Splitting of the normal bundle to $\mathbb{P}^1$ after blowup

Let $Y$ be a smooth projective subvariety of $X$ also smooth projective. Let $C\simeq\mathbb{P}^1$ be a smooth projective rational curve in $X$ meeting $Y$ in a single, reduced point. Assume we know the splitting type of the normal, $N_CX \simeq \mathcal{O}_C(a_1)\oplus\mathcal{O}_C(a_2)\oplus\ldots$ and of the tangent bundle $\left.TX\right|_C\simeq\mathcal{O}_C(b_1)\oplus\mathcal{O}_C(b_2)\oplus\ldots$ Is it possible to retrieve the corresponding decompositions for the blowup $X'$ of $X$ along $Y$?

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I don't have time to write down a detailed answer right now, but I think that you can figure out the degree of these bundles, that is the sum of the $a_i$'s and the sum of the $b_i$'s, but not the individual terms. For that you would have to know something more about the intersection of $C$ and $Y$. Something that sort of amounts to the "angle" at which they intersect, or more precisely which of the terms of the decomposition point in the direction of $Y$. I'll try to come back in a few days and flash this out unless someone comes up with another answer. –  Sándor Kovács Dec 12 '13 at 6:06