A sharp bound for the slope of double cover fibrations

Abstract. Let f: X → B be a fibration of genus g whose general fiber is a double cover of a smooth curve of genus γ. We show that 4(g−1)/(g−γ) is a sharp lower bound for the slope of f when g> 4γ +1, proving a conjecture of Barja. Moreover, we give a characterization of the fibered surfaces that reach the bound. In the case g = 4γ + 1 we obtain the same sharp bound under the additional assumption that the involutions on the general fibers glue to a global involution on X. Introduction and preliminaries A fibered surface, or simply a fibration, is a proper surjective morphism with connected fibers f from a smooth surface X to a smooth complete curve B. Call F the general fiber of f. A fibration is said to be relatively minimal if the fibers 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 fibers are