Different Aspects of Blowing-up
Resolution of singularities plays an essential role in biratonal geometry. One approach is the blowing-up. In this notes, we will explore different approaches to blowing-up.
1 Introduction
To illustrate the idea, we start with a simple example.
Let \(\mathbb{A}^2\) be the affine plane over an algebraically closed field \(k\). Consider the curve \(C\) defined by the equation \(y^{2}=x^{2}(x+1)\). The curve \(C\) has a singularity at the origin \(O\). We can resolve the singularity of \(C\) by blowing-up the origin of \(\mathbb{A}^2\). Let \(\mathbb{P}^1=\mathbb{A}^2\setminus\{0\}/\sim\) be the projective line whose points can be considered as lines passing through the origin \(O\) of \(\mathbb{A}^2\). Consider the subset \(\Bl_O{\mathbb{A}^2}=\{(P, L)\mid x\in \mathbb{A}^2, L=\overline{OP}\}\subset\mathbb{A}^2\times \mathbb{P}^1\), where \(\overline{OP}\) is the line passing through the origin \(O\) and \(P\). Then there is a natural projection \(\pi: \Bl_O{\mathbb{A}^2}\to \mathbb{A}^2\). As there is a unique line passing throught two distinct points, \(\pi: \pi^{-1}(\mathbb{A}^2\setminus\{O\})\to \mathbb{A}^2\setminus\{O\}\) is bijective and \(\pi^{-1}(O)=\mathbb{P}^1\). Let \((x, y)\) be the coordinate of \(\mathbb{A}^2\) and \([u, v]\) the homogeneous coordinates of \(\mathbb{P}^1\). It can be checked that \(\Bl_O{\mathbb{A}^2}\) is defined by the equation \(xv-yu=0\). We call \(\pi: \Bl_O{\mathbb{A}^2}\to \mathbb{A}^2\) the blowing-up of \(\mathbb{A}^2\) at the origin \(O\) and \(\pi^{-1}(O)=\mathbb{P}^1\) the exceptional divisor of the blowing-up.
The pre-image \(\pi^{-1}(C)\) is called the total transform of \(C\) which is defined by \(xv-yu=0\) and \(y^{2}=x^{2}(x+1)\). If \(u\neq 0\), let \(t=v/u\), then \(y=xt\) and \(\pi^{-1}(C)\) is defined by \(x^{2}(x+1)=x^{2}t^2\) and \(y=xt\). It is clear that \(\pi^{-1}(C)\) consists of two irreducible components \(x+1-t^{2}=0\) and \(x=y=0\). The component \(x+1-t^2=0\) is the strict transform (also known as the proper transform) of \(C\) in the sense that it is the Zariski closure of \(\pi^{-1}(C\setminus\{0\})\). If \(v\neq 0\), let \(u/v=s\), then \(x=ys\) and \(\pi^{-1}(C)\) is defined by \(y^{2}=y^{2}s^2(ys+1)\) and \(x=ys\). It is clear that \(\pi^{-1}(C)\) consists of two irreducible components \(s^2(ys+1)-1=0\) and \(x=y=0\). The component \(x=y=0\) is the exceptional divisor of \(\Bl_O{\mathbb{A}^2}\). The component \(x+1-t^2=0\) and \(s^2(ys+1)-1=0\) agree on the open subset \(uv\ne 0\) of \(\Bl_O{\mathbb{A}^2}\) via the tranformation \(s=\frac{1}{t}=\frac{x}{y}\). The coordinate ring of \(\tilde{C}\) is \[k[x, y, u, v]/(xv-yu, y^{2}-x^{2}(x+1), xu^2+u^2-v^2).\]
Note that the total transform \(\pi^{-1}(C)\) of \(C\) is reducible even though \(C\) is irreducible. Comparing with the blowing-up \(\Bl_O\mathbb{A}^2\) which share the same irreducibility as \(\mathbb{A}^2\), it’s reasonable to define the Zariski closure \(\Bl_O{C}=\pi^{-1}(C\setminus\{0\})\) together with the projection \(\pi_C=\pi|_{\Bl_O{C}}\) as the blowuing-up of \(C\). The exceptional divisor of \(\Bl_OC\) consists of the points \((0, \pm 1)\) which are the intersection of the exceptional divisor \(\pi^{-1}(O)\) and \(\Bl_OC\). Those points are the slopes of the tangent lines of \(C\) at the origin \(O\).
