Blowing up

Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI), in Codex at Ultra. Public domain (CC0).

Blowing up makes an ideal locally principal in a universal way. At the origin of the plane, the new exceptional curve records a direction of approach. The construction also works when the center is singular, has multiplicities, or lies on a nonreduced scheme. In those cases the direction picture must be supported by the actual algebra: the Rees algebra, its affine charts, and the torsion removed in a strict transform.

We use effective Cartier divisors from Effective Cartier divisors and invertible sheaves, separatedness of Proj from The diagonal and separated morphisms, and projectivity from Projective morphisms and Chow's lemma. Relative Proj and its affine charts are the subject of the planned Projective space, relative Proj and maps to projective space lesson of Sheaves and schemes. Our projective-bundle convention is \(\mathbb P(\mathcal F)=\operatorname{Proj}_X\operatorname{Sym}\mathcal F\), parametrizing invertible quotients. No finite-type condition on the ideal is needed until projectivity is asserted.

1. The Rees algebra and its charts

Let \(Z\subset X\) be a closed subscheme with quasi-coherent ideal \(\mathcal I\). Its Rees algebra is the graded algebra

with multiplication inherited from the structure sheaf. Define the blow-up and its structural map by

On \(U=\operatorname{Spec}A\), write \(I\subset A\) for the ideal. It is helpful to identify the Rees algebra with \(A[IT]\subset A[T]\), where \(T\) has degree one. For \(a\in I\), its standard Proj open has coordinate ring

Here the last ring is the subring generated by the image of \(A\) and all fractions \(u/a\), \(u\in I\). This notation is valid even when \(a\) is a zero divisor in \(A\). The map from the degree-zero localization sends \(uT^n/(aT)^n\) to \(u/a^n\), for \(u\in I^n\). It is injective: if the latter is zero, some power of \(a\) kills \(u\), and the same power of \(aT\) kills the homogeneous numerator. Its image is precisely the ring described in (1.2), since products of degree-one fractions generate it.

The opens \(\operatorname{Spec}R_a\) cover \(b^{-1}U\), because the Rees algebra is generated in degree one. If \(I=(a_1,\ldots,a_r)\), those \(r\) charts suffice. On the \(a\)-chart the overlap with the \(c\)-chart is \(D(c/a)\). Inverting this ratio gives the same ring as inverting \(a/c\) in the \(c\)-chart. This supplies explicit transition maps. See Stacks, Tag 0804.

For computations, a chart can be presented by symbols \(u_i=a_i/a\) and relations \(a u_i=a_i\), but these relations alone may leave \(a\)-power torsion. The chart is their quotient by all such torsion. Indeed, the presented algebra localized at \(a\) is \(A_a\), and its image there is exactly \(A[I/a]\); the kernel of localization is the \(a\)-power torsion. This is why one should compute the Rees algebra rather than automatically replace it by the symmetric algebra.

2. The exceptional divisor and its sign

On the \(a\)-chart, the pulled-back center ideal is

Moreover \(a\) acts injectively on \(R_a\), because \(R_a\) embeds in \(A_a\), where \(a\) is a unit. Thus the inverse-image closed subscheme

is an effective Cartier divisor, called the exceptional divisor. It may be empty. On \(X\setminus Z\), the ideal is the unit ideal, its Rees algebra is \(\mathcal O[T]\), and Proj is the base itself. Therefore \(b\) is an isomorphism over \(X\setminus Z\).

There is a canonical identification

To verify it without a sign convention guess, map the degree-one tautological module to the pulled-back ideal by sending the local generator \(aT\) to \(a\). On the \(a\)-chart both modules are free of rank one. On an overlap their generators change by the same ratio \(c/a\), so the identifications glue. Dualizing the ideal gives the second equality. This proves all assertions of Stacks, Tag 02OS.

