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".
Best Answer
Consider a morphism $f:\mathbb{P}^1 \to \mathbb{P}^1$ of degree $e> 2$. Using Grothendieck's splitting lemma, it is straightforward to compute that $f_*\mathcal{O}(1)$ is isomorphic to $\mathcal{O}^{\oplus 2} \oplus \mathcal{O}(-1)^{\oplus (e-2)}$. Thus the determinant is $\mathcal{O}(-(e-2))$, which is anti-ample. Thus, I don't see any reason at all that $\text{det}(f_*L)$ should be ample in general.