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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".

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One of the common situations which arise is the following. Assume $f$ is flat and projective and let $L=f^*A\otimes B$ where $A$ is ample on $S$. Further assume that $f_*B$ is globally generated and higher direct images vanish. Then $f_*B$ is a vector bundle and $f_*L=A\otimes f_*B$ is ample and thus, so is its determinant. So, typically, if $A$ is ample on $S$, $B$ is $f-ample on $X$, then for all large $n$, $f^*A\otimes B^n$ will have this property. –  Mohan Apr 12 '13 at 16:55
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1 Answer 1

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.

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