Blow-ups and orders of vanishing

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Blow-ups and orders of vanishing

Fabian Carlström

June 13, 2012

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Abstract

We study blow-ups and their relation to orders of vanishing in algebraic geometry. In particular, we study the exceptional divisor and the strict transform of a blow-up. We use the order of vanishing to measure the severity of singularities, and show that if we blow up a closed point on a hypersurface we obtain points of equal or lower order above it.

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1 Introduction

The blow-up is a classic tool in algebraic geometry. It can be used, for example, to blow up nonsingular surfaces to obtain new nonsingular surfaces. The blow-up is also useful for resolving singularities.

Given a schemeX and a closed subscheme Z of X, the blow-up gives a new scheme ̃X together with a proper morphismπ∶ ̃X → X. An important property of the blow-up is that the morphism π is an isomorphism outside ofZ. The inverse image π⁻¹(Z) is called the exceptional divisor and it is of special interest.

As an example, let us consider the blow-up of the affine plane in the origin. It may be visualized as a twisted surface that is isomorphic to the plane outside of the origin, but above the origin, there is a projective line. This is the exceptional divisor. See Figure 1 below for an illustration of this blow-up. Let us describe the relationship between the new surface ̃X and the plane. A line through the origin in the plane corresponds to the line in ̃X that intersects the exceptional divisor at the height of the original line’s tangent direction. The effect of blowing up the origin is to take the plane and “inflate” the origin into a projective line. In this way the blow-up separates the lines in the plane by their tangent direction. We compute this blow-up in detail in Example 3.2.

P¹

π x

y

Figure 1: Blow-up of the affine plane in the origin.

In general it can be difficult to compute the blow-up of a given scheme. For computing certain blow-ups, however, the strict transform is a useful tool. For instance, say that we have a curve in the plane that we want to blow up in the origin. Instead of computing the blow-up directly we can use our already computed blow-up of the plane in the origin. We pull back the entire curve, except for the origin, through the blow-up, and then take the closure. This is the strict transform of the curve, and it is isomorphic to the blow-up of the curve in the origin. We prove the corresponding general result in Proposition 3.18.

Now, what is a singularity? A point on our geometrical object is singular if its tangent space is of a higher dimension than the object itself. A nonsingular object corresponds to a manifold.

The order of vanishing, defined in Section 4, can be used to measure the severity of singularities.

We will be interested in closed points on hypersurfaces in affinen-space, and in that case, the order has a natural interpretation. Hypersurfaces are defined by the zero set of a polynomial inn variables. Say thatY is the hypersurface defined by the equation f (x₁, . . . ,x_n) =0. The order of vanishing ofY at a point is the degree of the lowest degree term in the Taylor series of f at that

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point. In other words, the order ofY at a point corresponds to the number of partial derivatives of f that vanish at that point. If the order is 0, it means that the hypersurface does not pass through that point. A point is nonsingular if and only if its order is 1. Points of higher order than 1 are singular, and the higher the order, the more severe we consider the singularity.

Using the order of vanishing we will study the effect of blowing up closed points on hypersurfaces.

Because of the difficulty involved in computing such blow-ups directly, we instead use the blow-up of affinen-space in the origin, which we compute in Section 3.5, together with the strict transform.

By measuring the order at the points in the strict transform which intersect the exceptional divisor, we see that the order is less than or equal to the order of the point below. This result is found in Proposition 4.14. Thus if a closed point on a hypersurface is singular, we do not increase the severity of the singularity by blowing it up. Indeed, we hope to get something less singular. In Section 4.2 we study a few examples where the blow-up resolves singularities.

1.1 Overview

In Section 3 we define the blow-up of an affine scheme along a closed subscheme. To introduce the blow-up a few preliminary notions are needed, which are presented in Section 2.

After defining blow-ups we work through a few examples in detail. We begin by determining the blow-up of the affine plane in the origin. This provides a useful starting point and we refer back to it many times as the text progresses. We also provide examples of blowing up a singularity, and what can happen if we blow up a thick point. The point of the examples is to acquaint ourselves with the blow-up and also to provide useful motivation for studying certain parts in a general setting. We then turn to study the exceptional divisor and the strict transform. We end Section 3 by computing the blow-up of affinen-space in the origin.

In Section 4 we define the order of vanishing. We also define what it means for a scheme to be nonsingular at a point, and then relate it to the order of vanishing. In this way we can use the order to measure the severity of singularities. We spend the remainder of the section studying the special case of blowing up closed points on hypersurfaces in affinen-space. We end by showing that if we blow up a closed point on a hypersurface we obtain points of equal or lower order above it.

1.2 Acknowledgments

I am very grateful to my supervisors David Rydh and Roy Skjelnes for all their help and support.

2 Preliminaries

Let us establish some notation and conventions that will be used throughout the text. Whenever we use the wordring we mean a commutative ring with unity. Now let A be a ring. We say that an element f ∈ A is regular if it is not a zero divisor. For a multiplicative monoid S ⊆ A we let S⁻¹A denote the ring of fractions. For f ∈ A we let A_f denote the ring of fractionsS⁻¹A with S = { fⁿ}_n≥0. For a prime ideal p ⊆ A we let A_p=S⁻¹A with S = A ∖ p, the localization of A at p.

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A Z-graded ring is a ring A together with subgroups {Ad}_d_∈Zof the additive group ofA such that

A = ⊕

d∈Z

A_d andA_mA_n ⊆A_m+nfor allm, n ∈ Z.

The elements inA_dare calledhomogeneous of degree d. We also write deg f = d for f ∈ A_d. We say thatA is positively graded if A = ⊕_d≥0A_d.

A Z-graded A-module is a module M over a Z-graded ring A together with subgroups {Md}_d_∈Z ofM such that

M = ⊕

d∈Z

M_d andA_mM_n⊆M_m+nfor allm, n ∈ Z.

As in the case of rings, the elements inM_dare said to be homogeneous of degreed. A homomorphism of Z-graded A-modules is an A-module homomorphism φ∶ M → N satisfying φ(Md) ⊆N_d for alld ∈ Z.

For a ringA, a Z-graded A-algebra is a Z-graded ring B that is also an A-algebra such that the image ofA lies in B₀.

Let A = ⊕_d≥0A_d be a positively graded ring. For a homogeneous element f ∈ A we have a natural induced Z-grading on the localization Af: let the elementa/ fⁿ ∈ A_f be of degree dega − n ⋅ deg f . We let A_{( f )}denote the elements of degree 0 inA_f, that is,

A_{( f )}= { a

fⁿ ∈A_f ∶dega = n ⋅ deg f }.

2.1 Spec of a ring

Let A be a ring and define Spec A to be the set of its prime ideals. For an ideal I ⊆ A define V(I) = {p ∈ Spec A ∶ I ⊆ p}. If an ideal is generated by f₁, . . . ,f_n we will writeV( f₁, . . . ,f_n) instead ofV(( f₁, . . . ,f_n)). We have the following identities [3, Lemma ii.2.1]:

V(A) = ∅, V(0) = Spec A, V(IJ) = V(I) ∪ V(J), V(∑ Iα) = ⋂

α V(Iα).

