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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


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Volume 2018, Issue 739

Issues

Xiao’s conjecture for general fibred surfaces

Miguel Ángel Barja
  • Departament de Matemàtiques, Universitat Politècnica de Catalunya (UPC-BarcelonaTECH), Av. Diagonal 647, 08028 Barcelona, Spain
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/ Víctor González-Alonso / Juan Carlos Naranjo
  • Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
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Published Online: 2016-01-14 | DOI: https://doi.org/10.1515/crelle-2015-0080

Abstract

We prove that the genus g, the relative irregularity qf and the Clifford index cf of a non-isotrivial fibration f satisfy the inequality qfg-cf. This gives in particular a proof of Xiao’s conjecture for fibrations whose general fibres have maximal Clifford index.

References

  • [1]

    A. Albano and G. P. Pirola, Dihedral monodromy and Xiao fibrations, Ann. Mat. Pura App. (2015), 10.1007/s10231-015-0514-y. Google Scholar

  • [2]

    M. Á. Barja and L. Stoppino, Linear stability of projected canonical curves with applications to the slope of fibred surfaces, J. Math. Soc. Japan 60 (2008), no. 1, 171–192. CrossrefWeb of ScienceGoogle Scholar

  • [3]

    J.-X. Cai, Irregularity of certain algebraic fiber spaces, Manuscripta Math. 95 (1998), no. 3, 273–287. CrossrefGoogle Scholar

  • [4]

    A. Collino and G. P. Pirola, The Griffiths infinitesimal invariant for a curve in its Jacobian, Duke Math. J. 78 (1995), no. 1, 59–88. Google Scholar

  • [5]

    O. Debarre, Inégalités numériques pour les surfaces de type général, Bull. Soc. Math. France 110 (1982), no. 3, 319–346, With an appendix by A. Beauville. Google Scholar

  • [6]

    T. Fujita, On Kähler fiber spaces over curves, J. Math. Soc. Japan 30 (1978), no. 4, 779–794. CrossrefGoogle Scholar

  • [7]

    T. Fujita, The sheaf of relative canonical forms of a Kähler fiber space over a curve, Proc. Japan Acad. Ser. A Math. Sci. 54 (1978), no. 7, 183–184. CrossrefGoogle Scholar

  • [8]

    A. Ginensky, A generalization of the Clifford index and determinantal equations for curves and their secant varieties, preprint (2010), http://arxiv.org/abs/1002.2023.

  • [9]

    V. González-Alonso, On deformations of curves supported on rigid divisors, Ann. Mat. Pura App. (2014), 10.1007/s10231-014-0455-x. Google Scholar

  • [10]

    K. Konno, Clifford index and the slope of fibered surfaces, J. Algebraic Geom. 8 (1999), no. 2, 207–220. Google Scholar

  • [11]

    X. Lu and K. Zuo, On the slope of hyperelliptic fibrations with positive relative irregularity, Trans. Amer. Math. Soc., to appear. Google Scholar

  • [12]

    G. P. Pirola, Curves on generic Kummer varieties, Duke Math. J. 59 (1989), no. 3, 701–708. CrossrefGoogle Scholar

  • [13]

    G. P. Pirola, On a conjecture of Xiao, J. reine angew. Math. 431 (1992), 75–89. Google Scholar

  • [14]

    G. P. Pirola and F. Zucconi, Variations of the Albanese morphisms, J. Algebraic Geom. 12 (2003), no. 3, 535–572. CrossrefGoogle Scholar

  • [15]

    F. Serrano, Isotrivial fibred surfaces, Ann. Mat. Pura Appl. (4) 171 (1996), 63–81. CrossrefGoogle Scholar

  • [16]

    G. Xiao, Fibered algebraic surfaces with low slope, Math. Ann. 276 (1987), no. 3, 449–466. CrossrefGoogle Scholar

  • [17]

    G. Xiao, Irregularity of surfaces with a linear pencil, Duke Math. J. 55 (1987), no. 3, 597–602. CrossrefGoogle Scholar

  • [18]

    G. Xiao, Problem list, Birational geometry of algebraic varieties: Open problems (XXIII International Symposium), Division of Mathematics, Taniguchi Foundation, Katata (1988), 36–40. Google Scholar

About the article

Received: 2015-03-18

Revised: 2015-07-07

Published Online: 2016-01-14

Published in Print: 2018-06-01


Funding Source: Ministerio de Economía y Competitividad

Award identifier / Grant number: MTM2012-38122-C03-01/FEDER

Award identifier / Grant number: MTM2012-38122-C03-02

Funding Source: Generalitat de Catalunya

Award identifier / Grant number: 2009-SGR-1284

Funding Source: H2020 European Research Council

Award identifier / Grant number: StG 279723 ‘Arithmetic of algebraic surfaces’ (SURFARI)

During the development of this work, the first and second authors were supported by the Spanish ‘Ministerio de Economía y Competitividad’ (project MTM2012-38122-C03-01/FEDER) and the ‘Generalitat de Catalunya’ (project 2009-SGR-1284). The third author was supported by the Spanish ‘Ministerio de Economía y Competitividad’ (project MTM2012-38122-C03-02). The second author was also supported by the Spanish ‘Ministerio de Educación’ (grant FPU-AP2008-01849) and by the ‘European Research Council’ (StG 279723 ‘Arithmetic of algebraic surfaces’, SURFARI).


Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2018, Issue 739, Pages 297–308, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2015-0080.

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