The exceptional scheme itself is

Indeed, quotient the Rees algebra by the degree-zero center ideal: its degree-\(n\) quotient is \(\mathcal I^n/\mathcal I^{n+1}\). Formula (2.3) retains scheme structure and multiplicities, rather than only a set of tangent directions.

3. The universal property

Theorem 3.1. The blow-up is final among \(X\)-schemes \(f:Y\to X\) for which \(f^{-1}Z\) is an effective Cartier divisor. Equivalently, every such \(f\) factors uniquely through \(b:B\to X\).

Proof. Section 2 shows that \(B\) is an object of the category. Work over an affine base open \(\operatorname{Spec}A\), and near a point of \(Y\) trivialize the pulled-back ideal as \(t\mathcal O_Y\), where \(t\) is a nonzerodivisor. The images of elements of \(I\) have the form \(t u_a\), and the coefficients \(u_a\) generate the unit ideal. In the local ring at the chosen point, at least one coefficient is a unit. After shrinking, choose \(a\in I\) whose image \(\alpha\) is itself a nonzerodivisor generator of the pulled-back ideal.

For \(u\in I\), the quotient \(f^\sharp(u)/\alpha\) is a well-defined regular function on this neighborhood. More generally, \(u\in I^n\) maps to a unique multiple of \(\alpha^n\). Sending \(u/a^n\) to that quotient defines a homomorphism \(A[I/a]\to\mathcal O_Y\) on the neighborhood. Any relation between fractions can be cleared by a power of \(a\); its image can then be canceled by the corresponding power of the regular element \(\alpha\). Thus the homomorphism respects all relations, and gives a local lift to the \(a\)-chart.

These local lifts agree on overlaps. To justify both agreement and uniqueness, use the open complement \(Y\setminus f^{-1}Z\). It is schematically dense by Proposition 1.2 of the Cartier lesson, and there the lift is forced by the isomorphism \(B\setminus E\cong X\setminus Z\). The map \(B\to X\) is separated: this is local on \(X\), where it is separatedness of Proj. The equalizer of any two lifts is consequently a closed subscheme. Its defining ideal vanishes on the schematically dense complement, and the injection into the direct image of the complement makes that ideal zero. Hence the lifts agree everywhere on the overlap and glue. The same reasoning gives uniqueness for any two global lifts. \(\square\)

This is Stacks, Tag 0806. Schematic density supplies uniqueness even for a nonreduced \(Y\); mere topological density would not suffice.

Notice the condition on the image ideal \(\mathcal I\mathcal O_Y\subset\mathcal O_Y\). Having an arbitrary invertible quotient of \(f^*\mathcal I\) is weaker and does not express this universal property. The canonical map \(b^*\mathcal I\to\mathcal I_E\) is surjective but need not be injective.

4. Basic properties

Theorem 4.1. The following hold.

1. Blowing up an effective Cartier divisor is an isomorphism.
2. A blow-up of a reduced scheme is reduced. A blow-up of an integral scheme in a nonzero quasi-coherent ideal is integral and birational.
3. If \(\mathcal I\) is of finite type, \(b\) is projective, hence proper, and \(\mathcal O_B(1)\) is relatively ample.
4. Blowing up commutes with flat base change.

Proof. For (1), the identity of \(X\) is itself an object satisfying the same final property; equivalently, locally \(I=(f)\) with \(f\) regular, and \(A[I/f]=A\). The local identity maps glue.

For (2), every chart is a subring of a localization of the reduced base ring, so is reduced. If the base is integral and the ideal nonzero, every nonempty affine base open has a nonzero restriction of the ideal. To check the latter, a quasi-coherent ideal zero on an affine neighborhood is zero on every affine open meeting it: localization on that nonempty intersection injects the domain. All nonempty affine opens meet, so a nonzero ideal cannot have such a zero restriction. Nonempty charts are domains, and every chart contains the same generic point: after inverting a nonzero chart generator, its ring has the base function field as fraction field. The complement of the center is a nonempty open, isomorphic to its inverse image and dense in every chart. The blow-up is therefore irreducible and reduced, with the same function field, proving the assertion. The nonzero qualification is necessary: for \(\mathcal I=0\), the Rees algebra has no positive-degree part and its Proj is empty. This is the blow-up of the whole scheme.

