--- date: 2022-02-23 18:45 modification date: Monday 14th March 2022 11:58:24 title: blowup aliases: [blowup, exceptional divisor, exceptional curve, exceptional set, proper transform, strict transform] --- --- - Tags - #AG/basics - Refs: - - Many explicit examples of blowups: - Links: - [[Cremona transformation]] - [self intersection](Unsorted/self%20intersection.md) - [[Castelnuevo's criterion]] - [Cartier divisor](Unsorted/Weil%20divisor.md) --- # blowup - Universal property: - $\tilde X \mapsvia\pi X$ is the blowup of $X$ along $Z\leq X$ a closed subvariety if $\pi\inv(Z)$ is a Cartier divisor and $\pi$ is universal wrt this, i.e. the closest to $X$, so if $Y\to X$ satisfies this then $\exists Y\to \tilde X$. - Note that if $Z$ is Cartier then $\pi = \id_X$. - Via the [Rees algebra](Rees%20algebra): - For $X = \spec(R) \in \Aff\Sch$ and $Z = \spec (R/I)\leq X$ a closed subscheme defined by $I\normal R$, $\Bl_Z(X) \da \Proj(\Rees(I))$, - Via graph closures, at a point $p$: - Take the Zariski closure of a graph $\Gamma \subseteq \AA^n\times \PP^{n-1}$ defined by the following: take a map $$\begin{align*}\gamma: \AA^n\smts{p}&\to \PP(\T_p \AA^n) \\ x &\mapsto g(x)\end{align*}$$ where $g(x)$ is the second lines through $x$ and $p$, where $p$ is regarded as a point in the vector space $\T_p \AA^n$. The blowup map is the restriction $\pi: \AA^n\times \PP^{n-1}\to \AA^n$. - Via graph closures, at a closed subvariety: - For $Z = V(I) \leq X$ a closed subvariety with $I = \gens{g_1,\cdots, g_k}_R$ fg, define a morphism $$\begin{align*}\gamma: X\sm Z &\to \PP^{k-1} \\x &\mapsto \tv{g_1(x) : \cdots : g_k(x)} \end{align*}$$ For $\tilde X$, take the Zariski closure of the graph $\Gamma \subseteq X\times \PP^{k-1}$ and the projection $\pi: X\times \PP^{k-1} \to X$. - Via equations: - For $X$ an affine variety and $Z = V(I) \leq X$ with $I = \gens{g_1,\cdots, g_k}_R$ (where $I\normal R = k[x]$) a [[regular sequence]], letting $\tv{x_1:\cdots: x_{k}}$ be projective coordinates on $\PP^{k-1}$, take the subvariety $\tilde X \subseteq X\times \PP^{k-1}$ defined by the equations $x_ig_j - x_jg_i$ for all $1\leq i,j\leq k$, i.e. the vanishing of all $2\times 2$ minors of $$\begin{bmatrix}x_1 & x_2 & \cdots & x_k \\ g_1 & g_2 & \cdots & g_k \end{bmatrix}$$ - Via affine charts: - For $X = \AA^n$ and $Z$ a coordinate subspace defined by $x_j = 0$ for some collection of $J\subseteq \ts{1,\cdots, n}$, set $U_j \cong \AA^n$ for $j\in J$, and glue $U_a$ to $U_b$ via $$\begin{align*}x_i &\mapsto x_i x_a\inv, \qquad i\neq a,b,\, \\x_a &\mapsto x_b\inv, \\ x_b&\mapsto x_a x_b, \\x_i &\mapsto x_i, \, \qquad i\not\in J\end{align*}$$ Call this $\tilde X$, and define projections $$\begin{align*} \pi_j: \AA^n &\to \AA^n\\ x_i &\mapsto x_i \quad i\not\in J\smts{a}\\x_i &\mapsto x_i x_a, \quad i\in J\smts{a} \end{align*}$$ - Via ring extensions: - Letting $I = \gens{r_1,\cdots, r_k}_R$, take the ring extensions $R\injects R_i \da R\adjoin{{r_1\over r_i}, \cdots, {r_k\over r_i}}$ inside the rings $R_{r_i} = R\adjoin{r_i\inv}$ glued pairwise. - Local blowup for germs: - For $\pi: \tilde X\to X$ a blowup along $Z$ and $x'$ lying above $x\in X$, the local blowup of $X$ along $Z$ at $a$ is the scheme morphism corresponding to the ring morphism $\pi^*: \OO_{X, x} \to \OO_{\tilde X, \tilde x}$ (or sometimes the morphism on their completions). Some definitions: ![](attachments/Pasted%20image%2020220905140853.png) ![](attachments/Pasted%20image%2020220914195953.png) ![](attachments/Pasted%20image%2020220810115657.png) ![](attachments/Pasted%20image%2020220727093815.png) ![](attachments/Pasted%20image%2020220526140204.png) ![](attachments/Pasted%20image%2020220526140258.png) ![](attachments/Pasted%20image%2020220526140337.png) # Examples ![](attachments/Pasted%20image%2020221110222840.png) # Transforms ![](attachments/Pasted%20image%2020220810145413.png) # Facts ![](attachments/Pasted%20image%2020220622002453.png) ![](attachments/Pasted%20image%2020220526140506.png) ![](attachments/Pasted%20image%2020220418093243.png) ![](attachments/Pasted%20image%2020220410221414.png) ![](attachments/Pasted%20image%2020220410221435.png) ![](attachments/Pasted%20image%2020220410221444.png) ![](attachments/Pasted%20image%2020210510013848.png) A **blow-up** is a [birational](Unsorted/dominant%20morphism.md) transformation that replaces a [closed subscheme](Unsorted/closed%20immersion.md) with an [effective](effective) [Cartier divisor](Unsorted/Weil%20divisor.md). Precisely, given a [Noetherian scheme](Unsorted/Noetherian%20scheme.md) $X$ and a closed subscheme $Z \subset X$, the **blow-up of $X$ along $Z$** is a [proper morphism](Unsorted/proper%20morphism.md) $\pi: \widetilde{X} \rightarrow X$ such that 1. $\pi^{-1}(Z) \hookrightarrow \widetilde{X}$ is an effective Cartier divisor, called the **exceptional divisor** and 2. $\pi$ is universal with respect to (1). Concretely, it is constructed as the [relative Proj](relative%20Proj) of the [Rees algebra](Rees%20algebra) of $O_{X}$ with respect to the [ideal sheaf](ideal%20sheaf.md) determining Z. For varieties: to blow up along functions $f_1,\cdots, f_n$, define a morphism \[ F:V(f_i)^c &\to \PP^{n-1} \\ x &\mapsto \tv{f_1(x),\cdots, f_n(x)} .\] Then $\tilde X = \Bl(X; f_1,\cdots, f_n) \da \cl_{X\times \PP^{n-1}}( \Gamma_F)$ 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 $\tilde X \birational X$ with a common open dense subset. Note that $\Gamma_F \isovia{\ro\pi{\Gamma_F}} V(f_i)^c$, so $V(f_i)^c$ can be identified with a dense open subset of $\tilde X$. The complement $\tilde X\sm\Gamma_F = \pi\inv(V(f_i))$ is the **exceptional set**. For $Y\leq X$ a closed subvariety, $\tilde Y\subseteq Y\times \PP^{n-1} \leq X\times \PP^{n-1}$ is a closed subvariety of the blowup $\tilde X$, and we refer to $\tilde Y$ as the **strict transform** of $Y$ in $\tilde X$. ![](attachments/Pasted%20image%2020220404234418.png) # Examples $\Bl(\AA^n; x_1,\cdots, x_n) = \ts{(x,y) \in \AA^n\times \PP^{n-1} \st y_i x_j = y_j x_i }$ replaces $\vector 0\in \AA^n$ with a copy of $\PP^{n-1}$. ![](attachments/Pasted%20image%2020220410221502.png) ![](attachments/Pasted%20image%2020220526140627.png) ![](attachments/Pasted%20image%2020220526140819.png) ![](attachments/Pasted%20image%2020220526140911.png)