If I have a quasiprojective variety $X$, and a subscheme $Z$, then the blowup $$f:Y = Bl(X,Z)\rightarrow X$$ is projective over $X$, since it is constructed by a relative Proj construction. Can I find a relatively ample bundle on $Y$ that is trivial on $$f^{-1}(X\backslash Z)?$$
At first I thought the construction of $O(1)$ in the Proj construction guarantees this property, but only under assumption of generation in degree 1. However, if I have a projective small contraction $Y \rightarrow X$, then it is a blowup map, but then certainly an ample bundle can't be trivial on the birational locus, since it would be trivial on $Y$.
So under what conditions can I find such a relatively ample bundle?
Edit: Karl's answer explains away my confusion: I have to blow up more than the image of the exceptional locus in case of a small contraction. It's still reasonable to ask when it's trivial on the birational locus, in which case Sandor's answer applies.
