(Blowup)
Let \(X\) be a Noetherian scheme and \(Z \subseteq X\) a closed subscheme with ideal sheaf \(\mathcal{I}\). The Rees algebra of \(\mathcal{I}\) is the graded quasicoherent \(\OO_X\)-algebra \(\bigoplus_{d \geq 0} \mathcal{I}^d\), with \(\mathcal{I}^0 = \OO_X\); it is generated in degree \(1\). The blowup of \(X\) along \(Z\) is the \(X\)-scheme \[\pi \colon \Bl_Z X \coloneqq \operatorname{\mathbf{Proj}}_X \bigoplus_{d \geq 0} \mathcal{I}^d \to X,\] using the relative Proj of D-SCHRELSPECPROJ. The exceptional divisor is the closed subscheme \(E = \pi^{-1}(Z)\) of \(\Bl_Z X\), with ideal sheaf \(\mathcal{I} \cdot \OO_{\Bl_Z X}\).
(Properties and universal property)
Let \(\pi \colon \Bl_Z X \to X\) be the blowup of a Noetherian scheme along a closed subscheme \(Z\).
\(\pi\) is projective, \(E\) is an effective Cartier divisor, and \(\pi\) restricts to an isomorphism \(\Bl_Z X \setminus E \to X \setminus Z\).
If \(X\) is integral and \(Z \neq X\), then \(\Bl_Z X\) is integral and \(\pi\) is birational.
For every morphism \(f \colon Y \to X\) such that \(f^{-1}(Z)\), the closed subscheme with ideal sheaf \(f^{-1}\mathcal{I} \cdot \OO_Y\), is an effective Cartier divisor on \(Y\), there is a unique morphism \(g \colon Y \to \Bl_Z X\) with \(f = \pi \circ g\).
Let \(X = \AA^2_k = \Spec k[x,y]\) and \(Z\) the origin, \(\mathcal{I} = (x, y)\). Since \(x, y\) is a regular sequence, the Rees algebra is \(k[x,y][s,t]/(xt - ys)\), with \(s \mapsto x\) and \(t \mapsto y\) in degree \(1\). So \(\Bl_0 \AA^2 = V(xt - ys) \subseteq \AA^2_k \times \PP^1_k\). On the chart \(s \neq 0\) with coordinate \(u = t/s\), the equation becomes \(y = xu\), the chart is \(\Spec k[x, u] \cong \AA^2\), and \(E\) is the line \(V(x)\) there; the two charts glue \(E\) to \(\PP^1_k\). This recovers the blowup of a variety at a point in D-VARBLOW.
(Blowdown)
A proper birational morphism \(\rho \colon Y \to X\) of integral Noetherian schemes is the blowdown of a closed subscheme \(F \subseteq Y\) if there are a closed subscheme \(Z \subseteq X\) and an isomorphism \(Y \cong \Bl_Z X\) over \(X\) carrying \(F\) to the exceptional divisor. Then \(X\) is obtained from \(Y\) by blowing down \(F\), and \(\rho\) contracts \(F\) to \(Z\).
If \(Y\) is a smooth projective surface over an algebraically closed field and \(F \subseteq Y\) is a curve with \(F \cong \PP^1\) and \(F^2 = -1\), then by Castelnuovo’s criterion T-SRFCAST there is a smooth projective surface \(X\) and a point \(p \in X\) with \(Y \cong \Bl_p X\) and \(F\) the exceptional curve, so \(F\) can be blown down.
Dependencies
D-SCHRELSPECPROJRelative Spec and relative Projuses