A sharp bound for the slope of double cover fibrations

Let f : X → B be a fibred surface of genus g whose general fibre is a double cover of a smooth curve of genus γ. We show that, for g > 4γ+1, 4 g−1 g−γ is a sharp lower bound for the slope of f , proving a conjecture of Barja. Moreover, we give a characterisation of the fibred surfaces that reach the bound. In the case g = 4γ+1 we obtain the same sharp bound under the assumption that the involutions on the general fibres glue to a global involution on X . Introduction and preliminaries A fibred surface, or simply a fibration, is a proper surjective morphism with connected fibres f from a smooth surface X to a smooth complete curve B. Call F the general fibre of f . A fibration is said to be relatively minimal if the fibres contain no (-1)-curves. The genus g of F is called the genus of the fibration. We say that f is smooth if all the fibres are smooth, isotrivial if all the smooth fibres are mutually isomorphic, and locally trivial if it is smooth and isotrivial. A hyperelliptic (respectively bielliptic) fibration is a fibred surface whose general fibre is a hyperelliptic (respectively bielliptic) curve. A fibration is said to be semistable if all its fibres are reduced nodal curves which are moduli semistable (any rational smooth component meets the rest of the curve in at least 2 points). Relative invariants As usual, the relative canonical sheaf of a fibration f : X → B is the line bundle ωf = ωX ⊗ (f ∗ωB) , where ωV is the canonical bundle of V . Remark 0.1. A relatively minimal fibration is a fibration such that ωf is f -nef. A semistable fibration is a relatively minimal fibration whose fibres are nodal and reduced. ∗Work partially supported by: 1) PRIN 2003: Spazi di moduli e teoria di Lie; 2) GNSAGA; 3) FAR 2002 (Pavia): Varietà algebriche, calcolo algebrico, grafi orientati e topologici.