For (3), the surjection \(\operatorname{Sym}\mathcal I\twoheadrightarrow\mathcal R(\mathcal I)\) gives a closed immersion \(B\hookrightarrow\mathbb P(\mathcal I)\). The finite-type hypothesis on \(\mathcal I\) is exactly the projectivity convention used in the projective-morphism lesson. Relative ampleness of the tautological sheaf follows there as well, or locally from the closed immersion into a finite-dimensional projective space supplied by finitely many ideal generators.

For (4), on a flat affine base change \(A\to C\), tensoring \(I^n\hookrightarrow A\) remains injective and identifies its image with \((IC)^n\). Thus

as graded algebras. The base-change identity for Proj follows on each standard affine chart by localization and taking degree zero. These identities glue to the claimed cartesian square. \(\square\)

See Stacks, Tags 0807, 0808, 02ND, 02NS, and 0805.

Proposition 4.2 (closure of a graph). On an affine scheme \(X=\operatorname{Spec}A\), let \(I=(a_1,\ldots,a_r)\). The blow-up is the schematic closure of the graph of the morphism

where the graph is viewed in \(X\times_A\mathbb P^{r-1}_A\).

Proof. Send the degree-one variable \(U_i\) of \(A[U_1,\ldots,U_r]\) to \(a_iT\). This graded surjection gives a closed immersion of the blow-up into that projective space over \(X\). On the complement of the center, some \(a_i\) is invertible locally; the \(i\)-chart has homogeneous ratios \(U_j/U_i=a_j/a_i\). Thus the closed immersion restricts to exactly the graph in question. The exceptional complement is schematically dense in the blow-up by its Cartier property. Any closed subscheme of the ambient projective space containing the graph has an ideal whose restriction to the blow-up vanishes on that complement, hence vanishes everywhere. Such a closed subscheme contains the blow-up. The blow-up itself is closed and contains the graph, proving the assertion. \(\square\)

This gives a concrete method for resolving a displayed rational map to projective space. For an integral affine source, multiply a homogeneous tuple of fractions by a common denominator and blow up the resulting nonzero coordinate ideal. The projective projection from the graph closure supplies an everywhere-defined map on the blow-up, agreeing with the original rational map on a dense open. This construction repairs that particular indeterminacy; it does not assert that the resulting scheme is nonsingular. The thickened-center example below shows why smoothness needs a separate check.

5. Strict transforms remove exceptional torsion

Let \(f:T\to X\) be any morphism. Its total transform is \(T\times_X B\). The exceptional inverse image there is locally defined by a chart generator \(a\), which can become a zero divisor. Define the strict transform \(T'\) as the closed subscheme obtained by quotienting its structure sheaf by sections supported on that exceptional inverse image. On an affine chart this is exactly the ideal of \(a\)-power torsion, the kernel of localization at \(a\). These kernels localize and agree on overlaps, so the definition glues. It is the schematic closure of \(T\times_X(B\setminus E)\) in the total transform.

Theorem 5.1. The strict transform of \(T\) is canonically the blow-up of \(T\) along \(f^{-1}Z\). In particular, this applies to every closed subscheme of \(X\), with no flatness assumption.

