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✓ 39 results · all verified · 25 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 14 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Blowups, Exceptional Divisors, and Strict Transforms

1 · Prerequisites

2 · Summary

This page develops the blowup of a scheme along a quasi-coherent ideal sheaf of finite type, together with its exceptional subscheme, its functorial universal property, its behaviour under base change and restriction, and the strict and total transforms of closed subschemes and divisors.

For a scheme $X$ and a finite-type quasi-coherent ideal $I$ one forms the Rees algebra sheaf $R (I) = ⨁_{n \geq 0} I^{n}$ and sets ${Bl}_{I} X = {\underset{}{Proj}}_{X} R (I)$ ; on an affine chart $Spec A$ with $I = (f_{0}, \dots, f_{r})$ the blowup is covered by the standard charts $Spec A [I / f_{i}]$ , and the chart rings are the affine blowup algebras, described by the normal form $b / a^{n}$ inside $A_{a}$ and by the polynomial presentation $A [x_{1}, \dots, x_{r}] / (a x_{i} - a_{i})$ modulo its $a$ -power torsion. The exceptional subscheme $E = π^{- 1} V (I)$ is the scheme-theoretic preimage of the centre; when the centre is a regular immersion its normal cone is the symmetric algebra of $I / I^{2}$ by the theory of regular sequences, so $E$ is the projectivised normal bundle of the centre in $X$ .

The page then specialises to a closed point on a regular finite-type surface of pure dimension two over a field, where the blowup is again regular, the exceptional curve is a projective line over the residue field of the centre, and its normal sheaf has degree $- 1$ . The chart computations are used to compute strict transforms of plane curves, to show that a point blowup separates tangent directions, to prove the multiplicity recurrence $π^{*} C = C^{'} + m E$ , and to track the normalization defect $δ_{k}$ and the contact order of regular branches under blowups. These are the local inputs to the final resolution theorem, which resolves a reduced projective plane curve over an arbitrary field into regular embedded normal-crossing support by repeated point blowups. Choice conventions, the case of inseparable residue fields, and the boundary of the claims (no resolution in higher dimension, no relative smoothness over imperfect fields) are recorded explicitly at the items where they occur.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Literature-sourcedjudge pass (gpt-6.1-sol)Open item page →

Rees algebra sheaf of a finite type ideal

Definition

Assume the Axiom of Choice, inherited from the quasi-coherence suppliers used below (The Axiom of Choice).

Let $X$ be a scheme and let $I \subseteq O_{X}$ be a quasi-coherent ideal sheaf of finite type (Quasi-coherent ideal sheaves). Put $I^{0} : = O_{X}$ , and for $n \geq 1$ let $I^{n}$ be the $n$ -fold product of the ideal sheaf $I$ inside $O_{X}$ , that is, the image of the multiplication map $I^{\otimes n} \to O_{X}$ (equivalently, the ideal sheaf generated by all local products of $n$ sections of $I$ ). The Rees algebra sheaf of $I$ is the sheaf of graded $O_{X}$ -algebras

$R (I) : = ⨁_{n \geq 0} I^{n},$

whose degree- $n$ piece is the ideal sheaf $I^{n}$ , whose multiplication $I^{m} \otimes_{O_{X}} I^{n} \to I^{m + n}$ is induced by multiplication in $O_{X}$ , and whose unit is the identification $O_{X} = I^{0}$ . Since multiplication of ideals is associative and commutative and $I^{m} I^{n} \subseteq I^{m + n}$ with equality for the product ideal, $R (I)$ is a commutative graded $O_{X}$ -algebra with $R (I)_{0} = O_{X}$ and $R (I)_{1} = I$ .

The construction is local on $X$ and agrees with the affine Rees algebra: if $U = Spec A \subseteq X$ is affine and $I ∣_{U} = \tilde{I}$ for an ideal $I \subseteq A$ (Quasi-coherent ideal sheaves), then $I^{n} ∣_{U} = \tilde{I^{n}}$ for every $n \geq 0$ , and taking sections on $U$ gives the affine Rees algebra $⨁_{n \geq 0} I^{n} t^{n}$ of (The Rees algebra of an ideal and the Rees module of a filtered module) with its degree- $n$ piece $I^{n}$ . Two affine covers therefore glue to canonically isomorphic sheaves, and the graded pieces are the ideal powers defined above.

The following properties are part of the definition and are used by the consumers of this item.

