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Compactifying the space of stable maps


Authors: Dan Abramovich and Angelo Vistoli
Journal: J. Amer. Math. Soc. 15 (2002), 27-75
MSC (2000): Primary 14H10, 14D20
DOI: https://doi.org/10.1090/S0894-0347-01-00380-0
Published electronically: July 31, 2001
MathSciNet review: 1862797
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study a notion of twisted stable map, from a curve to a tame Deligne-Mumford stack, which generalizes the well-known notion of stable map to a projective variety.


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  • [$\aleph$-C-V] D. Abramovich, A. Corti and A. Vistoli, Twisted bundles and admissible covers, preprint, 2001, math.AG/0106211.
  • [$\aleph$-G-V] D. Abramovich, T. Graber and A. Vistoli, Algebraic orbifold quantum products, in preparation.
  • [$\aleph$-J] D. Abramovich and T. Jarvis, Moduli of twisted spin curves, preprint, math.AG/0104154.
  • [$\aleph$-O] Dan Abramovich and Frans Oort, Alterations and resolution of singularities, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 39–108. MR 1748617, https://doi.org/10.1007/978-3-0348-8399-3_3
  • [$\aleph$-V1] D. Abramovich and A. Vistoli, Complete moduli for families over semistable curves, preprint, math.AG/9811059.
  • [$\aleph$-V2] D. Abramovich and A. Vistoli, Complete moduli for fibered surfaces. In Recent Progress in Intersection Theory, G. Ellingsrud, W. Fulton, A. Vistoli (eds.), Birkhäuser, 2000.
  • [Ar] M. Artin, Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165–189. MR 0399094, https://doi.org/10.1007/BF01390174
  • [B-M] K. Behrend and Yu. Manin, Stacks of stable maps and Gromov-Witten invariants, Duke Math. J. 85 (1996), no. 1, 1–60. MR 1412436, https://doi.org/10.1215/S0012-7094-96-08501-4
  • [B-G-I] Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6); Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. MR 0354655
  • [C-R] W. Chen and Y. Ruan, Orbifold Gromov-Witten theory, preprint, math.AG/0103156.
  • [D-M] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 0262240
  • [E-H-K-V] D. Edidin, B. Hassett, A. Kresch and A. Vistoli, Brauer groups and quotient stacks, American Journal of Mathematics, to appear, math.AG/9905049.
  • [E-G] Dan Edidin and William Graham, Equivariant intersection theory, Invent. Math. 131 (1998), no. 3, 595–634. MR 1614555, https://doi.org/10.1007/s002220050214
    Angelo Vistoli, The Chow ring of ℳ₂. Appendix to “Equivariant intersection theory” [Invent. Math. 131 (1998), no. 3, 595–634; MR1614555 (99j:14003a)] by D. Edidin and W. Graham, Invent. Math. 131 (1998), no. 3, 635–644. MR 1614559, https://doi.org/10.1007/s002220050215
  • [F-G] B. Fantechi and L. Göttsche, Orbifold cohomology for global quotients, preprint, math.AG/0104207.
  • [F-P] W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 45–96. MR 1492534, https://doi.org/10.1090/pspum/062.2/1492534
  • [G-EGA] A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228. MR 0217083
    A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961), 222. MR 0217084
    A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 167. MR 0217085
    A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. II, Inst. Hautes Études Sci. Publ. Math. 17 (1963), 91 (French). MR 0163911
  • [G-FGA] Alexander Grothendieck, Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957–1962.], Secrétariat mathématique, Paris, 1962 (French). MR 0146040
  • [G-SGA1] Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1); Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud. MR 0354651
  • [Il] Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680
    Luc Illusie, Complexe cotangent et déformations. II, Lecture Notes in Mathematics, Vol. 283, Springer-Verlag, Berlin-New York, 1972 (French). MR 0491681
    Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680
    Luc Illusie, Complexe cotangent et déformations. II, Lecture Notes in Mathematics, Vol. 283, Springer-Verlag, Berlin-New York, 1972 (French). MR 0491681
  • [dJ-O] A. J. de Jong and F. Oort, On extending families of curves, J. Algebraic Geom. 6 (1997), no. 3, 545–562. MR 1487226
  • [K-M] Seán Keel and Shigefumi Mori, Quotients by groupoids, Ann. of Math. (2) 145 (1997), no. 1, 193–213. MR 1432041, https://doi.org/10.2307/2951828
  • [K-S] G. M. Kelly and Ross Street, Review of the elements of 2-categories, Category Seminar (Proc. Sem., Sydney, 1972/1973) Springer, Berlin, 1974, pp. 75–103. Lecture Notes in Math., Vol. 420. MR 0357542
  • [Kn] Donald Knutson, Algebraic spaces, Lecture Notes in Mathematics, Vol. 203, Springer-Verlag, Berlin-New York, 1971. MR 0302647
  • [Ko] Maxim Kontsevich, Enumeration of rational curves via torus actions, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 335–368. MR 1363062, https://doi.org/10.1007/978-1-4612-4264-2_12
  • [Kr] Andrew Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), no. 3, 495–536. MR 1719823, https://doi.org/10.1007/s002220050351
  • [La] G. La Nave, Stable reduction for elliptic surfaces with sections, Ph.D. Thesis, Brandeis University, 2000.
  • [L-MB] Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, 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. 39, Springer-Verlag, Berlin, 2000 (French). MR 1771927
  • [Ma] Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
  • [Mi] James S. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
  • [Mo] Shinichi Mochizuki, Extending families of curves over log regular schemes, J. Reine Angew. Math. 511 (1999), 43–71. MR 1695789, https://doi.org/10.1515/crll.1999.511.43
  • [Vi] Angelo Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613–670. MR 1005008, https://doi.org/10.1007/BF01388892
  • [We] S. Wewers, Construction of Hurwitz spaces, Institut für Experimentelle Mathematik preprint No. 21 (1998).

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Additional Information

Dan Abramovich
Affiliation: Department of Mathematics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215
Email: abrmovic@math.bu.edu

Angelo Vistoli
Affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40127 Bologna, Italy
Email: vistoli@dm.unibo.it

DOI: https://doi.org/10.1090/S0894-0347-01-00380-0
Received by editor(s): May 11, 2000
Published electronically: July 31, 2001
Additional Notes: The first author’s research was partially supported by National Science Foundation grant DMS-9700520 and by an Alfred P. Sloan research fellowship
The second author’s research was partially supported by the University of Bologna, funds for selected research topics
Article copyright: © Copyright 2001 American Mathematical Society