In Kollár and Mori's Birational Geometry of Algebraic Varieties, lemma2.60 claim some multiple of a big divisor induced birational morphism onto its image in a projective space. But the proof only show it can be written as a sum of an ample divisor and an effective divisor, and say the result is obvious for the latter. I try to find the details for the latter but have no clues. Thank you for any answer or comments. Furthermore the lemma assume the scheme is a projective variety, does it must be integral? Can the result be done for proper varieties?
The second question: The authors also claim a divisor is big iff its birational pullback is big. I know when the varieties are integral normal and proper, this can be done by Zariski' main theorem and projective formula. Are these conditions necessary?