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.

flag	asked Aug 12 at 4:35 MZWang 403●4

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). – Yusuf Mustopa Aug 12 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). – Jason Starr Aug 12 at 14:46

Thanks to Yusuf and Jason. – MZWang Aug 25 at 9:37