TITLE: Algebraic definition of blow-ups
QUESTION [5 upvotes]: Let $X$ be a scheme. Choose $C\subset X$ be a subscheme of $X$ and let $\mathcal{I}\subset \mathcal{O}$ be the corresponding ideal sheaf. Then $\mathcal{B}=\oplus_{d\ge0}\mathcal{I}^d$ is a sheaf of $\mathcal{O}$-algebra The blow-up of $X$ along $C$ is defined as 
$$
Y=Proj \mathcal{B} \rightarrow X. 
$$
My question is, how can one understand $Proj \mathcal{B}$ to $see$ geometric description of blow-up? More precisely, when both $X$ and $C$ are smooth complex variety, $Y$ is obtained by replacing $C$ by $\mathbb{P}(N_{C/X})$, but I cann't really see this description from $Proj \mathcal{B}$. 
THank you for your help.

REPLY [2 votes]: The fiber of $Y\to X$ above $C$ is 
$$ Y\times_X C=\mathrm{Proj}(\mathcal B\otimes_{O_X} O_X/\mathcal I).$$ 
We have
$$ \mathcal B\otimes_{O_X} O_X/\mathcal I = \oplus_{d\ge 0} (\mathcal I^d\otimes_{O_X} O_X/\mathcal I)=\oplus_{d\ge 0} (\mathcal I^d/\mathcal I^{d+1}).$$ 
As $C$ is locally complete intersection in $X$, $N_{C/X}:=\mathcal I/\mathcal I^2$ is locally free and 
$$\mathcal I^d/\mathcal I^{d+1} \simeq  \mathcal{Sym}^d_{O_X}(N_{C/X})$$ 
(symetric power). Therefore 
$$ \mathcal B\otimes_{O_X} O_X/\mathcal I\simeq \mathcal{Sym}_{O_X}(N_{C/X})$$ 
(symetric algebra). So the fiber of $Y\to X$ above $C$ is the projective bundle $\mathbb P(N_{C/X})$.