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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie


IMPACT FACTOR 2018: 1.859

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1435-5345
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Volume 2019, Issue 747

Issues

Néron models of jacobians over base schemes of dimension greater than 1

David HolmesORCID iD: https://orcid.org/0000-0002-6081-2516
Published Online: 2016-06-16 | DOI: https://doi.org/10.1515/crelle-2016-0014

Abstract

We investigate to what extent the theory of Néron models of jacobians and of abel–jacobi maps extends to relative curves over base schemes of dimension greater than 1. We give a necessary and sufficient criterion for the existence of a Néron model. We use this to show that, in general, Néron models do not exist even after making a modification or even alteration of the base. On the other hand, we show that Néron models do exist outside some codimension-2 locus.

References

  • [1]

    E. Artin, Algebraic numbers and algebraic functions, American Mathematical Society, Providence 1967. Google Scholar

  • [2]

    M. Artin, Algebraization of formal moduli. I, Global analysis, University of Tokyo Press, Tokyo (1969), 21–71. Google Scholar

  • [3]

    A. Bellardini, On the log-Picard functor for aligned degenerations of curves, preprint (2015), http://arxiv.org/abs/1507.00506.

  • [4]

    O. Biesel, D. Holmes and R. de Jong, Néron models and the height jump divisor, preprint (2014), http://arxiv.org/abs/1412.8207.

  • [5]

    S. Bosch, W. Lütkebohmert and M. Raynaud, Néron models, Springer, Berlin 1990. Google Scholar

  • [6]

    P. Deligne, Le lemme de Gabber, Séminaire sur les pinceaux arithmétiques: La conjecture de Mordell (Paris 1983/84), Astérisque 127, Société Mathématique de France, Paris (1985), 131–150. Google Scholar

  • [7]

    P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Publ. Math. Inst. Hautes Études Sci. 36 (1969), 75–109. CrossrefGoogle Scholar

  • [8]

    B. Edixhoven, On Néron models, divisors and modular curves, J. Ramanujan Math. Soc. 13 (1998), no. 2, 157–194. Google Scholar

  • [9]

    B. Fantechi, L. Göttsche, L. Illusie, S. L. Kleiman, N. Nitsure and A. Vistoli, Fundamental algebraic geometry, Math. Surveys Monogr. 123, American Mathematical Society, Providence 2005. Google Scholar

  • [10]

    A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Publ. Math. Inst. Hautes Études Sci. 8 (1961), 1–222. Google Scholar

  • [11]

    A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. II, Publ. Math. Inst. Hautes Études Sci. 17 (1963), 137–223. Google Scholar

  • [12]

    A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Publ. Math. Inst. Hautes Études Sci. 24 (1965), 1–231. Google Scholar

  • [13]

    A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Publ. Math. Inst. Hautes Études Sci. 28 (1966), 1–255. Google Scholar

  • [14]

    A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. IV, Publ. Math. Inst. Hautes Études Sci. 32 (1967), 1–361. Google Scholar

  • [15]

    A. Grothendieck and M. Demazure, Schémas en groupes I (SGA 3-1), Lecture Notes in Math. 151, Springer, Berlin 1970. Google Scholar

  • [16]

    A. Grothendieck, M. Raynaud and D. S. Rim, Seminaire de géométrie algébrique Du Bois-Marie 1967–1969. Groupes de monodromie en géométrie algébrique (SGA 7 I), Lect. Notes in Math. 288, Springer, Berlin 1972. Google Scholar

  • [17]

    R. Hain, Normal functions and the geometry of moduli spaces of curves, Handbook of moduli. Volume I, Adv. Lect. Math. (ALM) 24, International Press, Sommerville (2013), 527–578. Google Scholar

  • [18]

    D. Holmes, A Néron model of the universal jacobian, preprint (2014), http://arxiv.org/abs/1412.2243.

  • [19]

    D. Holmes, Quasi-compactness of Néron models, and an application to torsion points, preprint (2016), http://arxiv.org/abs/1604.01155.

  • [20]

    D. Holmes, Torsion points and height jumping in higher-dimensional families of abelian varieties, preprint (2016), http://arxiv.org/abs/1604.04563.

  • [21]

    A. J. de Jong, Smoothness, semi-stability and alterations, Publ. Math. Inst. Hautes Études Sci. 83 (1996), 51–93. CrossrefGoogle Scholar

  • [22]

    G. Laumon and L. Moret-Bailly, Champs algébriques, Ergeb. Math. Grenzgeb. (3) 39, Springer, Berlin 2000. Google Scholar

  • [23]

    Q. Liu, Algebraic geometry and arithmetic curves, Oxf. Grad. Texts Math. 6, Oxford University Press, Oxford 2002. Google Scholar

  • [24]

    W. Lütkebohmert, On compactification of schemes, Manuscripta Math. 80 (1993), no. 1, 95–111. CrossrefGoogle Scholar

  • [25]

    A. Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Publ. Math. Inst. Hautes Études Sci. 21 (1964), 1–128. Google Scholar

  • [26]

    M. Raynaud, Faisceaux amples sur les schémas en groupes et les espaces homogènes, Lect. Notes in Math. 119, Springer, Berlin 1970. Google Scholar

  • [27]

    M. Raynaud, Spécialisation du foncteur de Picard, Publ. Math. Inst. Hautes Études Sci. 38 (1970), 27–76. CrossrefGoogle Scholar

  • [28]

    M. Raynaud, Jacobienne des courbes modulaires et opérateurs de Hecke, Courbes modulaires et courbes de Shimura, Astérisque 196–197, Société Mathématique de France, Paris (1991), 9–25. Google Scholar

  • [29]

    M. Raynaud and L. Gruson, Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math. 13 (1971), 1–89. CrossrefGoogle Scholar

  • [30]

    J. H. Silverman, Heights and the specialization map for families of abelian varieties, J. reine angew. Math. 342 (1983), 197–211. Google Scholar

  • [31]

    Stacks project, 2013, http://stacks.math.columbia.edu.

  • [32]

    J. Tate, Variation of the canonical height of a point depending on a parameter, Amer. J. Math. 105 (1983), no. 1, 287–294. CrossrefGoogle Scholar

About the article

Received: 2014-12-31

Revised: 2016-02-10

Published Online: 2016-06-16

Published in Print: 2019-02-01


Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2019, Issue 747, Pages 109–145, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2016-0014.

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