Show Summary Details
More options …

# 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

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2017: 1.49

Online
ISSN
1435-5345
See all formats and pricing
More options …
Volume 2018, Issue 739

# 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
• Email
• Other articles by this author:
/ Víctor González-Alonso
/ Juan Carlos Naranjo
• Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
• Email
• Other articles by this author:
Published Online: 2016-01-14 | DOI: https://doi.org/10.1515/crelle-2015-0080

## Abstract

We prove that the genus g, the relative irregularity ${q}_{f}$ and the Clifford index ${c}_{f}$ of a non-isotrivial fibration f satisfy the inequality ${q}_{f}\le g-{c}_{f}$. 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.

• [3]

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

• [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.

• [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.

• [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.

• [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.

• [15]

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

• [16]

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

• [17]

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

• [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

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,

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.