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blowup
Some definitions:
Examples
Transforms
Facts
A blow-up is a birational transformation that replaces a closed subscheme with an effective Cartier divisor. Precisely, given a Noetherian scheme \(X\) and a closed subscheme \(Z \subset X\), the blow-up of \(X\) along \(Z\) is a proper morphism \(\pi: \widetilde{X} \rightarrow X\) such that
- \(\pi^{-1}(Z) \hookrightarrow \widetilde{X}\) is an effective Cartier divisor, called the exceptional divisor and
- \(\pi\) is universal with respect to (1).
Concretely, it is constructed as the relative Proj of the Rees algebra of \(O_{X}\) with respect to the ideal sheaf determining Z.
For varieties: to blow up along functions \(f_1,\cdots, f_n\), define a morphism \begin{align*} F:V(f_i)^c &\to {\mathbf{P}}^{n-1} \\ x &\mapsto {\left[ {f_1(x),\cdots, f_n(x)} \right]} .\end{align*} Then \(\tilde X = \operatorname{Bl}(X; f_1,\cdots, f_n) \coloneqq{ \operatorname{cl}} _{X\times {\mathbf{P}}^{n-1}}( \Gamma_F)\) is the closure of the graph. Note that there is a morphism \(\pi:\tilde X\to X\) given by projection onto the first component. If \(X\) is irreducible, then \(\tilde X \overset{\sim}{\dashrightarrow}X\) with a common open dense subset.
Note that \(\Gamma_F \underset{{ \left.{{\pi}} \right|_{{\Gamma_F}} }}{ { \, \xrightarrow{\sim}\, }} V(f_i)^c\), so \(V(f_i)^c\) can be identified with a dense open subset of \(\tilde X\). The complement \(\tilde X\setminus\Gamma_F = \pi^{-1}(V(f_i))\) is the exceptional set.
For \(Y\leq X\) a closed subvariety, \(\tilde Y\subseteq Y\times {\mathbf{P}}^{n-1} \leq X\times {\mathbf{P}}^{n-1}\) is a closed subvariety of the blowup \(\tilde X\), and we refer to \(\tilde Y\) as the strict transform of \(Y\) in \(\tilde X\).
# Examples
\(\operatorname{Bl}({\mathbf{A}}^n; x_1,\cdots, x_n) = \left\{{(x,y) \in {\mathbf{A}}^n\times {\mathbf{P}}^{n-1} {~\mathrel{\Big\vert}~}y_i x_j = y_j x_i }\right\}\) replaces \(\mathbf{0}\in {\mathbf{A}}^n\) with a copy of \({\mathbf{P}}^{n-1}\).
