 1771927 (2001f:14006)
- M. Olsson and J. Starr. Quot functors for Deligne-Mumford stacks. Comm. Algebra 31 (2003), no. 8, 4069–4096. MR 2007396 (2004i:14002)
- M. Romagny. Group actions on stacks and applications. Michigan Math. J. 53 (2005), 209–236, MR 2125542 (2005m:14005)
- B. Totaro. The resolution property for schemes and stacks. J. Reine Angew. Math. 577 (2004), 1–22, MR 2108211 (2005j:14002)
- A. Vistoli. The deformation theory of local complete intersections. ArXiv:alg-geom/9703008.
Additional Information
Dan Abramovich
Affiliation:
Department of Mathematics, Brown University, 151 Thayer Street, Providence, Rhode Island 02912
MR Author ID:
309312
ORCID:
0000-0003-0719-0989
Email:
abrmovic@math.brown.edu
Tom Graber
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication:
Department of Mathematics, California Institute of Technology, Mathematics 253-37, Caltech, Pasadena, California 91125
Email:
graber@caltech.edu
Martin Olsson
Affiliation:
School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540
Address at time of publication:
Department of Mathematics, University of California, Berkeley, California 94720
MR Author ID:
718643
Email:
molsson@math.utexas.edu
Hsian-Hua Tseng
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, B.C. V6T 1Z2, Canada
Email:
hhtseng@math.ubc.ca
Received by editor(s):
November 7, 2005
Received by editor(s) in revised form:
November 24, 2005
Published electronically:
March 7, 2007
Additional Notes:
Research of the first author was partially supported by NSF grant DMS-0335501. Research of the second author was partially supported by NSF grant DMS-0301179 and an Alfred P. Sloan Research Fellowship. Research of the third author was partially supported by an NSF post-doctoral research fellowship.