Based on this example, generalizations and equivalent definitions of blowing-up are frequently seen. In this notes, we will explore those definitions with focus on the blowing-up of an affine variety at a point.
2 Blowing-up defined by equations
Genearlizing the example in Section 1, we define the blowing-up of an affine variety at a point.
Definition 1 (Embedded blowing-up) Let \(\mathbb{A}^n\) be an \(n\)-dimensional affine space over an algebraically closed field \(k\). Consider the quasi-projective variety \(\Bl_O{\mathbb{A}^n}\) in \(\mathbb{A}^n\times \mathbb{P}^{n-1}\) defined by the ideal of maximal minors of the matrix \[\begin{pmatrix} x_1 & \cdots & x_n \\ u_1 & \cdots & u_{n} \end{pmatrix},\] where \(x_i\) are the coordinates of \(\mathbb{A}^n\) and \(u_i\) are the homogeneous coordinates of \(\mathbb{P}^{n-1}\).
The projection \(\pi: \Bl_O{\mathbb{A}^n}\to \mathbb{A}^n\) is called the blowing-up of \(\mathbb{A}^n\) at the origin \(O\). The subvariety \(\pi^{-1}(O)\) is called the exceptional divisor of the blowing-up.
Let \(X\) be an irreducible affine variety in \(\mathbb{A}^n\) passing through the origin \(O\). Denote by \(\Bl_O{X}\) the Zariski closure of \(\pi^{-1}(X\setminus O)\) in \(\Bl_O{\mathbb{A}^n}\). The projection \(\pi_X: \Bl_O{X}\to X\) is called blowing-up of \(X\) at the origin \(O\). The exceptional divisor of the blowing-up of \(X\) at \(O\) is \(\pi_X^{-1}(O)\) in \(\Bl_O{X}\).
Note that \(\Bl_O{X}\) is the non-exceptional irreducible component of the pre-image \(\pi^{-1}(X)\subset\Bl_O{\mathbb{A}^n}\). The variety \(\pi^{-1}(X)\) is called the total transform of \(X\). The total transform \(\pi^{-1}(X)\) can be identified with the fiber product \(X\times_{\mathbb{A}^n}\Bl_O{\mathbb{A}^n}\). The defining ideal of the total transform \(\pi^{-1}(X)\) is generated by the ideal of \(X\times \mathbb{P}^{n-1}\) together with the ideal of \(\Bl_O{\mathbb{A}^n}\) in \(\mathbb{A}^n\times \mathbb{P}^{n-1}\).
One can define the blowing-up of a subvariety \(Y\) of an affine variety \(X\subset \mathbb{A}^n\) in a similar way. Suppose that \((f_1, \cdots, f_r)\) is the defining ideal of \(Y\) in \(\mathbb{A}^n\). Then the blowing-up \(\Bl_Y\mathbb{A}^n\) of \(\mathbb{A}^n\) at \(Y\) is a subvariety in \(\mathbb{A}^n\times \mathbb{P}^r\) defined by the ideal of maximal minors of the matrix \[\begin{pmatrix} f_1 & \cdots & f_r \\ u_1 & \cdots & u_{r} \end{pmatrix}.\] Denote by \(\pi: \Bl_Y\mathbb{A}^n\to \mathbb{A}^n\) the projection. Let \(\Bl_YX\) be the Zariski closure of \(\pi^{-1}(X\setminus Y)\). Then the projection \(\pi_X: \Bl_YX\to X\) is the blowing-up of \(X\) at \(Y\).
The above definition seems dependent on the choice of defining equations of the blowing-up center and the embedding in \(\mathbb{A}^n\). However, however, it can be shown that the blowing-up \(\Bl_YX\) is independent of the choice of defining equations and the embedding in \(\mathbb{A}^n\), see for example (Lopez-Benito 2017, chap. 1).