Proof. Take \(X=\operatorname{Spec}A\), \(T=\operatorname{Spec}C\), and \(J=IC\). On the \(a\)-chart the total transform has ring

Let \(\alpha\) be the image of \(a\) in \(C\). The natural map

is surjective: an element of \(J\) is a finite \(C\)-linear combination of images of elements of \(I\), so its ratio by \(\alpha\) is in the image. After inverting \(a\), both rings in (5.1) become \(C_\alpha\). Since the target embeds in that localization, the kernel of (5.1) is exactly the \(a\)-power torsion in \(H\). The quotient defining the strict transform is therefore the blow-up chart of \(C\) along \(J\). Ratios on overlaps are the same, so these identifications glue. The generators from \(I\) generate \(J\) as a \(C\)-ideal, ensuring these charts cover its blow-up. \(\square\)

This supplies the ring computation in Stacks, Tag 080E. The definition is Tag 080D. It also works for quasi-coherent modules: quotient their pullbacks by exceptional power torsion. Formula (5.1) kills exactly that torsion, so taking this quotient before or after passing to the blow-up of \(T\) gives the same module. Indeed, for any module \(M\), localization of \(M/\ker(M\to M_a)\) is \(M_a\), and quotienting by the kernel ideal in (5.1) does not change that localization; a second torsion quotient removes exactly the same kernel.

Flat base change gives total transform equal to strict transform, because tensoring preserves the injective multiplication by the chart generator. Without flatness the two can be very different. For the origin point mapping to the affine plane, its total transform under the blow-up is \(\mathbb P^1\). Its pulled-back center is the whole point, so its own blow-up and its strict transform are empty.

6. Examples and exercises

Exercise 6.1 (easy). Compute the blow-up of \(\mathbb A^2_k\) at the origin, its exceptional divisor, and the normal line bundle of that divisor.

Solution. For \(I=(x,y)\), the two charts are

On the overlap, \(s=t^{-1}\) and \(y=xt\). The exceptional equations are \(x=0\) and \(y=0\). Their two affine lines glue by \(s=t^{-1}\) to \(E\cong\mathbb P^1_k\). Globally the blow-up is the incidence subscheme \(xV=yU\) of \(\mathbb A^2\times\mathbb P^1\), as verified on these two charts. The tautological \(\mathcal O_B(1)\) restricts to \(\mathcal O_{\mathbb P^1}(1)\); hence (2.2) gives \(\mathcal O_B(E)|_E=\mathcal O_{\mathbb P^1}(-1)\). The minus sign describes the normal line bundle. Both charts are affine planes, so the blown-up surface is smooth over every field.

Exercise 6.2 (easy). Show that blowing up an effective Cartier divisor does nothing, and compute the blow-up along \((x)\) in \(k[x,y]/(xy)\).

Solution. A regular local equation \(f\) has chart \(A[I/f]=A\), proving the first assertion. In the second case \(x\) is a zero divisor. There is one chart, whose ring is the image of \(A\) in \(A_x\), since \(I=(x)\). Its kernel is the \(x\)-power torsion, namely \((y)\): the localization is \(k[x,x^{-1}]\), and the polynomial part in \(x\) injects. Thus the blow-up is \(\operatorname{Spec}k[x]\to\operatorname{Spec}A\), keeping the component \(y=0\). Its exceptional divisor is \(x=0\). A locally principal center defined by a zero divisor need not give the identity.

Exercise 6.3 (medium). Assume \(\operatorname{char}k\ne2\). Resolve the node \(C=V(y^2-x^2(x+1))\) by blowing up the origin, and compute its intersection with the exceptional divisor. Describe the same equations in characteristic two.

Solution. On the \(x\)-chart, substitute \(y=xt\). The total-transform equation is \(x^2(t^2-x-1)=0\). Removing \(x\)-power torsion gives the strict transform \(t^2=x+1\), an affine line with parameter \(t\). On the \(y\)-chart, substitute \(x=sy\); the strict equation is \(1-s^2-s^3y=0\). This forces \(s\) invertible, because reducing modulo \(s\) would give \(1=0\); this chart is therefore the open \(t\ne0\) of the first chart. The whole strict transform is the affine line mapping by

It is smooth, and the morphism is proper as the blow-up of the curve, birational, and an isomorphism off the origin. Thus it is a resolution. Its exceptional fibre is \(k[t]/(t^2-1)\), two reduced points \(t=1,-1\) in the stated characteristic. In characteristic two the strict transform is still smooth, but the exceptional fibre is one double point \(k[t]/(t-1)^2\); the original singularity is not a two-branch node. The equation computation preserves that distinction.