The powers are quasi-coherent. Each $I^{n}$ is a quasi-coherent $O_{X}$ -module (Quasi-coherent module on a scheme). Indeed, cover $X$ by affine opens $U = Spec A$ with $I ∣_{U} = \tilde{I}$ ; then $I^{n} ∣_{U} = \tilde{I^{n}}$ is the associated sheaf of an $A$ -module, and quasi-coherence is local on $X$ . Alternatively, for $n \geq 1$ the multiplication map $I^{n - 1} \otimes_{O_{X}} I \to I^{n}$ is surjective from a quasi-coherent source by Tensor product preserves quasi-coherence.
The Rees algebra sheaf is quasi-coherent. On each affine chart $U = Spec A$ one has $R (I) ∣_{U} = \tilde{⨁_{n \geq 0} I^{n}}$ , the associated sheaf of the graded $A$ -module $⨁_{n \geq 0} I^{n}$ ; since quasi-coherence is local on $X$ , the direct sum of the quasi-coherent sheaves $I^{n}$ is quasi-coherent.
Degree-one generation. $R (I)_{0} = O_{X}$ and $R (I)_{1} = I$ generate $R (I)$ as an $O_{X}$ -algebra: for every $n \geq 1$ the product map $I^{\otimes n} \to I^{n}$ is surjective, because $I^{n}$ is by construction generated by products of $n$ local sections of $I$ . Equivalently, the canonical graded $O_{X}$ -algebra homomorphism $⨁_{n \geq 0} {Sym}_{O_{X}}^{n} (I) \to R (I)$ is surjective. The finite type hypothesis is used in subsequent finite-type structural results, not in the relative Proj construction; the Rees algebra sheaf and its relative Proj are defined for any quasi-coherent ideal.

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

Blowup of a scheme along an ideal sheaf

Definition

Assume the Axiom of Choice as inherited from the relative Proj construction (The Axiom of Choice). Let $X$ be a scheme and let $I \subseteq O_{X}$ be a quasi-coherent ideal sheaf of finite type (Quasi-coherent ideal sheaves), with zero scheme $Z = V (I)$ , the closed subscheme of $X$ cut out by $I$ . The blowup of $X$ along $I$ (or along $Z$ ) is the $X$ -scheme

${Bl}_{I} X : = {Proj}_{X} R (I),$

the relative Proj of Relative Proj of a graded quasi-coherent algebra applied to the Rees algebra sheaf $R (I) = ⨁_{n \geq 0} I^{n}$ of Rees algebra sheaf of a finite type ideal, equipped with its structural morphism

$π ⁣ : {Bl}_{I} X ⟶ X$

and its relative twists $O (n)$ , $n \in Z$ , both as in Relative Proj of a graded quasi-coherent algebra. The notation records the ideal sheaf $I$ , not merely the closed subscheme $Z$ ; the finite type hypothesis is part of the definition because the later structural results for blowups (invertibility of the pullback of $I$ , the exceptional divisor, and base change) are proved under it.

In the affine case $X = Spec A$ with $I = \tilde{I}$ for an ideal $I \subseteq A$ , the Rees algebra sheaf is $R (I) ≅ \tilde{⨁_{n \geq 0} I^{n}}$ (Rees algebra sheaf of a finite type ideal), and the absolute case of the relative Proj construction identifies ${Bl}_{I} X$ with the absolute Proj of the Rees algebra $R (I) = ⨁_{n \geq 0} I^{n} t^{n}$ (Relative Proj of a graded quasi-coherent algebra); the structural morphism is then the Proj structural morphism to $Spec A$ .

The exceptional subscheme of the blowup is denoted $E$ and is introduced separately; no property of $E$ is assumed here.

Remarks

The Axiom of Choice is inherited from the affine-local Proj construction used by Relative Proj of a graded quasi-coherent algebra; the blowup selects no further data beyond that interface.
This item only sets up the construction. Projectivity of $π$ , the universal property of the blowup, the invertibility of the pullback of $I$ , and flat base change are supplied by later items of this page and are not asserted here.

