New slope inequalities for families of complete intersections

We prove $f$-positivity of $\mathcal{O}_X(1)$ for arbitrary dimension fibrations over curves $f\colon X\to B$ whose general fibre is a complete intersection. In the special case where the family is a global complete intersection, we prove numerical sufficient and necessary conditions for $f$-positivity of powers of $\mathcal{O}_X(1)$ and for the relative canonical sheaf. From these results we also derive a Chow instability condition for the fibres of relative complete intersections in the projective bundle of a $\mu-$unstable bundle.