On the slope of bielliptic fibrations (Q2718948)

One considers a relatively minimal bielliptic fibration \(\pi : S\to B\) of genus \(g\geq 6\) (\(S\) surface, \(B\) curve). Let \(\omega:=\omega _{S/B}\) be the relative canonical bundle and let \(\Delta (\pi)\) be the degree of \(\pi _\ast (\omega)\). One shows that the slope \(\lambda (\pi) := \omega ^2 /\Delta (\pi)\) of \(\pi\) is at least \(4\) and it is \(4\) iff \(S\) is the minimal desingularization of a double cover of a smooth elliptic surface \(V\), such that the branch locus of the double cover has only negligible singularities and the fibres of \(\tau : V\to B\) are smooth and isomorphic.