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

Exceptional subscheme of a blowup

Definition

Let $X$ be a scheme, let $I \subseteq O_{X}$ be a quasi-coherent ideal sheaf (Quasi-coherent ideal sheaves) with zero scheme $Z = V (I) ↪ X$ , the closed subscheme cut out by $I$ (Closed immersions of schemes), and let $π ⁣ : {Bl}_{I} X \to X$ be the blowup of Blowup of a scheme along an ideal sheaf. The exceptional subscheme of the blowup is the scheme-theoretic inverse image

$E : = π^{- 1} (Z) = Z \times_{X} {Bl}_{I} X$

of Scheme-theoretic inverse images of subschemes (Base change of objects, morphisms and properties); it is a closed subscheme $E ↪ {Bl}_{I} X$ , its ideal sheaf is the inverse image ideal $I O_{{Bl}_{I} X} : = Im (π^{*} I \to O_{{Bl}_{I} X})$ , and the structural morphism of the blowup restricts to a morphism $E \to Z$ . Set-theoretically, $E$ is the preimage $π^{- 1} (Z)$ of the underlying set of $Z$ .

Remarks

For an ideal not of finite type, the notation here extends Blowup of a scheme along an ideal sheaf by using ${Proj}_{X} (⨁_{n \geq 0} I^{n})$ directly. Its graded algebra is quasi-coherent by the affine ideal-power calculation of Rees algebra sheaf of a finite type ideal, and no finite generation is required by Relative Proj of a graded quasi-coherent algebra.

The exceptional subscheme is defined for every quasi-coherent ideal sheaf $I$ on $X$ ; no smoothness of $X$ , regularity of $Z$ , or invertibility of $I$ is assumed.
The construction inherits the Axiom of Choice from the blowup (Blowup of a scheme along an ideal sheaf), and no further data is chosen.
Nothing is asserted here about the components of $E$ , its codimension in the blowup, or the invertibility of its ideal sheaf; those are supplied by later items of this page.

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Associated graded algebra of an ideal generated by a regular sequence

Statement

Let $R$ be a commutative ring and let $f_{1}, \dots, f_{c}$ be an $R$ -regular sequence (Regular Sequence On A Module), with $J = (f_{1}, \dots, f_{c})$ (The ideal generated by a subset and principal ideals) and associated graded ring ${gr}_{J} R = ⨁_{n \geq 0} J^{n} / J^{n + 1}$ (The associated graded ring and associated graded module of an ideal-adic filtration). The canonical graded homomorphism

$(R / J) [X_{1}, \dots, X_{c}] ⟶ {gr}_{J} R, X_{i} ⟼ f_{i} m o d J^{2},$

is an isomorphism. In particular $J / J^{2}$ is free over $R / J$ on the classes of $f_{1}, \dots, f_{c}$ , and ${Sym}_{R / J} (J / J^{2}) = {gr}_{J} R$ . No Noetherian or domain hypothesis is required.

Facts & Assumptions

Given: A commutative ring $R$ , an $R$ -regular sequence $f_{1}, \dots, f_{c} \in R$ (Regular Sequence On A Module) and the ideal $J = (f_{1}, \dots, f_{c})$ (The ideal generated by a subset and principal ideals) with its associated graded ring ${gr}_{J} R$ (The associated graded ring and associated graded module of an ideal-adic filtration).

[F1]

Regular Sequence On A Module: A finite ordered sequence $f_{1}, \dots, f_{c}$ in $R$ is $R$ -regular when $R / (f_{1}, \dots, f_{i - 1}) \neq 0$ and multiplication by $f_{i}$ is injective on $R / (f_{1}, \dots, f_{i - 1})$ for every $i$ , and $R / (f_{1}, \dots, f_{c}) \neq 0$ . In particular the truncated sequence $f_{1}, \dots, f_{c - 1}$ is likewise $R$ -regular, and $f_{c}$ is a nonzerodivisor on $R / J^{'}$ for $J^{'} = (f_{1}, \dots, f_{c - 1})$ .

[F2]

The associated graded ring and associated graded module of an ideal-adic filtration: For an ideal $J \subseteq R$ , the associated graded ring is ${gr}_{J} R = ⨁_{n \geq 0} J^{n} / J^{n + 1}$ with multiplication $(a + J^{m + 1}) (b + J^{n + 1}) = a b + J^{m + n + 1}$ ; in particular it is commutative and generated in degree one.

[F3]

The ideal generated by a subset and principal ideals: $J = (f_{1}, \dots, f_{c})$ is the ideal generated by the $f_{i}$ : it consists of the finite sums $\sum_{i} c_{i} f_{i}$ with $c_{i} \in R$ , and each $f_{i} \in J$ .