These define the closed sets of a topology on SpecA. This topology is called the Zariski topology on Spec A.

For an element f ∈ A we define D( f ) = Spec A ∖ V( f ). These are called the principal open sets of SpecA and they constitute a basis for the topology. This is because every open set of Spec A is of the form SpecA ∖ V(I), and by the identities above

SpecA ∖ V(I) = Spec A ∖ V(∑

f∈I

(f )) = ⋃

f∈I

D( f ).

Ifφ∶ A → B is a ring homomorphism we get an induced map φ^∗∶SpecB → Spec A by p ↦ φ⁻¹(p).

Furthermore, (φ^∗)⁻¹(V(I)) = V(IB) so that φ^∗is continuous [5, §4]. Ifφ is surjective the induced mapφ^∗is a homeomorphism onto the closed subsetV(ker φ) ⊆ Spec A.

Lemma 2.1. Let A be a ring and f an element of A. Then D( f ) is canonically homeomorphic to SpecA_f.

Proof. See Liu [4, Lemma 2.1.7 (c)].

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2.2 A brief aside on schemes

It will not be enough to considerX = Spec A as a topological space. By endowing X with the sheaf of ringsO^Xinduced by lettingO^X(D( f )) = A_f (see Liu [4, Section 2.3.1] for the details of the construction) we consider (X,OX)as a locally ringed topological space. Anaffine scheme is a ringed topological space that is isomorphic to some (SpecA,OSpecA). By abuse of notation we will write SpecA to refer to the affine scheme (Spec A,O^Spec^A).

Informally, ascheme is a ringed topological space that is locally an affine scheme. We will not make much use of the machinery of scheme theory, although we find it a convenient terminology to adopt. Hence we will speak of, for example,morphisms of schemes, and open and closed subschemes.

A reassuring fact for the reader with a background in commutative algebra is that studying affine schemes is equivalent to studying rings (assuming, as we do, that they are commutative with unity). Specifically, the category of rings is equivalent to the category of affine schemes [2, Corollary i-41]. We will use this equivalence and proceed using familiar methods.

Below we have collected a few results regarding schemes that will be useful to us. Note that Lemma 2.2 extends Lemma 2.1 from topological spaces to schemes.

Lemma 2.2. Let A be a ring and f an element of A. The open subset D( f ) ⊆ Spec A is an affine scheme isomorphic to Spec A_f.

Proof. See Liu [4, Lemma 2.3.7].

Proposition 2.3. Let (X, OX)be a scheme and let U ⊆ X be any open subset. Then (U,OX∣_U)is also a scheme. We say that (U,OX∣_U), or U, by abuse of notation, is an open subscheme of X.

Proof. See Liu [4, Proposition 2.3.9].

Below we define closed subschemes of anaffine scheme. The general definition is different but one can show that it agrees with our definition in the affine case. See Definition 2.3.19 and Proposition 2.3.20 in Liu [4].

Definition 2.4. Let X = Spec A be an affine scheme. A closed subscheme of X is an affine scheme Y = Spec A/I for some ideal I ⊆ A together with the morphism of schemes Y → X induced by the canonical ring homomorphismA → A/I. We let V(I) denote the closed subscheme Y and refer to it as the closed subscheme corresponding to the idealI.

Definition 2.5. A morphism f ∶ X → Y of schemes is a morphism of locally ringed topological spaces.

IfX is a scheme together with a morphism f ∶ X → S to a scheme S, we say that X is an S-scheme.

Proposition 2.6.

(i) A ring homomorphism φ∶ A → B induces a morphism f ∶ Spec B → Spec A of affine schemes.

(ii) Conversely, any morphism f ∶ Spec B → Spec A of affine schemes is induced by a ring homomorphism φ∶ A → B.

Proof. See Hartshorne [3, Proposition ii.2.3].

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By the Nullstellensatz there is a natural correspondence betweenkⁿand the maximal ideals of Speck[x1, . . . ,xn]ifk is algebraically closed. Motivated by this we make the following definition.

Definition 2.7. Let A be a ring. Define AⁿA=SpecA[x₁, . . . ,x_n], theaffine space of dimension n over A.

In casen = 1 or 2 we refer to AⁿAas theaffine line or affine plane, respectively.

2.3 Proj of a graded ring

As we noted above, the spectrum of a polynomial ring gives an analogue to the classic affine space.

Similarly, we introduce Proj of a graded ring which will turn out to be a suitable analogue to the classic projective space. As in the case of Spec, we begin by defining Proj as a set, after which we endow it with a topology, and finally describe how to consider it as a scheme. We then consider an example in detail to motivate a new definition of projective space.

Definition 2.8. Let A = ⊕_d≥0A_d be a positively graded ring. Define ProjA to be the set of homogeneous prime ideals ofA that do not contain the irrelevant ideal A₊= ⊕d>0A_d.

Let A be a positively graded ring A = ⊕_d≥0A_d. For a homogeneous ideal I ⊆ A we define V₊(I) = {p ∈ Proj A ∶ I ⊆ p}. We have the following identities [3, Lemma ii.2.4]:

V₊(A) = ∅, V₊(0) = ProjA, V₊(IJ) = V₊(I) ∪ V₊(J), V₊(∑

α

I_α) = ⋂

α

V₊(I_α).

We define a topology on ProjA by letting the closed sets be V₊(I) for homogeneous ideals I ⊆ A.

This topology is called theZariski topology on Proj A.

For a homogeneous element f ∈ A we define D₊(f ) = Proj A ∖ V₊(f ). These are the principal open sets and they form a basis of open sets on Proj A. It actually suffices to consider the sets D₊(f ) for f ∈ A₊to form a basis since

ProjA = Proj A ∖ V₊(A₊) = ⋃

f∈A+

D₊(f ).

Note that for two homogeneous elements f and д in A we have D₊(f д) = D₊(f ) ∩ D₊(д), an identity that will be useful when performing calculations.

Lemma 2.9. Let A = ⊕_d≥0A_dbe a positively graded ring. If f ∈ A is a homogeneous element, the principal open set D₊(f ) is canonically homeomorphic to Spec A_{( f )}.

Proof. See Liu [4, Lemma 2.3.36].

The above lemma is used to give ProjA the structure of a scheme.

Proposition 2.10. Let A = ⊕_d≥0A_d be a positively graded ring and f ∈ A a homogeneous element.

Then Proj A can be endowed with a unique scheme structure such that D₊(f ) is an affine scheme isomorphic to Spec A_{( f )}.

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This is the structure that we will consider Proj to have in the remainder of the text. The following lemma will be useful for studying closed subschemes of ProjA.

Lemma 2.11. Let A and B be graded rings and let φ∶ A ↠ B be a graded surjective homomorphism.

Then φ induces a closed immersion Proj B → Proj A onto V₊(kerφ).

