Normal bundle to an exceptional sphere in a blowup along a smooth subvariety Let $M$ be a smooth algebraic variety of dimension $m$ over $\mathbb{C}$, $S \subset M$ a smooth embedded subvariety of codimension two. Let $M_S$ denote the blowup along $S$ and $E$ denote the exceptional divisor. Let $\pi_E: E \to S$ denote the canonical morphism and $E_s=\pi_E^{-1}(s)$ over some point $s \in S$. Is it true that the normal bundle $N_{E_s/M_S}$ to $E_s$ is always isomorphic to $$ O_{\mathbb{P}^1}(-1)\oplus O_{\mathbb{P}^1}^{m-2}$$ in the algebraic category? I can see this in the topological category. 
Edit: If I'm correct, I believe one way to see this would be to use the fact that holomorphically locally around s, we can find a polydisc chart where $S$ is cut out by $z_1=z_2=0$. Since taking blowups is compatible with these charts the blow up is locally the blowup at a point times a polydisc of dimension $m-2$. Still, I would like to see a way to do this that is canonical and doesn't invoke complex analysis.  
 A: This is true. Some local maneuvering seems unavoidable, and there are ways to say essentially what you did under the guise of algebraic geometry (etale-locally, etc.). Here's an attempt to do it as you request.
We'll use the standard exact sequence for normal bundles: $$0\to N_{E_s/E}\to N_{E_s/M_S}\to N_{E/M_S}\otimes \mathcal{O}_{E_s}\to 0$$
From the general theory of blowups, we know that the third normal bundle is $\mathcal{O}_E(-1)$: Let $\mathcal{I}$ be the ideal sheaf of $S\subset M$. Then the blow up $M_S$ is defined as $\textbf{Proj}(\mathcal{I})$ with a natural invertible sheaf $\mathcal{I}'\cong \pi^{-1}(\mathcal{I})\cdot\mathcal{O}_{M_S}\cong \mathcal{O}_{M_S}(1)$ where $\pi:M_S\to M$ is the obvious map. Clearly $\mathcal{I}'/\mathcal{I}'^2\cong \mathcal{O}_E(1)$, so $N_{E/M_S}\cong \mathcal{O}_E(-1)$ as this is the dual of $\mathcal{I}'/\mathcal{I}'^2$.
Our exact sequence has become $$0\to N_{E_s/E}\to N_{E_s/M_S}\to \mathcal{O}_{E_s}(-1)\to 0$$ and so it remains to determine $N_{E_s/E}$. We'll use the following exact sequence: $$0\to T_{E_s}\to T_E\otimes \mathcal{O}_{E_s} \to N_{E_s/E}\to 0$$
Now, via the isomorphism $E_s \cong \mathbb{P}^1$, we have that $T_{E_s}\cong T_{\mathbb{P}^1}$. We also have that since $E$ is locally isomorphic to $S\times \mathbb{P}^1$ (cf Hartshorne II.8.24), we have that $T_E\otimes \mathcal{O}_{E_s}\cong (T_S\oplus T_{\mathbb{P^1}})\otimes\mathcal{O}_{E_s}\cong \mathcal{O}_{E_s}^{m-2}\oplus T_{\mathbb{P}^1}$ and clearly $N_{E_s/E}\cong \mathcal{O}_{E_s}^{m-2}$. So we have that $N_{E_s/M_S}$ is an extension of $\mathcal{O}_{\mathbb{P}^1}(-1)$ by $\mathcal{O}_{\mathbb{P}^1}^{m-2}$ (the first isomorphism only holds locally, but since we're restricting to $E_s$, this is okay). But by Grothendieck's theorem, all such extensions are split, and we have the desired result.