Proof

1.1F2F3

The degree- $n$ monomials in the $f_{i}$ generate $J^{n}$ , so the displayed graded map is surjective. To prove injectivity it suffices to show that every homogeneous relation $\sum_{∣ I ∣ = n} a_{I} f^{I} = 0$ has all $a_{I} \in J$ : a relation lying in $J^{n + 1}$ can be made zero by subtracting a degree- $n$ expression whose coefficients lie in $J$ . We prove this coefficient assertion by induction on the sequence length $c$ . For $c = 0$ , $J = 0$ and the associated graded ring has only degree zero, where the map is the identity.

2.1F1step 1.1

Suppose $c > 0$ and the assertion holds for $J^{'} = (f_{1}, \dots, f_{c - 1})$ . Fix $n$ and write the relation as $\sum_{e = 0}^{l} H_{e} f_{c}^{e} = 0$ , where $H_{e} = \sum_{∣ I^{'} ∣ = n - e} a_{I^{'}, e} f^{I^{'}}$ and $l \leq n$ . We induct on $l$ . If $l = 0$ , the assertion for $J^{'}$ puts all coefficients in $J^{'} \subset J$ . If $l > 0$ , reduction modulo $(J^{'})^{n - l + 1}$ gives $f_{c}^{l} H_{l} \in (J^{'})^{n - l + 1}$ . By the coefficient assertion for $J^{'}$ (also applicable to an expression lying in the next power), $f_{c}^{l} a_{I^{'}, l} \in J^{'}$ for every $I^{'}$ . Since $f_{c}$ is a nonzerodivisor on $R / J^{'}$ , every $a_{I^{'}, l}$ lies in $J^{'}$ .

3.1F3step 2.1

Consequently $H_{l} \in (J^{'})^{n - l + 1}$ ; express it as $\sum_{∣ I^{''} ∣ = n - l + 1} b_{I^{''}} f^{I^{''}}$ . Absorb $f_{c} H_{l}$ into $H_{l - 1}$ and remove the term with exponent $l$ . This gives a relation of the same degree with largest exponent $l - 1$ . Its modified coefficients are $a_{I^{''}, l - 1} + f_{c} b_{I^{''}}$ and its other coefficients are unchanged. The induction on $l$ puts all modified coefficients in $J$ , hence all original ones in $J$ , since $f_{c} b_{I^{''}} \in J$ . Together with the coefficients of $H_{l}$ , this proves the assertion and completes the induction on $c$ .

4.1F1step 1.1step 3.1∎

The coefficient assertion proves injectivity in every degree. In degree one, the isomorphism identifies $(R / J)^{c}$ with $J / J^{2}$ on the displayed classes. The symmetric algebra of this free module is the polynomial algebra, yielding the canonical symmetric-algebra identification. Only regularity of the ordered sequence and ideal-power generation were used.

Remarks

The double induction is on the length $c$ of the sequence (step 3.1) and, inside a fixed length, on the highest power $l$ of $f_{c}$ occurring in the relation (step 4.1). The lemma is the algebraic input to the computation of the normal cone of a regular immersion and to the identification of the associated graded algebras ${gr}_{J} R$ used in the deformation $δ_{k} (C^{'}) = δ_{k} (C) - r (\binom{m}{2})$ of the resolution theorem.

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Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains

Statement

Let $A$ be a commutative ring, let $I \subseteq A$ be an ideal and let $a \in I$ . Write $R (I) = ⨁_{n \geq 0} I^{n} t^{n} \subseteq A [t]$ for the Rees algebra of $I$ , graded by the degree of $t$ (The Rees algebra of an ideal and the Rees module of a filtered module, Nonnegatively graded rings and modules, homogeneous elements, and twists). The affine blowup algebra is

$A [I / a] : = (R (I))_{(a)},$

the degree-zero part of the localization of $R (I)$ at the multiplicative set generated by the degree-one element $a t \in I t$ (Multiplicative subsets and the localisation $S^{- 1} R$ as equivalence classes of fractions). Then:

the image of $a$ in $A [I / a]$ is a nonzerodivisor, $I A [I / a] = a A [I / a]$ , and $(A [I / a])_{a} = A_{a}$ ;
if $I = (a_{0}, \dots, a_{r})$ with $a = a_{0}$ (The ideal generated by a subset and principal ideals), the homomorphism $A [x_{1}, \dots, x_{r}] / (a x_{i} - a_{i}) ⟶ A [I / a]$ sending $x_{i} \mapsto a_{i} / a$ is surjective with kernel the $a$ -power torsion;
if $A$ is reduced then $A [I / a]$ is reduced; if $A$ is a domain and $a \neq 0$ then $A [I / a]$ is a domain;
the construction is independent of the generating set and of the representative used for the homogeneous localization, up to canonical $A$ -algebra isomorphism.