Proof. A homogeneous prime ideal p ∈ Spec B pulls back to a homogeneous prime ideal φ⁻¹(p) ∈ SpecA. Since φ is a graded surjective homomorphism it follows that φ(A₊) =B₊. Hence p ⊉ B₊⇔ φ⁻¹(p) ⊉ A₊, that is,φ induces a map f ∶ Proj B → Proj A. Furthermore, A/ ker φ ≅ B, and the homogeneous prime ideals inA/ ker φ are in one-to-one correspondence with the homogeneous prime ideals containing kerφ in Proj A. We conclude that f is a homeomorphism from Proj B ontoV₊(kerφ) ⊆ Proj A.

Now note that, forд ∈ A₊,

f⁻¹(D₊(д)) = {p ∈ Proj B ∶ f (p) ∌ д} = D₊(φ(д)).

We therefore see that the restriction f ∣_D₊_(φ(д))∶D₊(φ(д)) → D₊(д) is induced by the ring homomorphismA_(д)→B_(φ(д)). But that is just the restriction of the localization homomorphism A_д→B_φ(д), which is surjective sinceφ is surjective and localization is exact. Hence f ∣_D

+(φ(д))is a closed immersion, so the map f^♯of sheaves is surjective on the stalks, hence it is surjective. We conclude that f is a closed immersion.

In practice we will determine ProjA by finding a cover {D₊(f ) ≅ Spec A_{( f )}}, then determine each affine patch separately and finally consider how they glue together. Let us turn to an example and determine Projk[x, y].

Example 2.12. Let A = k[x, y] for an algebraically closed field k. Consider A to be graded by the degree, that is, let degx = deg y = 1. To determine Proj A we note that

D₊(x) ∪ D₊(y) = Proj A ∖ V₊(x, y) = Proj A.

We therefore determine the patchesD₊(x) ≅ Spec A_(x)andD₊(y) ≅ Spec A_(y), and then determine how they are glued together along their intersectionD₊(x y) ≅ Spec A_{(x y)}.

To begin with we haveAx=k[x, x⁻¹,y]. The ring A is generated by the elements r = cxⁱy^jwith c ∈ k, and by considering the subset with i + j = n we see that the degree 0 part of A_xis

A_(x)= { r

xⁿ ∈Ax∶degr = n} = k[y

x] ⊆Ax.

It follows by symmetry thatA_(y)=k[x/y]. Similarly, A_{x y}=k[x, x⁻¹,y, y⁻¹], and the elements of degree 0 inA_{(x y)}are linear combinations ofcxⁱ⁻ⁿy^j−nwithi − n = −( j − n), hence

A_{(x y)}=k[x y,y

x].

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Letu = y/x so that

A_(x)=k[u], A_(y)=k[u⁻¹], A_{(x y)}=k[u, u⁻¹] =k[u]_u.

It is then clear that SpecA_(x)≅Speck[u] = A¹kand SpecA_(y)≅ A¹k. Their intersection SpecA_{(x y)} can be regarded as the affine line minus a point (the point corresponding to the maximal ideal (u)).

We will consider the geometry of the patches separately, from which it will be clear how they glue together. Since the closed points of SpecA_(x)are in one-to-one correspondence with

{(y

x −a) ∶ a ∈ k},

we can naturally identify SpecA_(x)with the linesy = ax through the origin in Spec A = A²k. By placing a vertical line inx = 1 we can identify each line y = ax with its coefficient a ∈ k, making it clear how SpecA_(x)= A¹k:

x y

a

y = ax

In the same way SpecA_(y)can be identified with the linesby = x through the origin. One can make the same observation regarding how SpecA_(y)= A¹_kusing the horizontal liney = 1. Note that SpecA_{(x y)}can be identified with the linesy = ax where a ≠ 0.

Ifa ≠ 0 we can let b = a⁻¹so that

y = ax ⇔ by = x,

and using this identification SpecA_(x)and SpecA_(y)are glued together along their intersection SpecA_{(x y)}to form ProjA. That they are glued together in this way means that the point y = ax in SpecA_(x)is considered equivalent to the pointa⁻¹y = x in Spec A_(y), they are both a representative of the same point in ProjA.

However, in SpecA_(y)there is the linex = 0 which does not correspond to anything in Spec A_(x). When gluing SpecA_(x)and SpecA_(y)together we can regard this as the liney = ax with a = ∞.

In this way we see that Projk[x, y] in some sense behaves like the classic projective line. Motivated by this we make the following definition.

Definition 2.13. Let A be a ring and consider A[x1, . . . ,x_n]to be graded by the degree, that is, degxi=1. We define Pⁿ⁻¹A =ProjA[x1, . . . ,xn], theprojective space of dimension n − 1 over A.

In Example 2.12 we constructed Projk[x, y] = P¹_k, the projective line overk.

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2.4 The fiber product

Before we proceed to define the blow-up we present a few auxiliary results regarding thefiber product. At first we will need the fiber product to define the exceptional divisor, a concept that will emerge as soon as we consider our first example of a blow-up. Later, the fiber product will play a central role in some important results about blow-ups.

Lemma 2.14. Let A be a ring, I an ideal of A and M an A-module.

(i) A/I ⊗_AM ≅ M/IM.

(ii) If M is flat over A, then I ⊗_AM ≅ IM.

Proof. The tensor product is right exact [1, Proposition 2.18], hence by tensoring the exact sequence 0 →I → A → A/I → 0 with M over A we obtain the exact sequence

I ⊗AM ^φ A ⊗AM ≅ M ^ψ A/I ⊗AM 0

whereφ∶ a ⊗Am ↦ am ∈ M; thus im φ ≅ IM. Considering that M/ ker ψ ≅ A/I ⊗AM and kerψ ≅ im φ, the isomorphism A/I ⊗_AM ≅ M/IM follows.

IfM is flat over A, then φ is injective, hence I ⊗_AM ≅ im φ ≅ IM.

Lemma 2.15. Let A be a ring, I an ideal of A and ̃A = ⊕_d≥0I^d. Let B be a flat A-algebra and B = ⊕̃ _d≥0(IB)^d. Then ̃B ≅ ̃A ⊗_AB.

Proof. Since B is flat over A we have I ⊗_AB ≅ IB by Lemma 2.14. Note that (IB)^d =I^dB, hence I^d⊗_AB ≅ (IB)^d. The tensor product commutes with the direct sum, we therefore conclude that B ≅ ̃̃ A ⊗_AB.

Definition 2.16. Let f ∶ X → S and д∶ Y → S be morphisms of schemes. The fiber product of X and Y over S is a scheme X ×_SY together with morphisms πX∶X ×_SY → X, πY∶X ×_SY → Y satisfying f ○ π_x= д ○ π_Y, such that given any schemeZ with morphisms to X and Y whose compositions toS to agree, these morphisms factor via a unique morphism through X ×_SY.

Z

X ×_SY Y

X S

∃!

