On the slope of fibred threefolds

Abstract

We study from a geographical point of wiew fibrations of threefolds over smooth curves $f: T \longrightarrow B$ such that the general fibre is of general type. We prove the non-negativity of certain relative invariants under general hypotheses and give lower bounds for $K^3_{T/B}$ depending on other relative invariants. We also study the influence of the relative irregularity $q(T)-g(B)$ on these bounds. A more detailed study of the lowest cases of the bounds is given.