Is there exist an irreducible subvariety dominating the given one?

Question

Let $f$ be a surjective morphism from a projective variety $X$ to a projective variety $Y$ and L a line bundle on $Y$. I want to prove that if the pull back of L is nef, then L is nef. I want use the projection formula: $f^*L.C' = L. f_*C $. So let $C$ be an irreducible curve on Y and I wish to find an irreducible curve $C'$ on X such that the push forward of $C'$ is some multiple of $C$. Question: does such $C'$ always exist ? Or more generally, assuming that $V$ is an irreducible subvariety of $Y$, is there exist an irreducible subvariety $W$ of $X$ mapping onto V and $dim W = dim V $ ?

Answer 1

To see the claim: it is enough to show that if we have a surjective morphism of a projective varieties $f:Y\to C$ where $C$ is an irreducible curve, then there is an irreducible curve $D\subset Y$ such that $f(D)=C$.

(By Chow's lemma, replacing $Y$ birationally, we can assume that $f$ is projective.) So $Y\subset \mathbb{P}^n_k$. Choose a linear subspace $H$ such that $Y\cap H$ is of dimension 1. Let $\eta$ be the generic point of $C$.

Now $f^{-1}(\eta)\cap (Y\cap H)\neq \emptyset$. So $$f|_{Y\cap H}:Y\cap H \to Y\to C$$ is dominant, hence sujective since $f$ is proper.

So now you have a curve $Y\cap H$ in $Y$ such that $f(Y\cap H)=C.$ So $f_*(Y\cap H)=\deg(f|_{Y\cap H})C$.

Now you are done.

After doing the reductions I came across Kleiman's article [1] (see page 303), where the argument is discussed.

Kleiman[1]: https://www.jstor.org/stable/pdf/1970447.pdf