We prove that the genus g, the relative irregularity and the Clifford index of a non-isotrivial fibration f satisfy the inequality . This gives in particular a proof of Xiao’s conjecture for fibrations whose general fibres have maximal Clifford index.
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)
Funding statement: 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).
We would like to thank Prof. Gian Pietro Pirola for the many stimulating discussions around this topic, especially for presenting to us several counterexamples to the original conjecture of Xiao and for suggesting the new proof of the result of Ginensky (Theorem 2.4). We are also grateful to the referees for their suggestions, which helped us to improve the exposition of our results.
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