Facts & Assumptions

Given: A commutative ring $A$ , an ideal $I \subseteq A$ and an element $a \in I$ ; the Rees algebra $R (I) = ⨁_{n \geq 0} I^{n} t^{n} \subseteq A [t]$ of (The Rees algebra of an ideal and the Rees module of a filtered module), localized at the multiplicative set generated by the degree-one element $a t$ (Multiplicative subsets and the localisation $S^{- 1} R$ as equivalence classes of fractions), with grades read in the sense of (Nonnegatively graded rings and modules, homogeneous elements, and twists).

[F1]

The Rees algebra of an ideal and the Rees module of a filtered module: For an ideal $I$ of a commutative ring $A$ , the Rees algebra is the graded subring $R (I) = ⨁_{n \geq 0} I^{n} t^{n} \subseteq A [t]$ , whose degree- $n$ piece is $I^{n}$ (so $R (I)_{n} = I^{n} t^{n}$ ), with $I^{0} = A$ .

[F2]

Multiplicative subsets and the localisation $S^{- 1} R$ as equivalence classes of fractions: For a commutative ring $R$ and a multiplicative subset $S \subseteq R$ , the localization $S^{- 1} R$ has elements written $r / s$ with $(r, s) \sim (r^{'}, s^{'})$ iff $u (r s^{'} - r^{'} s) = 0$ for some $u \in S$ ; the operations are $r / s + r^{'} / s^{'} = (r s^{'} + r^{'} s) / (s s^{'})$ and $(r / s) (r^{'} / s^{'}) = r r^{'} / (s s^{'})$ ; every $s \in S$ maps to a unit; if $0 \in S$ the localization is the zero ring.

[F3]

Nonnegatively graded rings and modules, homogeneous elements, and twists: A nonnegatively graded ring is a commutative ring $S = ⨁_{n \geq 0} S_{n}$ with $S_{n} S_{m} \subseteq S_{n + m}$ ; an element of $S_{n}$ is homogeneous of degree $n$ . In particular the degree-zero part of a $Z$ -graded ring is closed under addition, multiplication and contains the unit class of the localization, and products of homogeneous elements add degrees.

[F4]

The ideal generated by a subset and principal ideals: If $I = (a_{0}, \dots, a_{r})$ then every element of $I$ is a finite sum $\sum_{i} c_{i} a_{i}$ with $c_{i} \in A$ , and conversely each $a_{i}$ lies in $I$ .

Proof

1.1F1F2F3

An element of the localization $R (I)_{a t}$ is a class $z / (a t)^{n}$ with $z \in R (I)$ and $n \geq 0$ ; writing $z = \sum_{i} b_{i} t^{i}$ , its degree-zero component is $b_{n} t^{n} / (a t)^{n}$ with $b_{n} \in I^{n}$ . Hence the degree-zero part $D = (R (I))_{(a)}$ consists of the classes of $b t^{n} / (a t)^{n}$ with $n \geq 0$ and $b \in I^{n}$ , so that $A [I / a] = D$ ; for $b \in I^{n}$ , $c \in I^{m}$ one has $b t^{n} / (a t)^{n} = c t^{m} / (a t)^{m}$ in $D$ if and only if $a^{k} (a^{m} b - a^{n} c) = 0$ for some $k \geq 0$ , because the defining relation is $(a t)^{k} ((a t)^{m} b t^{n} - (a t)^{n} c t^{m}) = 0$ and $t$ is a nonzerodivisor of $A [t]$ .

2.1F1F2step 1.1

The assignment $ι ⁣ : D \to A_{a}$ , $b t^{n} / (a t)^{n} \mapsto b / a^{n}$ , is a well-defined injective ring homomorphism: well-defined and injective because $b / a^{n} = c / a^{m}$ in $A_{a}$ holds exactly when $a^{k} (a^{m} b - a^{n} c) = 0$ for some $k \geq 0$ , i.e. exactly under the equality criterion of step 1.1; compatible with addition because $(b t^{n}) / (a t)^{n} + (c t^{m}) / (a t)^{m} = (a^{m} b + a^{n} c) t^{n + m} / (a t)^{n + m}$ and with multiplication because the product is $b c t^{n + m} / (a t)^{n + m}$ . Its image is the $A$ -subalgebra $B = A [b / a : b \in I] \subseteq A_{a}$ generated by the fractions $b / a$ : every $b / a$ is the image of $b t / (a t)$ , and conversely every $b \in I^{n}$ is a finite sum of products of $n$ elements of $I$ , so $b / a^{n}$ is a sum of products of these generators (with coefficients from $A$ ). Hence $A [I / a] ≅ B$ as $A$ -algebras and all computations may be performed in $B \subseteq A_{a}$ .

