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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.

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    You are right. I'll fixe the typo. Thanks.2012-11-18

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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 $\mathbb P(N_{C/X})$.

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    A great answer! This bridges the two definitions! Thank you very much.2012-11-19