Definition
Let \(\iota:Z\hookrightarrow X\) be a closed immersion with ideal sheaf \(\mathcal I\). Form the Rees algebra of the pair \((\mathcal O_X, \mathcal I)\) by \[\mathcal B = \bigoplus_{n\geq 0}\mathcal I^n\] by setting \(\mathcal I^0 = \mathcal O_X\). This is a graded \(\mathcal O_X\)-algebra.
The blowup of \(X\) along \(Z\) is denoted \(\Bl_Z(X)\) and defined \[\Bl_Z(X) = \Proj(\mathcal B).\] The exceptional divisor \(E\) is the fiber of \(Z\) along the map \(\pi:\Bl_Z(X) \to X\). Since \(\Proj\) is compatible with base change, we have \[E = \Proj(\mathcal B)\times_X Z \cong \Proj(\pi^*\mathcal B) = \Proj(\mathcal B \otimes_{\mathcal O_X}\mathcal O_X/\mathcal I) = \Proj\left(\bigoplus_{n\geq 0} \mathcal I^n/\mathcal I^{n+1}\right).\]