πY

πX д

f

Proposition 2.17. Let S be a scheme, and let X and Y be two S-schemes. The fiber product X ×SY exists and is unique up to unique isomorphism. If A is a ring and B and C two A-algebras, we have a canonical isomorphism

Spec(B ⊗_AC) ^∼ SpecB ×_Spec_ASpecC.

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Proposition 2.18. Let A be a ring, B = ⊕_d≥0B_da positively graded A-algebra and C an A-algebra.

Decompose B ⊗AC as ⊕_d≥0(B_d⊗_AC) and consider B ⊗AC as a positively graded C-algebra with degree d part B_d⊗_AC. There exists a canonical isomorphism

Proj(B ⊗AC) ^∼ ProjB ×SpecASpecC.

Lemma 2.19. Let S be a scheme and let X, Y and Z be S-schemes. We then have canonical isomor- phisms

X ×_SS ≅ X, X ×_SY ≅ Y ×_SX, (X ×_SY) ×_SZ ≅ X ×_S(Y ×_SZ).

Proof. Follows immediately from the definition.

We consider a specific situation. LetX be a scheme, Y an X-scheme via the morphism f ∶ Y → X, and letU be an open subscheme of X. We then have the isomorphism of schemes

f⁻¹(U) ≅ Y ×XU.

This follows by considering the diagram below: letZ be given with compatible morphisms д and h to U and Y respectively. Since f is continuous, f⁻¹(U) is an open set and f⁻¹(U) is naturally an open subscheme ofY. From f ○ h = ι₁○д it follows that im h ⊆ f⁻¹(U). Therefore we immediately have that f⁻¹(U) satisfies the universal property of the fiber product with h being the unique map Z → f⁻¹(U).

Z

f⁻¹(U) Y

U X

д

h

ι2

f f

ι1

Now letX = Spec A be an affine scheme and let Y = Spec A/J be a closed subscheme of X. The schemeY is isomorphic to V(J) ⊆ X. Let д∶ Z → X be a morphism of schemes and assume that Z = Spec C.

As topological spaces,д⁻¹(Y) = д⁻¹(V(J)) = V(JC), which is homeomorphic to Spec C/JC.

Moreover, we have the isomorphism C/JC ≅ A/J ⊗AC by Lemma 2.14, hence Spec C/JC is homeomorphic to Spec(A/J ⊗_AC) ≅ Y ×_XZ. However, for д⁻¹(Y) to be a scheme we need to endow it with a sheaf of rings fulfilling certain properties, and this involves a choice. For any closed subschemeZ we therefore define

д⁻¹(Y) = Y ×_XZ.

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3 Blow-ups

Now that we have acquired some familiarity with Proj, we turn to blow-ups. As we will see, they will turn out to be useful for resolving singularities. The definition will probably seem esoteric at first, we will therefore immediately apply it to an example in an effort to shed some light on the situation. In the example we will observe a few useful properties of the blow-up which we will later see are true in general.

Definition 3.1. Let X = Spec A be an affine scheme and let Z be the closed subscheme of X corresponding to a finitely generated ideal I ⊆ A. Let ̃A be the positively graded A-algebra A = ⊕̃ _d≥0I^d(called theRees algebra), where I⁰=A, and set ̃X = Proj ̃A. The canonical morphism X → X is called the blow-up of X along Z. We also call Z the center of the blow-up.̃

Remark. By abuse of notation we sometimes refer to the scheme ̃X as the blow-up of X along Z.

Remark. A closed subscheme Z of an affine scheme Spec A is finitely presented if and only if it corresponds to a finitely generated idealI ⊆ A.

Remark. Because the irrelevant ideal ̃A₊is generated as anA-algebra by the degree 1 part I = ̃A₁, Proj ̃A is covered by the principal open sets {D₊(fi) ≅Spec ̃A_{( f}_i₎}as fivaries over the generators ofI. The A-algebra homomorphism A → ̃A_{( f}_i₎induces the morphism Spec ̃A_{( f}_i₎→SpecA of affine schemes, and these morphisms glue together to form the canonical morphism Proj ̃A → Spec A.

3.1 Introductory examples and results

We begin by computing the blow-up of the affine plane in the origin. This leads us to define the exceptional divisor and observe some of its behavior in the blow-up computed. We then proceed to compute two more examples, in an effort to familiarize ourselves with the blow-up. In the sections following this one we take a more general approach and consider some global properties.

Example 3.2. Let A = k[x, y] for an algebraically closed field k and let I = (x, y). Define the positively gradedA-algebra

A = ⊕̃

d≥0

I^d (I⁰=A).

We will determine the blow-up of SpecA along the closed subscheme defined by I, or in other words, the blow-up of the affine plane in the origin. LetX, Y ∈ I = ̃A1denote the elementsx, y considered as elements of degree 1 in ̃A (whereas x, y ∈ A = ̃A₀have degree 0). Note that this means thatyX = xY ∈ ̃A₁is the elementx y considered as an element of degree 1 in ̃A.

We observe that

D₊(X) ∪ D₊(Y) = Proj ̃A ∖ V₊(X, Y) = Proj ̃A.

Hence we will determine the affine schemesD₊(X) ≅ Spec ̃A_(X)andD₊(Y) ≅ Spec ̃A_(Y)and then consider how they are glued together alongD₊(XY) ≅ Spec ̃A_(XY)to form Proj ̃A.

Introduce the ringA[U, V] and consider it to be graded by the degree (deg U = deg V = 1). We have a surjective homomorphism of gradedA-algebras

φ∶ A[U, V] ↠ ̃A defined by U ↦ X and V ↦ Y .

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It follows thatA[U, V]/ ker φ ≅ ̃A, and since ̃A is graded this implies that ker φ is a homogeneous ideal, hence generated by homogeneous elements. We claim that kerφ = (xV − yU). Indeed, (xV − yU) ⊆ ker φ since φ(xV − yU) = xY − yX = 0 (as we noted above). Thus φ factors uniquely through theA-algebra

B = A[U, V]/(xV − yU).

Letφ denote the induced morphism B ↠ ̃A. If we can show that φ is injective, the conclusion follows. Take a homogeneous elementf ∈ ker φ of degree d. The elements UⁱV^jform a basis for A[U, V] over A, hence

f (U, V) = f₀(x, y)U^d+f₁(x, y)U^d−1V + ⋯ + f_d(x, y)V^d. InB we have the relation xV = yU, so in B we can write f as

f (U, V) = д0(x)U^d+д1(x)U^d−1V + ⋯ + д_d−1(x)UV^d−1+д_d(x, y)V^d

(note that the indeterminatey only occurs in the coefficient of V^d). The image off under φ lies in Ã_dand can be expressed as the degreed element

φ( f ) = д0(x)x^d+д1(x)x^d−1y + ⋯ + д_d−1(x)x y^d−1+д_d(x, y)y^d =0.

The last term can be decomposed as

д_d(x, y)y^d=д_d,0(x)y^d+д_d,1(x)y^d+1+ ⋯ +д_d,m(x)y^d+m.