3.1F1step 2.1

The element $a$ is invertible in $A_{a}$ , hence a nonzerodivisor on $A_{a}$ , hence a nonzerodivisor on the subring $B$ ; in particular the image of $a$ in $A [I / a]$ is a nonzerodivisor. Moreover $a B \subseteq I B$ because $a \in I$ , and $I B \subseteq a B$ because for $b \in I$ and $c / a^{n} \in B$ with $c \in I^{n}$ one has $b \cdot c / a^{n} = a \cdot b c / a^{n + 1}$ with $b c \in I^{n + 1}$ ; hence $I A [I / a] = a A [I / a]$ .

3.2F2step 2.1

The image of $A$ lies in $B$ , since $c = a c / a$ for $c \in A$ and $a c \in I$ ; thus $A_{a} = A [1 / a] \subseteq B_{a}$ , and the inclusion $B \subseteq A_{a}$ induces $B_{a} \subseteq (A_{a})_{a} = A_{a}$ . Hence $B_{a} = A_{a}$ , that is, $(A [I / a])_{a} = A_{a}$ .

3.3F4step 2.1

Let $φ ⁣ : C = A [x_{1}, \dots, x_{r}] / (a x_{i} - a_{i}) \to B$ send $x_{i}$ to $a_{i} / a$ . Its image contains $A$ and every $a_{i} / a$ ; since $I$ is generated by the $a_{i}$ , every $b \in I$ is a finite sum $\sum_{i} c_{i} a_{i}$ , whence $b / a = \sum_{i} c_{i} (a_{i} / a)$ lies in the image; as the image is an $A$ -subalgebra of $A_{a}$ containing all $b / a$ it contains $B$ . So $φ$ is surjective.

3.4F2step 2.1

If $A$ is reduced, then so is the localization $A_{a}$ , and $B \subseteq A_{a}$ is a subring of a reduced ring, hence reduced: if $x^{N} = 0$ in $B$ then $x = 0$ already in $A_{a}$ . If $A$ is a domain and $a \neq 0$ , then $A_{a} \subseteq Frac (A)$ is a nonzero subring of a field (nonzero because $a \cdot a^{- 1} = 1$ with $a \neq 0$ ), hence a domain, and so is its subring $B$ .

4.1F2step 3.3

Localizing $C$ at the multiplicative set generated by $a$ gives $C_{a} = A_{a} [x_{1}, \dots, x_{r}] / (a x_{i} - a_{i})$ in which $a$ is a unit and hence $x_{i} = a_{i} a^{- 1}$ ; the $A_{a}$ -algebra homomorphism $C_{a} \to A_{a}$ sending $x_{i} \mapsto a_{i} / a$ is therefore surjective and has zero kernel, because evaluation at $x_{i} = a_{i} / a$ identifies the quotient $A_{a} [x_{i}] / (x_{i} - a_{i} / a)$ with $A_{a}$ . Since the image of $C$ in $A_{a}$ is $B$ by step 3.3, $\ker (C \to B) = \ker (C \to C_{a})$ , and by the defining relation of the localization an element $f \in C$ lies in this kernel exactly when $a^{N} f = 0$ for some $N \geq 0$ , i.e. exactly when $f$ is $a$ -power torsion.

5.1step 1.1step 2.1step 3.3step 4.1∎

The description $B = {b / a^{n} : n \geq 0, b \in I^{n}}$ of step 2.1 uses only the pair $(I, a)$ : no generating family is chosen, and a class in $D$ is compared with another by the equality criterion of step 1.1, so the use of a particular fraction representative is immaterial. A finite generating family $I = (a_{0}, \dots, a_{r})$ with $a = a_{0}$ only produces the presentation $C$ of step 3.3, whose kernel is described in step 4.1; the image $B$ and hence the algebra are the same for every such family, with the identity of $B$ as the canonical isomorphism. Finally, for $a^{'} \in I$ with $a^{'} \neq a$ the algebras $B \subseteq A_{a}$ and $B^{'} \subseteq A_{a^{'}}$ lie in different localizations of $A$ ; the statement asserts no identification between them, and the transition maps between the corresponding charts are supplied separately by the standard-chart theorem.

