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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 ?

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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.

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  • $\begingroup$ 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. $\endgroup$ – user125056 Nov 21 '20 at 18:25
  • $\begingroup$ @user125056: The projection formula holds in any dimension. $\endgroup$ – abx Nov 22 '20 at 5:42
  • $\begingroup$ 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. $\endgroup$ – user125056 Nov 26 '20 at 19:41
  • $\begingroup$ Have a look at a book on Chow rings, e.g. Fulton. $\endgroup$ – abx Nov 26 '20 at 20:02
  • $\begingroup$ Do you mean Fulton - Intersection Theory? $\endgroup$ – user125056 Nov 28 '20 at 18:24

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