4.1 Blowing up in general

Here the basic reference is [ 4 , II.7 and V.3 ] .

Suppose \(X\) is a noetherian scheme and \(\mathcal{I}\) is a coherent sheaf of ideals in \(X\). Set \(\mathcal{S} = \oplus _{d\geq 0}\mathcal{I}^d\) and set \(\widetilde{X} = \operatorname{\mathbf{Proj}}\mathcal{S}\). We get a projective morphism \(\pi :\widetilde{X}\to X\) called the blow up of \(X\) at the ideal sheaf \(\mathcal{I}\). The scheme \(\widetilde{X}\) is also called the blow up of \(X\) along \(Y\), where \(Y\) is the closed subscheme of \(X\) corresponding to \(\mathcal{I}\).

Example 4.1

Take \(X = \mathbb {A}^2 = \operatorname{\mathrm{Spec}}A = k[x,y]\) and take \(I = (x,y)\) with \(\mathcal{I}\) the corresponding ideal sheaf. Then \(\mathcal{S}\) is the sheaf of algebras corresponding to the \(A\)-algebra \(S = \oplus _{d\geq 0} I^d\). To avoid confusion, let’s write \(u\) (resp. \(v\)) for the image of \(x\) (resp. \(y\)) in \(S_1\). Then \(S\) is generated by \(x,y, u\) and \(v\), with \(\deg x = \deg y = 0\) and \(\deg u = \deg v = 1\). We have the relations that \(xv = yu\), and, in fact, it isn’t too hard to see that we have that

\begin{equation} \label{pres} S = k[x,y,u,v]/(xv-yu). \end{equation}

4.2

Now, \(S\) is the quotient of the graded ring \(R = k[x,y,u,v]\) by the homogeneous ideal \((xv-yu)\). It follows that \(\widetilde{X} = \operatorname{\mathrm{Proj}}S\) is the closed subscheme of \(\operatorname{\mathrm{Proj}}R = \mathbb {A}^2\times \mathbb {P}^1\) cut out by homogeneous ideal \((xv-yu)\). Alternatively, \(\widetilde{X}\) can be seen as the closure of \(X\setminus \{ 0\} \) in \(X\times \mathbb {P}^1\) under the map \(j\) given by \(j(x,y) = ((x,y), [x,y])\).

The map \(\pi :\widetilde{X}\to X\) is given by the projection on the first factor \(p_1:\mathbb {A}^2\times \mathbb {P}^1\to \mathbb {A}^2\). The restriction of \(\pi \) to \(\pi ^{-1}(X\setminus \{ 0\} )\) is an isomorphism. (The inverse is just the map \(j\).)

On the other hand, \(E := \pi ^{-1}(0)\) is just \(\mathbb {P}^1\). This set \(E\), a divisor in \(\widetilde{X}\) is called the exceptional locus.

We have two affine charts covering \(\widetilde{X}\), \(U := D(v)\), where \(v = 1\) and \(V = D(u)\), where \(u = 1\). On \(U = \operatorname{\mathrm{Spec}}k[u, y]\) and \(V = \operatorname{\mathrm{Spec}}k[x,v]\). Moreover the maps \(\pi :U\to X\) and \(\pi :V\to X\) are given by \((u,y)\mapsto (uy, y)\) and \((x,v)\mapsto (x, xv)\) respectively.

It’s interesting to note that

\begin{equation} \label{notopen} \pi (U) = (\mathbb {A}^2\setminus V(y))\cup \{ (0,0)\} . \end{equation}

4.3

So \(\pi (U)\) is not an open (or closed) subset of \(X\). It is just a constructible set. (This came up in a seminar talk recently.)

It’s also interesing to note that the Jacobian determinant of the maps \(\pi \) is given in coordinates by

\begin{align*} \Bigm |\frac{\partial (x,y)}{\partial (u,y)}\Bigm | & = y,\\ \Bigm |\frac{\partial (x,y)}{\partial (x,v)}\Bigm | & = x,\\ \end{align*}

The blowup has a nice universal property, but to recall it, it helps to recall the definition of the inverse image of an ideal from Hartshorne.

