# A question on pull back of a nef and big divisor

In his book Higher-Dimensional Algerbraic Geometry, Debarre claimed that the pull back of a nef and big divisor under a generically finite morphism is still nef and big, but he only state the result and no proof. Can somebody tell me why or show me a reference? Thanks.

• This is a nice exercise. Some hints: Consider nefness and bigness separately. For preservation of bigness, look at the Stein factorization of your morphism, and use the characterizations of bigness given in Volume 1 of Lazarsfeld's book (Section 2.2, if I recall correctly). Aug 12, 2012 at 5:39
• I agree with Yusuf. Please note, the "generically finite morphism" here also must be dominant (when restricted to every irreducible component of the domain). Aug 12, 2012 at 14:46

One can consider the following characterization of nef and big divisors:

Let $$D$$ a divisor on an irreducible projective variety $$X$$. Then $$D$$ is nef and big if and only if there exist an effective divisor $$E$$ and a rational number $$0<\epsilon\ll1$$ such that $$D-\epsilon E$$ is ample.

proof: Let $$D$$ be a nef and big divisor. Since $$D$$ is big there exist an ample divisor $$A$$, an effective divisor $$E$$, and a positive integer $$k$$ such that $$kD\equiv A+E$$. If $$h>k$$ we can write $$hD\equiv(h-k)D+A+E$$. The divisor $$D^{\prime}=(h-k)D+A$$ is a sum of a nef and an ample divisor. Therefore $$D^{\prime}$$ is ample. If $$\displaystyle\epsilon=\frac{1}{h}$$ we get that $$D-\epsilon E\equiv\epsilon D^{\prime}$$ is ample.

You have a generically finite morphism $$\phi:X\rightarrow Y$$, and a nef and big divisor $$D$$ on $$Y$$. Consider the Stein factorization $$\phi=f\circ b$$, where $$b$$ has connected fibers and $$f$$ is finite. In you case, since $$\phi$$ is generically finite, $$b$$ is birational and $$f$$ is finite. Since $$D$$ is nef and big there exist an effective divisor $$E$$ and a rational number $$0<\epsilon\ll1$$ such that $$D-\epsilon E$$ is ample. Since $$f$$ is finite $$f^{*}(D-\epsilon E)=f^{*}D-\epsilon f^{*}E$$ is ample. Now, $$f^{*}E$$ is effective and then $$f^{*}D$$ is nef and big. Finally $$\phi^*D=b^*f^*D$$ is nef and big bacause $$b$$ is birational.

Onother approach:

• If $$\phi:X\rightarrow Y$$ is generically finite of degree $$d$$, then $$Vol(\phi^{*}D) = d\cdot Vol(D).$$ (volume of big line bundles under finite morphisms).

In particular $$D$$ is big $$\Leftrightarrow Vol(D)>0\Leftrightarrow Vol(\phi^{*}D)=d\cdot Vol(D)>0\Leftrightarrow\phi^{*}D$$ is big.

• If $$\phi:X\rightarrow Y$$ is a proper morphism, $$D$$ nef $$\Rightarrow\phi^{*}D$$ nef. If $$C$$ is an integral curve on $$X$$, by the projection formula we have $$\phi^*D\cdot C=D\cdot\phi_*C\geq0$$.