I meet some problem when reading Lazarsfeld's Positivity in Algebraic Geometry, Cor 1.2.28.
Corollary 1.2.28 Let $f:Y\longrightarrow X$ be a finite surjective mapping of projective schemes, and let $L$ a line bundle on $X$, if $f^*L$ is ample on $Y$, then $L$ is ample on $X$.
The proof states as below.
Proof. Let $V\subseteq X$ be an irreducible variety. Since $f$ is surjective and finite, there exists irreducible variety $W\subseteq Y$ mapping finite onto $V$: we start with $f^{-1}(V)$, one constructs $W$ by taking irreducible components and $\color{blue}{\textrm{cutting down by general hyperplanes}}.$ Then by the projection formula:
$$\int_W {c_1(f^*L)}^{\text{dim}W}=\text{deg}(W\longrightarrow V)\cdot\,\int_V{c_1(L)}^{\text{dim}(V)}$$
By applying the Nakai-Moishezon ampleness criterion, we are done.
What confuses me is the $\color{blue}{\textrm{'blue'}}$ sentence, why we need the general hyperplane cut-down procedure?
Any help would be appreciated. Thanks a lot.