Using this expansion of the last term, we can considerφ( f ) as a polynomial in y with coefficients ink[x]. Since φ( f ) = 0 and the elements yⁱ are linearly independent in ̃A_d, it must be the case thatд_i(x)x^d−i =0 andд_d,i(x) = 0 for all i. But these are polynomials in x with coefficients in k, hence they are 0 if and only if all coefficients are 0. We conclude that f = 0 in B, therefore φ is injective and kerφ = (xV − yU).

To return to the problem of determining the local patches of Proj ̃A, note that localization is exact [1, Proposition 3.3] so that

Ã_(X)≅ (A[U, V]/ ker φ)_(U)≅A[U, V]_(U)/(kerφ)_(U).

Let ψ∶ A[U, V] → A[U, V]_U = A[U, U⁻¹,V] be the canonical localization homomorphism.

Doing the same observations as in Example 2.12 leads us to conclude that (A[U, V])_(U)= { r

Uⁿ ∈A[U, V]U ∶degr = n} = A[V U].

By restrictingψ to ker φ ⊆ A[U, V] we observe that (ker φ)_(U) = (xVU⁻¹−y) as an ideal in A[U, V]_(U). Hence

Ã_(X)≅ A[V

U]/(xV

U −y) = k[x, y,V

U]/(xV

U −y) = k[x,V

U] =k[x, u] with u = V U.

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Completely analogous reasoning yields Ã_(Y)≅ k[x, y,U

V]/(yU

V −x) = k[y,U

V] =k[y, v] with v = U V.

Therefore Spec ̃A_(X) ≅ Speck[x, u] = A²_k and Spec ̃A_(Y) ≅ Speck[y, v] = A²_k. By performing calculations similar to the ones above we see that

A[U, V]_(UV)=A[V U,U

V] and (kerφ)_(UV)= (xV U −y), hence

Ã_(XY)≅ k[x, y,V U,U

V]/(xV

U −y) ≅ k[x,V U,U

V].

Letu = V/U and v = U/V so that

Ã_(X)≅k[x, y, u]/(ux − y) = k[x, u], Ã_(Y)≅k[x, y, v]/(v y − x) = k[y, v],

Ã_(XY)≅k[x, y, u, v]/(ux − y, uv − 1) = k[x, u, v]/(uv − 1).

To understand the geometry let us consider the spectrum of ̃A_(X)≅k[x, y, u]/(ux−y). Imagine the surfacey = ux embedded into Spec k[x, y, u] = A³k. Consider the cross sections parallel to the x y-plane: for each value of u we have the line y = ux. If we imagine travelling along the u-axis in the positive direction we successively obtain steeper lines. Analogously in the negative direction.

An attempt to illustrate this is shown below. It might help to think of thex y-plane as having au-axis pointing outward from the page. To the right we have the three lines shown in the left picture but illustrated as viewed parallel to thexu-plane. We could also think of this in terms of Speck[x, y, u]/(ux − y) = Spec k[x, u] = A²ksince every point in Speck[x, u] can be naturally identified with a point on the surfacey = ux.

x y

y = ux

x u

A similar analysis applies to the spectrum of ̃A_(Y)≅k[x, y, v]/(v y − x). By the natural identification ofv with u⁻¹ in Spec ̃A_(XY) ≅ Speck[x, y, u, u⁻¹]/(ux − y) we see that the effect of gluing Spec ̃A_(X)and Spec ̃A_(Y)together is to add a “point at infinity” on theu-axis above (recall Example 2.12). These two charts glued together constitutes Proj ̃A.

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A geometrically clear way to consider Proj ̃A is as the subset of A²_k× P¹kthat separates lines through the origin in A²_kby their tangent direction (recall Figure 1). Imagine Proj ̃A lying over SpecA = A²k. The blow-up morphism is just the projection down onto the plane. Over the origin we have the projective line (theu-axis in the chart illustrated above), and over any other point there is a unique corresponding point in Proj ̃A. So Proj ̃A is isomorphic to Spec A outside of the origin (a result that we will revisit generally in Corollary 3.13). However, the entire projective line is mapped onto the origin.

The inverse image of the center of the blow-up is of special interest.

Definition 3.3. Let π∶ ̃X → X be the blow-up of X along Z. The inverse image π⁻¹(Z) = ̃X ×XZ of the center is called theexceptional divisor of the blow-up.

In the example above the exceptional divisor is P¹k. It is locally generated by a regular element:

in Spec ̃A_(X) we make the identification y = ux, hence (x, y) = (x, ux) = (x). In the same manner we obtain that (x, y) = (y) in Spec ̃A_(Y). Indeed, the exceptional divisor is always locally generated by a regular element, as we will later see in Proposition 3.10.

A natural question to ask is then: what happens if we blow up along a closed subscheme corresponding to an ideal that is already generated by a regular element? Before we answer this question we need some better notation for keeping track of the degree of an element in the Rees algebra.

Let A be a ring and I an ideal of A. We will, when necessary, consider ̃A as the A-algebra A = ⊕̃ _d≥0t^dI^d, wheret is a dummy variable and t^d merely follows along to keep track of the degree. If we apply this to Example 3.2, we would writetx y as the element x y of degree 1 instead ofxY or yX.

Lemma 3.4. Let A be a ring and I = ( f ) an ideal of A generated by a regular element f . Then the blow-up Proj ̃A → Spec A along V(I) is an isomorphism.

Proof. We have a surjective homomorphism φ∶ A[U] ↠ ̃A = ⊕_d≥0t^dI^d of gradedA-algebras defined byU ↦ t f ∈ ̃A1. Takeд ∈ ker φ of degree d. Then for some a ∈ A,

д = aU^d ↦at^df^d=0 ∈ ̃A_d,

and since f is regular this implies that д = 0. Hence ker φ = 0 and A[U] ≅ ̃A. Therefore Ã_{(t f )}≅A[U]_(U)≅A, so Proj ̃A ≅ Spec A and the blow-up is an isomorphism.

Example 3.5. We slightly generalize the calculation of ker φ in Example 3.2. Let A = k[x, y] and let f be a non-constant element of k[x]. Set I = ( f , y) and ̃A = ⊕d≥0t^dI^d. We have a surjective homomorphism of gradedA-algebras

φ∶ A[U, V] ↠ ̃A defined by U ↦ t f and V ↦ t y.

By arguing exactly as in Example 3.2 we see that kerφ = ( f V − yU).

The following lemma will be useful in the computations below.

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Lemma 3.6. If A is a positively graded ring and x, y ∈ A two homogeneous elements of the same degree, then

A_{(x y)}≅ (A_(x))_y/x ≅ (A_(y))_x/y. Proof. Let d = deg x = deg y. We want to show that

A_{(x y)}= { a

(x y)ⁿ ∈Ax y∶dega = 2dn}

and

(A_(x))_y/x = { a/x^m

(y/x)ⁿ ∈ (A_x)_y/x ∶dega = dm}

are isomorphic. Consider the map

φ∶ A_x→A_{x y} defined by a

xⁿ ↦ ayⁿ (x y)ⁿ = a

xⁿ.