3 Blowing-up defined by the closure of graph
In (Hironaka and Rossi 1964), a geometric construction of blowing-up is given. The construction is independent of the choice of the embedding in \(\mathbb{A}^n\).
Definition 2 (Blowup as the closure of graph) Let \(Y\) be a subvariety of an affine variety \(X\). Suppose that the defining ideal \(I(Y)\) of \(Y\) is generated by \(f_1, \cdots, f_r\), where not all of them vanishes identically on \(X\). The blowing-up \(\Bl_YX\) is the zariski closure of the graph of the morphism \(X\setminus Y\to \mathbb{P}^{r-1}\), \(x\mapsto [f_x(x), \cdots, f_r(x)]\) in \(X\times \mathbb{P}^{r-1}\).
It is not so hard to see that this definition is equivalent to the previous one at least in the case of blowing up the origin of \(\mathbb{A}^n\). In (Hironaka and Rossi 1964, Remark 2), the authors show that the blowing-up defined in this definition satisfies the university property (see Definition 6).
Nash blowing-up and higher Nash blowing-up are defined in a similar way. Instead of defining the graph using the defining equations of \(Y\), one defines a graph using Gauss map from the smooth locus to a Grassminian. See (Yasuda 2007) and references therein for more details.
4 Proj of Rees Algebra
Instead of defining blowing-up using equations or geometrically, one can also use the coordinate ring which is considered a Rees algebra.
Definition 3 (Rees Algebra) Let \(R\) be a ring and \(I\) an ideal. The Rees algebra (also called blowing-up algebra) of \(R\) at \(I\) is the \(\mathbb{N}_0\)-graded \(R\)-algebra \[\Rees(R, I) = \bigoplus\limits_{n\geq 0}I^n\] where \(I^0:=R\).
Recall that an \(R\)-algebra \(A\) is \(N_0\)-graded algebra if \(A=\bigoplus\limits_{n\ge 0}A_n\) where \(A_n\) are \(R\)-modules such that \(A_0=R\) and \(A_iA_j\subset A_{i+j}\). The elements of \(A_n\) are called homogeneous elements of degree \(n\).
Definition 4 Given a graded \(R\)-algebra \(A=\bigoplus\limits_{n\geq 0}A_n\), the Proj of \(A\), denoted as \(\Proj A\) is the set of homogeneous prime ideals of \(A\) not containing \(A_+=\bigoplus\limits_{n>0} A_n\).
The Proj of the polynomial ring \(k[x_0,\cdots, x_n]\) is the projective space \(\mathbb{P}^n\). In general, \(\Proj A\) is projective scheme over \(\Spec R\). The scheme structure on \(\Proj A\) can be given by gluing together the affine schemes \(\Spec A_{(f)}\), where \(f\in A_+\) is homogeneous. For more details, see for example (The Stacks project 2023, sec. 00JM). Using the fact that \(A\) is an \(R\)-algebra if and only if there is an action of the multiplicative group \(k^*\) on \(A\) (see for example (Brion 2009 Exmple 1.7)), one can show that \(\Proj A\) is quotient of the action of the multiplicative group \(k^*\) on \(\Spec A\setminus \Spec R\) (see the MO discussions on(Saal Hardali 2016) on group action and quotients).
Definition 5 (Blowing-up as Proj) Let \(X\) be an affine variety and \(I\) an ideal of a subvareity \(Y\subset X\). The blowing-up of \(X\) at \(Y\) is the projective scheme \(\Bl_Y{X}=\Proj(\Rees(X, I))\) where \(\Rees(X, I)\) is the Rees algebra of \(X\) at \(I\).
This approach is intrinsic and can be generalized to the blowing-up of an ideal sheaf.
To see the proj construction genearlizes the first approach, we first consider a more heuristic description of the Rees algebra \(\Rees(R, I)\).
