# Push-forward of nef divisors via finite morphisms

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 ?

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.
• I have a question. Why does the formula $f_{*}D\cdot C = D\cdot f^{*}C$ hold? If $X$ and $Y$ are surfaces it is the projection formula but why does it hold in general? Thank you very much. – user125056 Nov 21 '20 at 18:25
• Yes but the version of the projection formula I know is this one: $f^{*}D\cdot C = D\cdot f^{*}C$. The divisor is pulled back and the curve is pushed forward. – user125056 Nov 26 '20 at 19:41