It is clearly a ring homomorphism. Note thatφ∶ (y/x)ⁿ↦ (y/x)ⁿ ∈A_{x y}, where the latter is a unit.

Hence by the universal property of localization [1, Proposition 3.1],φ induces a homomorphism φ∶ (Ax)_y/x →Ax y such that a/x^m

(y/x)ⁿ ↦ axⁿ x^myⁿ.

SinceA_(x)is a subring ofAxwe can consider the restrictionψ∶ (A_(x))_y/x →Ax yofφ. Then (A_(x))_y/x ∋ a/x^m

(y/x)ⁿ ↦ axⁿ

x^myⁿ ∈A_{(x y)}⊆A_{x y}

since dega = dm (and therefore deg(axⁿ) =deg(x^myⁿ)). We thus have a homomorphism ψ∶ (A_(x))_y/x →A_{(x y)}.

It is injective since ψ∶ a/x^m

(y/x)ⁿ ↦ axⁿ

x^myⁿ =0 ⇔ (x y)^jaxⁿ=0 ∈A, some j ≥ 0

⇔ x^j+ny^ja = 0 ∈ A, some j ≥ 0

⇔ (y

x)

j a

x^m =0 ∈A_x, somej ≥ 0

⇔ a/x^m

(y/x)ⁿ =0 ∈ (Ax)_y/x, so kerψ = 0. Surjectivity is immediate:

(A_(x))_y/x ∋ a/x²ⁿ (y/x)ⁿ ↦

axⁿ x²ⁿyⁿ =

a

(x y)ⁿ ∈A_{(x y)}.

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Example 3.7. We consider an example where the blow-up becomes singular. For an algebraically closed fieldk, let A = k[x, y], I = (x²,y) and ̃A = ⊕_d≥0t^dI^d. Once again we have a surjective homomorphismφ∶ A[U, V] ↠ ̃A of graded A-algebras, this time defined by U ↦ tx²andV ↦ t y.

By Example 3.5 we conclude that kerφ = (x²V − yU). Let u = V/U and v = U/V. By proceeding exactly as in Example 3.2 we obtain

Ã_(tx2)≅k[x, y, u]/(x²u − y) = k[x, u] and Ã_(ty)≅k[x, y, v]/(yv − x²).

We therefore have that one of the patches is isomorphic to A²_kwhile the other patch is isomorphic to Speck[x, y, v]/(yv − x²). Consider the latter surface: as we travel along thev-axis looking at the cross sections parallel to thex y-plane, we get parabolas. If we instead consider the cross sections parallel to theyv-plane for varying x, we find hyperbolas. See the pictures below where a few of the real projections have been drawn.

x y

yv = x²

v y

yv = x²

If we consider cross sections parallel to the plane y = −v we obtain circles (since for some constantc ∈ k, (−v + c)v − x² = 0 ⇔ (v − c/2)²+x² = c²/4). These are conic sections, and Speck[x, y, v]/(yv − x²)is a cone.

v y

Let us investigate the intersection of the patches. By Lemma 3.6 we have Ã_(t2x²y)≅ ( ̃A_(tx2))_{t y/tx}₂ ≅k[x, u]_u =k[x, u, u⁻¹] and

Ã_(t2x²y)≅ ( ̃A_(ty))

tx²/ty≅ (k[x, y, v]/(yv − x²))_s=k[x, y, v, v⁻¹]/(yv − x²) =k[x, v, v⁻¹],

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the final equality following since (yv − x²) = (y − x²v⁻¹)ink[x, y, v, v⁻¹]. Using the identities u = v⁻¹andv = u⁻¹we glue the patches along their intersection Spec ̃A_(t2x²y). Explicitly, we have the homomorphisms

k[x, u]

k[x, u, u⁻¹] k[x, y, u⁻¹]/(yu⁻¹−x²)

ι1

ι2

whereι₂∶y ↦ x²u, and the other elements are canonically included into k[x, u, u⁻¹]. Then Specι₁ and Specι₂determine the gluing maps.

The exceptional divisor of the blow-up is indeed locally generated by a regular element: in the chart Speck[x, u] we have (x²,y) = (x²,ux²) = (x²)and in the chart Speck[x, y, v]/(yv − x²) we have (x²,y) = (yv, y) = (y).

A point of interest on Speck[x, y, v]/(yv − x²): m = (x, y, v) is singular. Consider m/m²as a vector space overk. Since yv − x²∈m²it follows thatx, y and v are linearly independent over k, hence dim_km/m²=3. Thus the cotangent space of m is of a higher dimension than that of the cone, meaning that m is a singular point. We will discuss singularities further in Section 4.

Example 3.8. Let us compute the blow-up of a curve that is singular in the origin. Let A = k[x, y]/(y²−x³)for an algebraically closed fieldk, let I = (x, y) ⊆ A and let ̃A = ⊕_d≥0t^dI^d. The curve SpecA is illustrated below.

x y

y²=x³

We have a surjective gradedA-algebra homomorphism

φ∶ A[U, V] ↠ ̃A defined by U ↦ tx and V ↦ t y,

and we claim that kerφ = (xV − yU, yV − x²U, V²−xU²). The inclusion ⊇ is clear. Thusφ factors uniquely through theA-algebra

B = A/(xV − yU, yV − x²U, V²−xU²).

Letφ∶ B → ̃A denote the induced homomorphism. If we can show that φ is injective it will follow that kerφ = (xV − yU, yV − x²U, V²−xU²).

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Take f ∈ ker φ such that deg f = d. Since UⁱV^jgenerateB as an A-algebra we can write f = д₀(x, y)U^d+д₁(x, y)U^d−1V + ⋯ + д_d(x, y)V^d.

Since we have the relations

y²=x³, yU = xV , yV = x²U, V²=xU², inB we can rewrite f using V²=xU²to obtain

f = д₀^′(x, y)U^d+д₁^′(x, y)U^d−1V . Then considering that yV = x²U we can further rewrite f as

f = д₀^′′(x, y)U^d+д₁^′′(x)U^d−1V

(note thatд^′′₁ is a polynomial only inx). Furthermore, y²=x³yields that f = (h₀(x) + h₁(x)y)U^d+д₁^′′(x)U^d−1V . Finally, sinceyU = xV we have

f = h0(x)U^d+ (д₁^′′(x) + xh1(x))U^d−1V = h0(x)U^d+h2(x)U^d−1V . This gives us that

φ( f ) = h0(x)x^d+h2(x)x^d−1y = 0 ∈ ̃A_d,

but since 1 andy are linearly independent in A as a k[x]-module, this means that h₀(x)x^d =0 andh₂(x)x^d−1=0, implying that f = 0 ∈ B. Hence φ is injective and

kerφ = (xV − yU, yV − x²U, V²−xU²).

