Behavior of divisors under push forward and pull back

Question

Consider a birational morphism between smooth projective varieties $f:X\to Y$. I would like to understand the behavior of push-pull/pull-push of effective divisors under $f$. I know that if $D$ is an effective divisor on $Y$, then $f_*f^*D-D$ is always effective. However, $f^*f_*E-E$ might not be effective when $E$ is an effective divisor on $X$ due to the existence of contracted-by-$f$ components in $E$. My question is:

If $|H|$ is a base-point free linear system on $X$, could we find an effective divisor $E$ in $|H|$ whose components are not contracted by $f$?

If this is affirmative, then $f^*f_*H-H$ is effective for any very ample divisor $H$ on $X$. I assume this is well-known, but I could not prove it or find any reference.

Thank you!

Answer 1

Of course, just choose a point on each (divisorial) component of the exceptional locus of $f$, then for each of these points the divisors in $|H|$ passing through form a hyperplane in $|H|$, and any divisor in the complement of (finitely many) these hyperplanes does not contain a component contracted by $f$.