# Different Aspects of Blowing-up

Algebraic Geometry

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.

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July 9, 2023

## 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 .

## 3 Blowing-up defined by the closure of graph

In , 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 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 . 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 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 .

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 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.

## References

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Hironaka, H., and H. Rossi. 1964. “On the Equivalence of Imbeddings of Exceptional Complex Spaces.” Mathematische Annalen 156 (4): 313–33. https://doi.org/10.1007/BF01361027.
Lopez-Benito, Pedro Nunez. 2017. “Blow-Ups in Algebraic Geometry.” B.{{Sc}}, LudwigMaximiliansUniversitat M\"unchen.
Saal Hardali. 2016. “Geometric Construcion of Proj as a Quotient by a $$\mathbb{G}_m$$ Action.” MathOverflow. https://mathoverflow.net/q/238380.
Simis, Aron. 2020. Commutative Algebra. Berlin: De Gruyter.
The Stacks project, Authors. 2023. “The Stacks Project.” 2023. https://stacks.math.columbia.edu.
Yasuda, Takehiko. 2007. “Higher Nash Blowups.” Compositio Mathematica 143 (6): 1493–1510. https://doi.org/10.1112/S0010437X0700276X.