Let $f:X\rightarrow Y$ be a finite morphism between smooth projective varieties, and let $D$ be an effective nef but not ample divisor on $X$. 


Consider the divisor $f\_{\*}D$ on $Y$. Is $f\_{\*}D$ nef but not ample as well ? 



Push-forward of nef divisors via finite morphisms
I suppose you want $f$ to be surjective, otherwise $f\_\*D$ is not defined. Then $f\_\*D$ is nef: for any curve $C\subset Y$, $\ (f\_\*D\cdot C)=(D\cdot f^\*C)\geq 0$. But it might very well be ample. Consider a smooth quadric $Q\subset \mathbb{P}^3$, and let $f:Q\rightarrow \mathbb{P}^2$ be the projection from a point outside $Q$. Let $D$ be a line contained in $Q$. Then $D$ is nef, not ample, but $f\_\*D$ is a line in $\mathbb{P}^2$, hence ample.