By proceeding as in earlier examples we obtain

Ã_(tx)≅A[U, V]_(U)/(xV − yU, yV − x²U, V²−xU²)_(U)

= A[V

U]/(xV

U −y, yV

U −x²,V² U² −x),

Ã_(ty)≅A[U, V]_(V)/(xV − yU, yV − x²U, V²−xU²)_(V)

= A[U

V]/(x − yU

V,y − x²U

V, 1 −xU² V²).

Letu = V/U and v = U/V and simplify the above expressions to

Ã_(tx) ≅k[x, u]/(x − u²) =k[u], Ã_(ty)≅k[y, v]/(yv³−1).

The charts of the blow-up are pictured below.

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x u

u²=x

v y

yv³=1

They are glued along their intersection Spec ̃A_(t2x y)≅Speck[u, v]/(uv − 1) = Spec k[u, u⁻¹], the affine line minus a point. Note thatv ≠ 0 in Spec k[y, v]/(yv³−1) since yv³ =1. Therefore this chart does not contain any information about the blow-up not already present in Speck[u]. We conclude that Proj ̃A ≅ Spec k[u] = A¹_k.

3.2 The exceptional divisor

Definition 3.9. Let X be a scheme and let E be a closed subscheme of X. We say that E is an effective Cartier divisor if there exists an affine neighborhood U = Spec A ⊆ X around any point x ∈ E such that U ∩ E = V( f ) for a regular element f ∈ A.

Proposition 3.10. The exceptional divisor of a blow-up is an effective Cartier divisor.

Proof. Let X = Spec A be an affine scheme and let π∶ ̃X → X be the blow-up of X along the closed subscheme corresponding to the finitely generated idealI = ( f₁, . . . ,f_n) ⊆A. Then ̃X = Proj ̃A where ̃A = ⊕d≥0t^dI^d. Since the irrelevant ideal ̃A₊is generated bytI = ̃A1as anA-algebra, it follows that

Proj ̃A =⋃ⁿ

i=1D₊(t f_i).

Considering thatD₊(t fi) ≅ Spec ̃A_{(t f}_i₎, the restriction ofπ to D₊(t fi)is induced by the ring homomorphismA → ̃A_{(t f}_i₎. If we letπi =π∣_D₊_{(t f}_i₎we obtain

π⁻¹_i (V(I)) = V(I ̃A_{(t f}_i₎) ⊆Spec ̃A_{(t f}_i₎. Since fj=t fj(t fi)⁻¹fi∈ ̃A_{(t f}_i₎, it follows thatI ̃A_{(t f}_i₎= (fi), hence

Spec ̃A_{(t f}_i₎∩π⁻¹(V(I)) = V( fi).

Furthermore,

д (t f_i)^m

fi

1 =0 ∈ ̃A_{(t f}_i₎ ⇔ (t fi)^kд fi =0 for somek ≥ 0.

This implies that

(t f_i)^kд = 0 for some k ≥ 0 ⇔ д (t fi)^m

=0, therefore f_i ∈ ̃A_{(t f}_i₎is a regular element.

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Lemma 3.11. Let E be an effective Cartier divisor of a scheme X. Then X ∖ E is schematically dense in X.

Proof. An effective Cartier divisor E is locally generated by one regular element. Specifically, we may take an open affine SpecA ⊆ X such that Spec A ∩ E = V( f ) for a regular element f ∈ A. If SpecA ∖ V( f ) ≠ Spec A there exists a closed subscheme Spec A/I of Spec A such that the canonical inclusion SpecA ∖ V( f ) = D( f ) ≅ Spec A_f →SpecA factors through Spec A/I. This would be induced by the ring homomorphisms

A → A/I → A_f,

butf is a regular element of A so A → A_f is injective. HenceI = 0 and Spec A ∖ E = Spec A.

The conclusionX ∖ E = X follows since there exists an affine cover of X where each patch satisfies the above.

Remark. By Proposition 3.10 and Lemma 3.11, the complement of the exceptional divisor is dense in the blow-up.

3.3 Fibers of blow-ups

Proposition 3.12. Let X = Spec A be an affine scheme, let Z be the closed subscheme of X corre- sponding to a finitely generated ideal I ⊆ A, and let π∶ ̃X → X be the blow-up of X along Z. Let Y = Spec B be a closed subscheme of X and let ̃Y → Y be the blow-up of Y along Y ∩ Z.

(i) The canonical morphism ̃Y → ̃X ×_XY is a closed immersion.

(ii) If B is a flat over A, then ̃Y ^∼ X ×̃ XY is an isomorphism.

Proof. We have ̃X = Proj ̃A where ̃A = ⊕_d≥0I^dand ̃Y = Proj ̃B where ̃B = ⊕_d≥0I^dB. Let C = ̃A⊗ B and considerC to be a graded A-algebra with degree d part C_d =I^d⊗_AB. We then have a canonical surjective graded homomorphism

φ∶ ̃A ⊗AB ≅ ⊕

d≥0

(I^d⊗_AB) → ⊕

d≥0

I^dB = ̃B.

By Lemma 2.11, this induces the closed immersion

ψ∶ Proj ̃B → Proj( ̃A ⊗_AB) defined by p ↦ φ⁻¹(p).

Then by Proposition 2.18 we have the isomorphism

Proj( ̃A ⊗_AB) ^∼ Proj ̃A ×_Spec_ASpecB, which combined withψ yields the first result.

If B is a flat A-algebra, Lemma 2.15 gives us ̃B ≅ ̃A ⊗_A B and the second result follows by Proposition 2.18.

Corollary 3.13. Let X be an affine scheme and let π∶ ̃X → X be the blow-up of X along Z. Then π⁻¹(X ∖ Z) ≅ X ∖ Z.

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Proof. Let X = Spec A and let I = ( f₁, . . . ,f_n)be the finitely generated ideal such thatZ = V(I).

WriteX ∖ Z as a union of principal open sets:

X ∖ V(I) =⋃ⁿ

i=1D( f_i).

By checking the isomorphism on each principal open set the global isomorphism follows. As usual we identifyD( fi) ≅SpecA_f_i. SinceA_f_i is flat overA [1, Corollary 3.6], Proposition 3.12 yields

π⁻¹(D( fi)) ≅ ̃X ×XSpecA_f_i ≅Proj ̃A_f_i,

where Proj ̃A_f_i is the blow-up of SpecA_f_i alongV(IA_f_i). ButIA_f_i = A_f_i since fi ∈ I, so the blow-up of SpecA_f_i is alongV(1). Then by Lemma 3.4 we have

Proj ̃A_f_i ≅SpecA_f_i.

Remark. Note the importance of the above corollary: the blow-up is an isomorphism outside of the center.

Lemma 3.14. Let X = Spec A be an affine scheme and let Z = V(I) and Y = V(J) be two closed subschemes, with I ⊆ A a finitely generated ideal. If π∶ ̃X → X is the blow-up of X along Z, then

π⁻¹(Y) ≅ Proj(⊕

d≥0

I^d/JI^d).