Definition 4.4

Suppose \(f:X\to Y\) is a morphism of schemes and \(\mathcal{I}\subseteq \mathcal{O}_Y\) is a sheaf of ideals. Then the inverse image ideal of \(\mathcal{I}\) is the ideal sheaf \(f^{-1}\mathcal{I}\cdot \mathcal{O}_X\) generated by \(f^{-1}\mathcal{I}\) in \(\mathcal{O}_X\).

Another way to say this is to say that \(f^{-1}\mathcal{I}\cdot \mathcal{O}_X\) is the image of \(f^*\mathcal{I}\) in \(\mathcal{O}_X\) under the natural map \(f^*\mathcal{I}\to \mathcal{O}_X\). In general, this map is not injective.

Proposition 4.5

Suppose \(\pi :\widetilde{X}\to X\) is the blow up of a noetherian scheme \(X\) along a closed subscheme \(Y\) with ideal \(\mathcal{I}\). Then

The inverse image ideal sheaf \(\widetilde{\mathcal{I}} := \pi ^{-1}\mathcal{I}\cdot \mathcal{O}_{\widetilde X}\) is invertible. In fact, \(\widetilde{\mathcal{I}} = \mathcal{O}_{\widetilde{X}}(1)\).
If \(U = X\setminus Y\), then \(\pi \) induces an isomorphism \(\pi ^{-1} U\to U\).

Explanation ▼

This is [ 4 , Proposition II.7.13 ] . So you can read the proof in there.

But let’s explain how it works in Example 4.1. Then \(\widetilde{X}\) is the closed subscheme of \(\mathbb {A}^2\times \mathbb {P}^1\) cut out by the equation \((xv-yu)\). The ideal sheaf \(\mathcal{I}\) is the ideal sheaf in \(\tilde{X}\) generated by \(x\) and \(y\). Locally, it is principal. To see this, notice that, in the chart \(U\), it corresponds to the ideal generated by \(x=uy\) and \(y\), or, equivalently, just by \(y\). And, similarly, in the chart \(V\), it corresponds to the ideal generated just by \(x\).

To see, in this example, that \(\widetilde{\mathcal{I}} = \mathcal{O}_{\widetilde{X}}(1)\), first note that the sections of \(\mathcal{O}_{\widetilde{X}}(1)\) are, by definition, the degree \(0\) elements of \(k[x,y,u,v]/(xv-yu)\), and the rational sections are the degree \(0\) elements with nonvanishing denominators in \(k[x,y]/(xv-yu)\). Since \(xv = yu\) on \(\widetilde{X}\), we have a rational section \(s = u/x = y/v\) of \(\mathcal{O}_{\widetilde{X}}(1)\). Let’s compute the divisor \(\operatorname{\mathrm{div}}s\) of \(s\).

On the open set \(U = D(v) \cong \operatorname{\mathrm{Spec}}k[u,y]\), where \(v = 1\), we have \(x = yu\). So \(s = 1/y\). And \(y=0\) is the exceptional divisor \(E\) (since the map \(U\to \mathbb {A}^2\) is given by \((u,y)\mapsto (uy,y)\).) So \(\operatorname{\mathrm{div}}s|_U = -E\cap U\). Similarly, on \(\operatorname{\mathrm{div}}s|_V = -E\cap V\). So \(\operatorname{\mathrm{div}}s = -E\).

Theorem 4.6

Suppose \(\widetilde{X}\) and \(\mathcal{I}\) are as in Proposition 4.5 and \(f:Z\to X\) is any morphism such that \(f^{-1}\mathcal{I}\cdot \mathcal{O}_Z\) is invertible. Then there exists a unique morphism \(g:Z\to \widetilde{X}\) such that \(f = \pi \circ g\).

Proof ▼

This is [ 4 , II.7.14 ] .