Remarks

The vanishing of $A [I / a]$ is allowed: if $a = 0$ then $0$ lies in the multiplicative set generated by $a t$ and the localization, and hence $A [I / a]$ , is the zero ring, consistently with $A_{a} = 0$ ; all four assertions then hold trivially. The description $B = {b / a^{n}}$ is the normal form used in the chart computations of the blowup.

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Affine blowup standard charts and overlaps

Statement

Assume the Axiom of Choice as inherited from the Proj construction. Let $A$ be a ring, $I = (f_{0}, \dots, f_{r}) \subseteq A$ , $S = R (I) = ⨁ I^{n} t^{n}$ and $B_{i} = A [I / f_{i}] = (S [(f_{i} t)^{- 1}])_{0}$ . The standard opens $U_{i} = D_{+} (f_{i} t) = Spec B_{i}$ cover ${Bl}_{I} Spec A$ . Put $u_{i j} = (f_{j} t) / (f_{i} t)$ in $B_{i}$ . Then $U_{i} \cap U_{j} = D (u_{i j})$ in $U_{i}$ , with canonical $A$ -algebra identifications $(B_{i})_{u_{i j}} = S [(f_{i} t)^{- 1}, (f_{j} t)^{- 1}]_{0} = (B_{j})_{u_{j i}}$ . They send $f_{l} / f_{i}$ to $(f_{l} / f_{j}) / (f_{i} / f_{j})$ , and $u_{i j}$ to $u_{j i}^{- 1}$ . These identifications satisfy the identity and cocycle conditions and preserve the structural maps to $Spec A$ . Different finite generating families give compatible chart covers of the same canonical blowup; no bijection between the chart families is asserted. The formulas include zero divisors and empty charts; nilpotent $f_{i}$ gives $B_{i} = 0$ . Localization at the base element $f_{j}$ is generally smaller than this overlap and is not its formula.

Facts & Assumptions

Given: A ring $A$ , an ideal $I = (f_{0}, \dots, f_{r}) \subseteq A$ , the Rees algebra $S = R (I) = ⨁_{n \geq 0} I^{n} t^{n}$ (Rees algebra sheaf of a finite type ideal), the blowup ${Bl}_{I} Spec A$ (Blowup of a scheme along an ideal sheaf), and the Axiom of Choice as inherited from the Proj construction (The Axiom of Choice).

[F1]

Blowup of a scheme along an ideal sheaf: For $X = Spec A$ and $I = \tilde{I}$ , the blowup is the absolute Proj of the Rees algebra $R (I) = ⨁_{n \geq 0} I^{n} t^{n}$ , with structural morphism to $Spec A$ .

[F2]

Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: For a ring $A$ , an ideal $I \subseteq A$ and $a \in I$ , the affine blowup algebra is $A [I / a] = (R (I))_{(a)}$ , the degree-zero part of the localisation of $R (I)$ at the multiplicative set generated by $a t$ .

[F3]

Proj carries a scheme structure: For a commutative nonnegatively graded ring $S$ , $Proj S$ carries open subscheme identifications $φ_{f} ⁣ : D_{+} (f) \to Spec S_{(f)}$ for homogeneous $f \in S_{+}$ , the $D_{+} (f)$ form an affine open cover, and for homogeneous $f, g$ of degrees $d, e$ the set $D_{+} (f) \cap D_{+} (g) = D_{+} (f g)$ is carried by $φ_{f}$ onto $D (g^{d} / f^{e}) \subseteq Spec S_{(f)}$ , with transition induced by $S_{(f)} [(g^{d} / f^{e})^{- 1}] ≅ S_{(f g)} ≅ S_{(g)} [(f^{e} / g^{d})^{- 1}]$ ; the underlying space is $Proj S$ with the standard-open basis, the scheme is unique for these identifications, and if $f$ is nilpotent then $D_{+} (f) = \emptyset$ and $S_{(f)} = 0$ .

[F4]

Standard opens of Proj: For homogeneous $f \in S_{+}$ , $D_{+} (f) = {p \in Proj S : f \notin p}$ is a standard open.

[F5]

Standard opens are affine: The canonical chart map $φ_{f} ⁣ : D_{+} (f) \to Spec S_{(f)}$ is an isomorphism, including the empty case: nilpotent $f$ gives $D_{+} (f) = \emptyset$ and $S_{(f)} = 0$ .