Exercise 6.4 (medium). Recover the universal-property lift from the charts, and explain why it is unique for a nonreduced source.

Solution. Locally write \(I\mathcal O_Y=t\mathcal O_Y\) with \(t\) regular. Images \(a_i=t u_i\) generate this ideal, so the \(u_i\) generate the unit ideal. On the cover \(D(u_i)\), \(a_i\) is a regular generator and the ratios \(a_j/a_i=u_j/u_i\) define a map to the \(a_i\)-chart. All relations can be canceled by powers of the regular element \(a_i\). On overlaps they define the same fractions, or agree by the schematically dense complement and the closed-equalizer proof of Theorem 3.1. This proves gluing and uniqueness without requiring the source to be reduced.

Exercise 6.5 (hard). Blow up the vertex of \(S=V(xy-z^2)\subset\mathbb A^3_k\). Identify the exceptional conic and its normal line bundle, in every characteristic.

Solution. On the \(x\)-chart set \(u=z/x\), \(v=y/x\). The ring is \(k[x,u]\), with \(v=u^2\), and the map is \((x,u)\mapsto(x,xu^2,xu)\). On the \(y\)-chart the ring is \(k[y,w]\), with map \((y,w)\mapsto(yw^2,y,yw)\). The \(z\)-chart has ring \(k[z,r,r^{-1}]\), with \(x=zr\) and \(y=z/r\); it lies in the overlap of the first two charts. Their overlap satisfies \(w=u^{-1}\), \(y=xu^2\). Thus the blow-up is smooth and its exceptional divisor is two lines glued to \(\mathbb P^1\).

To see its given projective embedding, the vertex ideal filtration is the degree filtration of this standard graded cone. Its associated graded algebra is \(k[X,Y,Z]/(XY-Z^2)\); by (2.3) the exceptional divisor is that conic in \(\mathbb P^2\). The parametrization \([a:b]\mapsto[a^2:b^2:ab]\) identifies it with \(\mathbb P^1\), by the two charts just computed. It is smooth even in characteristic two: the partial derivatives are \(Y,X,-2Z\), and simultaneous vanishing on the conic would require \(X=Y=0\), contradicting its equation at a projective point. Its hyperplane bundle pulls back to \(\mathcal O_{\mathbb P^1}(2)\). Formula (2.2) therefore gives normal bundle \(\mathcal O_{\mathbb P^1}(-2)\).

These computations also show that the center's scheme structure matters. In the plane, the ideals \((x,y)\) and \((x,y^2)\) have the same support. The first blow-up is smooth. The \(x\)-chart for the second is \(k[x,y,v]/(xv-y^2)\), embedded in \(k[x,y]_x\) by \(v=y^2/x\). At its origin this ring has dimension two and cotangent-space dimension three, so is not regular. The morphisms cannot be isomorphic over the plane. Blowing up a thickened center can introduce a singularity even in a smooth ambient scheme.

References and proof dependencies

The construction, affine charts, exceptional Cartier property, universal property, base change and strict-transform theorem are proved above for arbitrary quasi-coherent center ideals. Finite type is used precisely for projectivity, and a nonzero ideal is required for integralness. The Stacks project authors, The Stacks project, are consulted in the AI Integrated Stacks Project edition at commit 565b10e987aba5969b21145a0833f42d69f96790; the linked proofs retain GNU FDL 1.2 and their expression is not reproduced here.

Ravi Vakil, The Rising Sea: Foundations of Algebraic Geometry, draft of 27 July 2024, Chapter 22, gives a complementary treatment by the universal property and computations. The chart proofs here keep the center's ideal, the exceptional line-bundle sign and the characteristic of the examples explicit.