Example
Let \(\pt = V(x,y) \subset \mathbb A^2_{x,y} = Y\). Set \(A = k[x,y]\) and \(I = (x,y)\subset A\). We construct the blowup \(\widetilde Y = \operatorname{Bl}_{\pt}Y\).
By definition, the blowup is \[\widetilde Y = \Proj(\mathcal B)\] where \(\mathcal B\) is the Rees algebra. Since everything in sight is affine, write \(B\) for \(\Gamma(Y, \mathcal B)\). It is \[B = \bigoplus_{n\geq 0} I^n = A\oplus I\oplus I^2 \oplus ...\] For \(f\in I\), write \(f^1\) when \(f\) is considered an element of \(I^1\) and \(f^0\) when \(f\) is considered an element of \(I^0 = A\). This is subtle: multiplying \(x\) by \(y\) in \(A\) is unambiguous, but in \(B\) it could mean any of the following:
- \(x^0\cdot y^0\), an element of \(I^0\) in degree \(0\),
- \(x^0\cdot y^1,\) an element of \(I^1\) in degree \(1\),
- \(x^1\cdot y^1\), a product of two degree \(1\) elements landing in \(I^2\).
The structure of this ring is made clearer by considering the surjective \(A\)-algebra morphism \[\varphi:A[X,Y] \to B, X\mapsto x^1, Y\mapsto y^1.\] Acting on \(y^1\) by \(x^0\) is the same as acting on \(x^1\) by \(y^0\), so \(\ker\varphi\) is generated by \(xY - yX\). Thus, taking Proj, we see \[\Proj(B) = \Proj(A[X,Y]/(xY - yX)) \hookrightarrow \mathbb P^1_{A} = \mathbb P^1_k\times_{\Spec k}Y.\] The exceptional divisor is \[E = \Proj\left(\bigoplus_{n\geq 0}I^n/I^{n+1}\right) = \Proj(k[x,y]) = \mathbb P^1_k \times_{\Spec k}\{\pt\} \hookrightarrow \mathbb P^1_{k}\times_{\Spec k}Y.\]