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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.

Author

Affiliation

Fei Ye

QCC-CUNY

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 ${\mathrm{Bl}}_O{\mathbb{A}}^2=\{(P,L)\mid x\in {\mathbb{A}}^2,L=\overline{OP}\}\subset {\mathbb{A}}^2\times {\mathbb{P}}^1$, where $\overline{OP}$ is the line passing through the origin $O$ and $P$. Then there is a natural projection $\pi :{\mathrm{Bl}}_O{\mathbb{A}}^2\to {\mathbb{A}}^2$. 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 ${\mathrm{Bl}}_O{\mathbb{A}}^2$ is defined by the equation $xv-yu=0$. We call $\pi :{\mathrm{Bl}}_O{\mathbb{A}}^2\to {\mathbb{A}}^2$ 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 $u\ne 0$, 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 $v\ne 0$, 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 ${\mathrm{Bl}}_O{\mathbb{A}}^2$. The component $x+1-t^2=0$ and $s^2(ys+1)-1=0$ agree on the open subset $uv\ne 0$ of ${\mathrm{Bl}}_O{\mathbb{A}}^2$ via the tranformation $s=\frac{1}{t}=\frac{x}{y}$. The coordinate ring of $\tilde{C}$ is $$k[x,y,u,v]/(xv-yu,y^2-x^2(x+1),x{u}^2+u^2-v^2).$$

Note that the total transform ${\pi }^{-1}(C)$ of $C$ is reducible even though $C$ is irreducible. Comparing with the blowing-up ${\mathrm{Bl}}_O{\mathbb{A}}^2$ which share the same irreducibility as ${\mathbb{A}}^2$, it’s reasonable to define the Zariski closure ${\mathrm{Bl}}_{O}C={\pi }^{-1}(C\setminus \{0\})$ together with the projection ${\pi }_C=\pi {|}_{{\mathrm{Bl}}_{O}C}$ as the blowuing-up of $C$. The exceptional divisor of ${\mathrm{Bl}}_{O}C$ consists of the points $(0,\pm 1)$ which are the intersection of the exceptional divisor ${\pi }^{-1}(O)$ and ${\mathrm{Bl}}_{O}C$. 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 ${\mathrm{Bl}}_O{\mathbb{A}}^n$ in ${\mathbb{A}}^n\times {\mathbb{P}}^{n-1}$ defined by the ideal of maximal minors of the matrix $$\left(\begin{array}{ccc}{x}_1& \cdots & x_n\\ u_1& \cdots & u_n\end{array}\right),$$ where $x_i$ are the coordinates of ${\mathbb{A}}^n$ and $u_i$ are the homogeneous coordinates of ${\mathbb{P}}^{n-1}$.

The projection $\pi :{\mathrm{Bl}}_O{\mathbb{A}}^n\to {\mathbb{A}}^n$ 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 ${\mathrm{Bl}}_{O}X$ the Zariski closure of ${\pi }^{-1}(X\setminus O)$ in ${\mathrm{Bl}}_O{\mathbb{A}}^n$. The projection ${\pi }_X:{\mathrm{Bl}}_{O}X\to X$ 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 ${\mathrm{Bl}}_{O}X$.

Note that ${\mathrm{Bl}}_{O}X$ is the non-exceptional irreducible component of the pre-image ${\pi }^{-1}(X)\subset {\mathrm{Bl}}_O{\mathbb{A}}^n$. 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 $X{\times }_{{\mathbb{A}}^n}{\mathrm{Bl}}_O{\mathbb{A}}^n$. 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 ${\mathrm{Bl}}_O{\mathbb{A}}^n$ 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 ${\mathrm{Bl}}_Y{\mathbb{A}}^n$ 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 $$\left(\begin{array}{ccc}{f}_1& \cdots & f_r\\ u_1& \cdots & u_r\end{array}\right).$$ Denote by $\pi :{\mathrm{Bl}}_Y{\mathbb{A}}^n\to {\mathbb{A}}^n$ the projection. Let ${\mathrm{Bl}}_{Y}X$ be the Zariski closure of ${\pi }^{-1}(X\setminus Y)$. Then the projection ${\pi }_X:{\mathrm{Bl}}_{Y}X\to X$ 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 ${\mathrm{Bl}}_{Y}X$ 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 ${\mathrm{Bl}}_{Y}X$ 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).

Remark 1. 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 $$\mathrm{Rees}(R,I)=\underset{n\ge 0}{⨁}{I}^n$$ where $I^0:=R$.

Recall that an $R$-algebra $A$ is $N_0$-graded algebra if $A=\underset{n\ge 0}{⨁}{A}_n$ where $A_n$ are $R$-modules such that $A_0=R$ and $A_i{A}_j\subset A_{i+j}$. The elements of $A_n$ are called homogeneous elements of degree $n$.

