Suppose $f: Y\rightarrow X$ is a birational morphism and $H$ is an ample divisor on $Y$. Suppose $X$ is $\mathbb{Q}$-factorial (so $f_{*}H$ is $\mathbb{Q}$-cartier) and we take $f^{*}f_{*}H$.
Why is the difference $f^{*}(f_{*}H)-H$ effective?
(Some easy observations:
1) $f^{*}(f_{*}H)-H$ intersects negatively with any curve that is contracted by $f$, so is anti-ample on every fiber if the map is projective.
2) Intuitively, we'd be worried about $H$ containing an exceptional divisor with high multiplicity, but then, since it must intersect positively with any contracted curve, it would also have components intersecting with that divisor properly, so the statement still seems plausible.)