Let $f:X\to S$ be a "nice" morphism of "nice" schemes. Let $L$ be an ample line bundle on $X$.

When is $\det f_\ast L$ also ample?

A "nice" morphism could be anything from "finite type separated" to "flat projective" or "birational proper surjective".

A "nice" scheme could be "integral non-singular", or "integral normal", or "with easy singularities".