Embedding Theorem for big line bundles 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 $f$? 
 A: 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 $X \setminus E$.
A: Colin, 
Francesco has already mentioned the "embedding theorem for big line bundles" that you were probably looking for, but I would like to add that this is pretty much the definition of big.
An alternative point one could make is that according to Kodaira's Lemma "big=ample+effective", so a high power of a big divisor is very ample on the complement of an effective divisor.
As for your last question, if $f:Y\to X$ is any proper morphism and $\mathscr L$ is an ample line bundle on $X$, then if $g$ denotes the morphism(!) defined by the global sections of very high powers of $f^*\mathscr L$, then it follows that $f$ factors through $g$. In particular, then $g$ is an isomorphism to its image (hence an embedding) wherever $f$ is (if $f$ is not birational, then this "wherever" is the empty set).
