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, Volume 129, Issue 3, pp 337–368 | Cite as

Quantitative Néron theory for torsion bundles

Article

Abstract

Let R be a discrete valuation ring with algebraically closed residue field, and consider a smooth, geometrically connected, and projective curve C K over the field of fractions K. For any positive integer r prime to the residual characteristic, we consider the finite K-group scheme \({{\rm Pic}_{C_K}[r]}\) of r-torsion line bundles on C K . We determine when there exists a finite R-group scheme, which is a model of \({{\rm Pic}_{C_K}[r]}\) over R; in other words, we establish when the Néron model of \({{\rm Pic}_{C_K}[r]}\) is finite. The obvious idea would be to study the points of the Néron model over R, but in general these do not represent r-torsion line bundles on a semistable reduction of C K . Instead, we recast the notion of models on a stack-theoretic base: there, we find finite Néron models, which represent r-torsion line bundles on a stack-theoretic semistable reduction of C K . This allows us to quantify the lack of finiteness of the classical Néron models and finally to provide an efficient criterion for it.

Mathematics Subject Classification (2000)

14h40 

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References

  1. 1.
    Abbes, A.: Réduction semi-stable des courbes d’après Artin, Deligne, Grothendieck, Mumford, Saito, Winters, ..., Courbes semi-stables et groupe fondamental en géométrie algébrique (Luminy, 1998), pp. 59–110. Progr. Math., 187. Birkhäuser, Basel (2000)Google Scholar
  2. 2.
    Abramovich D., Corti A., Vistoli A.: Twisted bundles and admissible covers. Comm. Algebra 31, 3547–3618 (2003)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Abramovich D., Vistoli A.: Compactifying the space of stable maps. J. Am. Math. Soc. 15, 27–75 (2002)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bacher R., de la Harpe P., Nagnibeda T.: The lattice of integral flows and the lattice of integral cuts on a finite graph. Bull. Soc. Math. France 125(2), 167–198 (1997)MATHMathSciNetGoogle Scholar
  5. 5.
    Biggs N.: Chip-firing and the critical group of a graph. Algebraic Combin. 9, 25–45 (1999)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Berman K.A.: Bicycles and spanning trees. SIAM J. Alg. Disc. Meth. 7, 1–12 (1986)MATHCrossRefGoogle Scholar
  7. 7.
    Bosch S., Lütkebohmert W., Raynaud M.: Néron Models, Results in Mathematics and Related Areas (3), vol. 21. Springer, Berlin (1990)Google Scholar
  8. 8.
    Busonero S., Melo M., Stoppino L.: Combinatorial aspects of nodal curves. Le Matematiche LXI(I), 109–141 (2006)MathSciNetGoogle Scholar
  9. 9.
    Caporaso L.: Néron models and compactified Picard schemes over the moduli stack of stable curves. Am. J. Math. 130, 1–47 (2005)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Chiodo A.: Stable twisted curves and their r-spin structures. Ann. Inst. Fourier 58(5), 1635–1689 (2008)MATHMathSciNetGoogle Scholar
  11. 11.
    Dhar D.: Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64(14), 1613–1616 (1990)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Deligne P., Mumford D.: Irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math. 36, 75–112 (1969)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Deschamps M.: Réduction semi-stable. Séminaire sur les pinceaux de courbes de genre au moins deux. Astérisque, Société Mathématique de France, Paris 86, 1–34 (1981)MATHGoogle Scholar
  14. 14.
    Grothendieck, A.: Technique de descente et théorèmes d’existence en géométrie algébrique. I-V. Séminaire Bourbaki, vol. 7, Exp. No. 236. Soc. Math. France, Paris, pp. 221–243 (1995)Google Scholar
  15. 15.
    Groupes de monodromie en géométrie algébrique. I. Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7I). Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D.S. Rim. Lecture Notes in Mathematics, vol. 288, p. viii+523. Springer, Berlin (1972)Google Scholar
  16. 16.
    Godsil C., Royle G.: Algebraic Graph Theory, Graduate Texts in Mathematics, vol. 207. Springer, Berlin (2001)Google Scholar
  17. 17.
    Keel S., Mori S.: Quotients by groupoids. Ann. Math. (2) 145(1), 193–213 (1997)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Laumon G., Moret-Bailly L.: Champs Algébriques, vol. xii+208. Springer, Berlin (2000)Google Scholar
  19. 19.
    Lorenzini D.: Arithmetical graphs. Math. Ann. 285(3), 481–501 (1989)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Lorenzini D.: A finite group attached to the Laplacian of a graph. Discret. Math. 91(3), 277–282 (1991)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Lorenzini, D.: Smith normal form and Laplacians. J. Combin. Theory Series B archive 98(6), 1271–1300, table of contents (2008)Google Scholar
  22. 22.
    Matsuki K., Olsson M.: Kawamata–Viehweg vanishing as Kodaira vanishing for stacks. Math. Res. Lett. 12, 207–217 (2005)MATHMathSciNetGoogle Scholar
  23. 23.
    Mumford, D.: Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Bombay (1970)Google Scholar
  24. 24.
    Olsson M.: On (log) twisted curves. Compos. Math. 143(2), 476–494 (2007)MATHMathSciNetGoogle Scholar
  25. 25.
    Raynaud M.: Spécialisation du foncteur de Picard. Inst. Hautes Études Sci. Publ. Math. 38, 27–76 (1970)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Romagny, M.: Sur quelques aspects des champs de revêtements de courbes algébriques. Ph.D. thesis, 120 (2002). http://www-fourier.ujf-grenoble.fr/THESE/ps/t120.ps.gz
  27. 27.
    Serre, J.-P.: Rigidité du foncteur de Jacobi d’échelon n ≥ 3. Application à l’exposé 17 du séminaire Cartan (1960/61)Google Scholar

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© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Institut Fourier, U.M.R. CNRS 5582, U.F.R. de MathématiquesUniversité de Grenoble 1Saint Martin d’HèresFrance

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