# Algebraic definition of blow-ups

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 $Proj \mathcal{B}$.

I believe the blow-up is actually $Proj\;\mathcal{B}$. – Jeff Tolliver Nov 18 '12 at 3:06
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 $\mathbb P(N_{C/X})$.