Definition 4 Given a graded $R$-algebra $A=\underset{n\ge 0}{⨁}{A}_n$, the Proj of $A$, denoted as $\mathrm{Proj}A$ is the set of homogeneous prime ideals of $A$ not containing $A_{+}=\underset{n>0}{⨁}{A}_n$.

The Proj of the polynomial ring $k[x_0,\cdots ,x_n]$ is the projective space ${\mathbb{P}}^n$. In general, $\mathrm{Proj}A$ is projective scheme over $\mathrm{Spec}R$. The scheme structure on $\mathrm{Proj}A$ can be given by gluing together the affine schemes $\mathrm{Spec}{A}_{(f)}$, 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 $k^{\ast }$ on $A$ (see for example (Brion 2009 Exmple 1.7)), one can show that $\mathrm{Proj}A$ is quotient of the action of the multiplicative group $k^{\ast }$ on $\mathrm{Spec}A\setminus \mathrm{Spec}R$ (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 ${\mathrm{Bl}}_{Y}X=\mathrm{Proj}(\mathrm{Rees}(X,I))$ where $\mathrm{Rees}(X,I)$ 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 $\mathrm{Rees}(R,I)$.

Define $R[tI]=\{\sum a_i{t}^i\mid a_i\in I^i\}$ 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 $$R[tI{]}_n=R[tI]\cap R[t{]}_n=\{a{t}^n\mid a\in I^n\}\cong I^n,$$ where the last isomorphism is given by the $R$-module morphism $a{t}^n\mapsto a$. It is easy to check that $R[tI]\to \mathrm{Rees}(R,I)$, $\sum a_i{t}^i\to \sum a_i$ is an isomorphism of graded $R$-algebras.

Note that ${\mathrm{Bl}}_O{\mathbb{A}}^2\setminus {\pi }^{-1}(O)$ can also be identifies with $$\{((x,y),(tx,ty))\in {\mathbb{A}}^2\setminus \{O\}\times {\mathbb{P}}^1\mid t\in k^{\ast }=k\setminus \{0\}\}.$$ Algebraically, regular functions on ${\mathrm{Bl}}_0{\mathbb{A}}^2$ are in the form $\sum _{a+b=r}{f}_{ab}(x,y)t^r{x}^a{y}^b$ 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 $$\begin{array}{rl}k[x,y,u,v]/(xv-yu)\to & k[x,y,tx,ty]\\ (x,y,u,v)\mapsto & (x,y,tx,tu)\end{array}$$ is isomorphism as $k[x,y]$-algebras. This shows that ${\mathrm{Bl}}_O{\mathbb{A}}^2$ is isomorphic to $\mathrm{Proj}(\mathrm{Rees}(k[x,y],(x,y)))$.

Moreover, one can also show that the coordinate ring $k[x,y,u,v]/(xv-yu,y^2-x^2(x+1),x{u}^2+u^2-v^2)$ of the blowing-up of $y^2=x^2(x+1)$ at the origin is isomorphic to the Rees algebra $k[\overline{x},\overline{y},t\overline{x},t\overline{y}]$, where $\overline{x}$, $\overline{y}$ are images of $x$ and $y$ in $k[x,y]/(y^2-x^2(x+1))$. This shows that ${\mathrm{Bl}}_O{C}^2$ is isomorphic to $\mathrm{Proj}(\mathrm{Rees}(k[\overline{x},\overline{y}],(\overline{x},\overline{y})))$.

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 $\mathrm{Proj}(R[tI])$, in other words, the exceptional divisor is $\mathrm{Proj}(R[tI]/IR[tI])$.

The quotient ring $$R[tI]/IR[tI]\cong \underset{r\ge 0}{⨁}{I}^r/I^{r+1}$$ is called the associated graded ring of $I$, denoted as ${\mathrm{gr}}_I(R)$.

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 ${\mathrm{Bl}}_Y(X)$ together with a morphism $\pi :{\mathrm{Bl}}_Y(X)\to X$ such that ${\pi }^{-1}(Y)$ is an effictive Cartier divisor on $X$ and $\pi$ satisfies the universal property: if $\varphi :Y\to X$ is another morphism such that ${\varphi }^{-1}(Y)$ is an effective Cartier divisor, then there is a unique morphism $\tilde{\varphi }:{\mathrm{Bl}}_Y(X)\to X$ such that $\tilde{\varphi }\circ \pi =\varphi$.

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.

References

Brion, Michel. 2009. “Introduction to Actions of Algebraic Groups.” https://www-fourier.ujf-grenoble.fr/~mbrion/notes_luminy.pdf.

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.