Define \(R[tI]=\{\sum a_it^i\mid a_i\in I^i\}\) to be the polynomial ring over \(R\) generated by \(tI\). As a subset of the graded polynomial ring \(R[t]\) over \(R\), \(R[tI]\) is a graded subring whose \(n\)-th graded component is \[R[tI]_n = R[tI]\cap R[t]_n = \{at^n\mid a\in I^n\} \cong I^n,\] where the last isomorphism is given by the \(R\)-module morphism \(at^n\mapsto a\). It is easy to check that \(R[tI]\to \Rees(R, I)\), \(\sum a_it^i\to \sum a_i\) is an isomorphism of graded \(R\)-algebras.
Note that \(\Bl_O\mathbb{A}^2\setminus \pi^{-1}(O)\) can also be identifies with \[\{((x, y), (tx, ty))\in\mathbb{A}^2\setminus\{O\}\times \mathbb{P}^1\mid t\in k^*=k\setminus\{0\}\}.\] Algebraically, regular functions on \(\Bl_0\mathbb{A}^2\) are in the form \(\sum\limits_{a+b=r}f_{ab}(x, y)t^rx^ay^b\) which forms the graded subring \(k[x, y][tx, ty]\) in the graded \(k[x, y][t]\) over \(k[x, y]\). This is exactly the polynomial ring \(k[x, y][t\mathfrak{m}_O]\). It can be checked that \[ \begin{aligned} k[x, y, u, v]/(xv-yu)\to & k[x, y, tx, ty]\\ (x, y, u, v)\mapsto & (x, y, tx, tu) \end{aligned} \] is isomorphism as \(k[x, y]\)-algebras. This shows that \(\Bl_O\mathbb{A}^2\) is isomorphic to \(\Proj(\Rees(k[x, y], (x, y)))\).
Moreover, one can also show that the coordinate ring \(k[x, y, u, v]/(xv-yu, y^{2}-x^{2}(x+1), xu^2+u^2-v^2)\) of the blowing-up of \(y^2=x^2(x+1)\) at the origin is isomorphic to the Rees algebra \(k[\bar{x}, \bar{y}, t\bar{x}, t\bar{y}]\), where \(\bar{x}\), \(\bar{y}\) are images of \(x\) and \(y\) in \(k[x, y]/(y^2-x^2(x+1))\). This shows that \(\Bl_OC^2\) is isomorphic to \(\Proj(\Rees(k[\bar{x}, \bar{y}], (\bar{x}, \bar{y})))\).
For the geometric root of Rees algebra and its relation with blowing-up, the reader may read the excellent exposition in (Simis 2020, sec. 7.3).
Another advantage of defining blowing-up using the Rees algebra \(R[tI]\) is that the exceptional divisor of the blowing-up is the zero locus of \(tI\) in \(\Proj(R[tI])\), in other words, the exceptional divisor is \(\Proj(R[tI]/IR[tI])\).
The quotient ring \[R[tI]/IR[tI]\cong\bigoplus\limits_{r\ge 0} I^r/I^{r+1}\] is called the associated graded ring of \(I\), denoted as \(\gr_I(R)\).
5 Universal property of blowing-up
We finally give the universal property of blowing-up. It is not the most intuitive way to define blowing-up, but it is the most useful way to define blowing-up.
Definition 6 (Universal property of blowing-up) Let \(X\) be a scheme and \(Y\) a subscheme of \(X\). The blowing-up of \(X\) at \(Y\) is a scheme \(\Bl_Y(X)\) together with a morphism \(\pi:\Bl_Y(X)\to X\) such that \(\pi^{-1}(Y)\) is an effictive Cartier divisor on \(X\) and \(\pi\) satisfies the universal property: if \(\phi: Y\to X\) is another morphism such that \(\phi^{-1}(Y)\) is an effective Cartier divisor, then there is a unique morphism \(\tilde{\phi}:\Bl_Y(X)\to X\) such that \(\tilde{\phi}\circ\pi=\phi\).
The existence of the blowing-up is guaranteed by the proj construction from the Rees algebra. The universal property is a consequence of the universal property of the Proj construction. For detials, see (The Stacks project 2023, sec. 01OF).
The universal property of blowing-up is very useful in deriving properties of blowing-up. For example, it can be used to show that the blowing-up of a smooth variety along a smooth subvariety is smooth. it can also be used to show that the blowing-up of an affine variety along a subvariety is independent of the choice of the embedding in an affine space.