Kodaira embedding theorem says that a positive line bundle is ample, i.e. high tensor powers are holomorphically embeddable into complex projective space of high dimension. 


However, ampleness is not stable under blow-ups. Usually a replacement is to consider big line bundles, which is stable under blow-ups. 


Is there an embedding theorem for big line bundles? One cannot hope that an embedding of high tensor powers of a big line bundle to be everywhere injective, but can we have injectivity almost everywhere?


Here's the precise question. Let $X$ be a complex compact manifold. Let $L$ be an ample line bundle over $X$. Let $f:Y\to X$ be a blow up, or series of blow ups. Is there some condition for the pullback $f^\*L$ to be an embedding outside the exceptional locus of $f$? 



Embedding Theorem for big line bundles
If $X$ is normal, then the Iitaka fibration theorem implies that $L$ is big if and only if the rational map


$\phi\_m \colon X \dashrightarrow \mathbb{P}H^0(X, L^{\otimes m})$


is birational onto its image for some $m >0$, see [Lazarsfeld, Positivity in Algebraic Geometry I, p. 139].


I guess this is the "embedding theorem for big line bundles" you are looking for.


Regarding your last question, since $L$ is ample by assumption and $f$ is an isomorphism outside its exceptional locus $E$, some power of $f^\*L$ will surely give an embedding of $X \setminus E$.


