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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