Proof. We defined the inverse image to be π⁻¹(Y) = ̃X ×XY. Then by Proposition 2.18 we have π⁻¹(Y) = ̃X ×_XY ≅ Proj( ̃A ⊗_AA/J),

and Lemma 2.14 yields

A ⊗̃ AA/J = (⊕

d≥0I^d) ⊗_AA/J ≅ ⊕

d≥0(I^d⊗_AA/J) ≅ ⊕

d≥0I^d/JI^d.

Example 3.15. Let us return to Example 3.2 and consider the fiber of π over the origin. Recall that we haveA = k[x, y], I = (x, y), ̃A = ⊕_d≥0I^dand the blow-upπ∶ Proj ̃A → Spec A. By Lemma 3.14 we immediately get that the exceptional divisor is

π⁻¹(V(I)) ≅ Proj(⊕

d≥0

I^d/I^d+1).

But ⊕_d≥0I^d/I^d+1≅k[x, y], hence π⁻¹(V(I)) ≅ Proj k[x, y] = P¹k, as expected.

Another way to see this is to instead use our affine cover of Proj ̃A. Remember that we have the patches Spec ̃A_(X)≅Speck[x, u] with y = ux and Spec ̃A_(Y)≅Speck[y, v] with x = v y. On the patch Spec ̃A_(X)we have the diagram

Spec ̃A_(X)×_Spec_ASpecA/I Spec ̃A_(X)

SpecA/I SpecA.

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By Proposition 2.17 we have Spec ̃A_(X)×_Spec_ASpecA/I ≅ Spec( ̃A_(X)⊗_AA/I), thus the diagram above is obtained by applying Spec to

Ã_(X)⊗_AA/I Ã_(X)

A/I A.

Since ̃A_(X)≅k[x, u] with y = ux, Lemma 2.14 gives us

Ã_(X)⊗_AA/I ≅ k[x, u] ⊗_AA/I ≅ k[x, u]/(x, ux) = k[u].

The same considerations over the patch Spec ̃A_(Y)yields ̃A_(Y)⊗_AA/I ≅ k[v] where v = u⁻¹. Hence in each patch the fiber over the origin is A¹k. These lines are then glued together as in Example 2.12, giving usπ⁻¹(V(I)) = P¹_k.

Example 3.16. Let us consider the fiber over the origin of the blow-up in Example 3.7. Recall the situation: we have the ringA = k[x, y], the ideal I = (x²,y) and the blow-up π∶ Proj ̃A → Spec A of SpecA along V(I). If m = (x, y), then by Lemma 3.14 the fiber over the origin is

π⁻¹(V(m)) ≅ Proj(⊕

d≥0

I^d/mI^d) =Projk[x²,y]

where degx²=degy = 1. Thus we obtain that the fiber over the origin is P¹_kin this case too.

We turn our attention to the exceptional divisorπ⁻¹(V(I)) = Proj ̃A×_Spec_ASpecA/I. To simplify the computations we use our affine cover of Proj ̃A which consists of the two patches

Spec ̃A_(tx)≅Speck[x, u] and Spec ̃A_(ty)≅Speck[y, v]/(yv − x²).

Proposition 2.17 leads us to compute

Ã_(tx)⊗_AA/I ≅ k[x, u]/(x²) ≅k[x]/(x²) ⊗_kk[u],

Ã_(ty)⊗_AA/I ≅ k[x, y, v]/(v y − x²,x²,y) ≅ k[x]/(x²) ⊗_kk[v].

Hence the fiber in the respective patch is

Spec ̃A_(tx)×_Spec_ASpecA/I ≅ Spec k[x]/(x²) ×_Spec_kSpeck[u], Spec ̃A_(ty)×_Spec_ASpecA/I ≅ Spec k[x]/(x²) ×_Spec_kSpeck[v].

The two affine lines glue together to form a projective line, and we conclude that the exceptional divisor is

π⁻¹(V(I)) ≅ P¹k×_Spec_kSpeck[x]/(x²).

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3.4 The strict transform

Definition 3.17. Let X be an affine scheme, Z a finitely presented closed subscheme of X and let π∶ ̃X → X be the blow-up of X along Z. Let Y be a closed subscheme of X and let ψ∶ ̃X ×XY → Y be the second projection from the fiber product. We define thestrict transform of Y under the blow-upπ to be the closure of ψ⁻¹(Y ∖ (Y ∩ Z)) in ̃X ×_XY. We define the total transform of Y underπ to be ψ⁻¹(Y) = ̃X ×XY.

Proposition 3.18. Let X be an affine scheme and let Y and Z be closed subschemes of X, with Z finitely presented. Let π∶ ̃X → X be the blow-up of X along Z. Then the strict transform of Y under π is isomorphic to the blow-up of Y along the closed subscheme Y ∩ Z.

ψ⁻¹(Y ∖ (Y ∩ Z)) ≅ ̃Y X ×̃ XY X̃

Y X

ψ π

Proof. Let π∶ ̃X → X be the blow-up of X along Z and let π^′∶ ̃Y → Y be the blow-up of Y along Y ∩ Z. By Proposition 3.12 we have a closed immersion f ∶ ̃Y → ̃X ×_XY, and by its construction we see that the following diagram commutes:

Ỹ X ×̃ XY X̃

Y X.

f

π^′ ψ π

ι

LetU = X ∖ Z and V = Y ∖ (Y ∩ Z). Note that V = ι⁻¹(U) ≅ Y ×XU. Furthermore, ψ⁻¹(V) ≅ ( ̃X ×XY) ×Y V and by Corollary 3.13 we have U ≅ π⁻¹(U) ≅ ̃X ×XU.

Combining the observations above together with Lemma 2.19 yields

ψ⁻¹(V) ≅ ̃X ×XV ≅ ̃X ×X(Y ×XU) ≅ Y ×X( ̃X ×XU) ≅ Y ×XU ≅ V ,

so f (π^′−1(V)) ≅ ψ⁻¹(V) since π^′ = ψ ○ f and π^′−1(V) ≅ V. Considering that f is a closed immersion andπ^′−1(V) = ̃Y by Lemma 3.11, we conclude that ψ⁻¹(V) is isomorphic to ̃Y.

Example 3.19. We will informally compute the blow-up of Spec k[x, y]/(y²−x³)in the origin using the strict transform (compare with Example 3.8). Later, in Proposition 4.10, we return to the problem a bit more generally and do the formal computations.

LetB = k[x, y]/(y²−x³)for an algebraically closed fieldk and let π∶ Proj ̃A → Spec A be the blow-up of A²kin the origin from Example 3.2. Then by Proposition 3.18 the desired blow-up is isomorphic to the closure ofπ⁻¹(SpecB ∖ V(x, y)).

Recall that Proj ̃A was constructed by gluing together two affine charts. One of the charts was Speck[x, u] with y = ux and the other chart was Spec k[y, v] with x = v y (where v = u⁻¹). In the former chart we have the total transform

Speck[x, u]/(y²−x³) ≅V(y²−x³) =V((x²)(u²−x)) = V(x²) ∪V(u²−x).

Blow-ups and orders of vanishing