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 $f$?