Proof

1.1F1F2F3F4F5

Put $g_{i} = f_{i} t \in S$ , a homogeneous element of degree one, so that $S = R (I) = A [I t]$ is generated as an $A$ -algebra by $I$ , and $S_{+}$ is generated as an ideal by $g_{0}, \dots, g_{r}$ ; hence no homogeneous prime $p$ of $S$ contains all $g_{i}$ without containing $S_{+}$ , and the standard opens $D_{+} (g_{i})$ cover $Proj S = {Bl}_{I} Spec A$ by [F1], [F3], [F4]. Moreover $B_{i} = A [I / f_{i}] = S_{(g_{i})}$ by [F2], so $U_{i} = D_{+} (g_{i}) = Spec B_{i}$ by [F5].

2.1F3F5step 1.1

For each pair $i, j$ , [F3] applied to the degree-one elements $g_{i}, g_{j}$ identifies $D_{+} (g_{i}) \cap D_{+} (g_{j}) = D_{+} (g_{i} g_{j})$ with $D (u_{i j}) \subseteq Spec B_{i}$ , where $u_{i j} = g_{j} / g_{i}$ , and identifies the two charts through the canonical isomorphisms $B_{i} [u_{i j}^{- 1}] = S_{(g_{i})} [(g_{j} / g_{i})^{- 1}] ≅ S_{(g_{i} g_{j})} ≅ S_{(g_{j})} [(g_{i} / g_{j})^{- 1}] = (B_{j})_{u_{j i}}$ .

3.1F3step 2.1

The identification of step 2.1 can be checked directly and torsion-safely: the canonical map $B_{i} [u_{i j}^{- 1}] \to (S [g_{i}^{- 1}, g_{j}^{- 1}])_{0}$ is surjective, because a degree-zero fraction $c / (g_{i}^{m} g_{j}^{n})$ with $c$ homogeneous of degree $m + n$ equals $(c / g_{i}^{m + n}) u_{i j}^{- n}$ with $c / g_{i}^{m + n} \in B_{i}$ ; and it is injective, because vanishing of the image means $g_{i}^{p} g_{j}^{q} c = 0$ in $S$ for some $p, q \geq 0$ , whence $u_{i j}^{q} (c / g_{i}^{m}) = g_{i}^{p} c g_{j}^{q} / g_{i}^{m + p + q} = 0$ in $B_{i} [u_{i j}^{- 1}]$ . No cancellation of $g_{i}$ in $S$ is used. The same identification sends $f_{l} / f_{i} = (f_{l} t) / (f_{i} t)$ to $(f_{l} t) / (f_{i} t)$ computed in $S_{(g_{i} g_{j})}$ , which equals $(f_{l} / f_{j}) / (f_{i} / f_{j})$ , and sends $u_{i j} = g_{j} / g_{i}$ to $u_{j i}^{- 1}$ .

4.1F3step 2.1step 3.1

The identifications satisfy the identity condition (for $i = j$ , $u_{i i} = 1$ and the transition is the identity) and the cocycle condition: on a triple overlap every transition is induced by the localisation map $S \to S [g_{i}^{- 1}, g_{j}^{- 1}, g_{k}^{- 1}]$ and taking degree zero, so the three compositions around the cycle coincide with the identity on $S_{(g_{i} g_{j} g_{k})}$ . They preserve the structural maps to $Spec A$ , because they are isomorphisms of $A$ -algebras for the structure maps $A = S_{0} \to S_{(g)}$ of the charts.

4.2F2step 2.1step 3.1

For $I = (x, y) \subseteq A = k [x, y]$ with $f_{0} = x$ , $f_{1} = y$ , one has $B_{0} = A [I / x] = k [x, y / x] = k [x, s]$ with $s = y / x$ and $y = x s$ , and $u_{01} = s$ , so the overlap $D (u_{01}) = D (s)$ in $Spec B_{0}$ retains the points with $x = 0$ and $s \neq 0$ . Localising instead at the base element $f_{1} = y$ gives $D (y) = D (x s) = D (x) \cap D (s)$ , which is strictly smaller than $D (s)$ and omits those points; hence localisation at the base element is not the overlap formula.

5.1F3step 4.1

A second finite generating family $I = (h_{0}, \dots, h_{r^{'}})$ gives the charts $D_{+} (h_{k} t) = Spec A [I / h_{k}]$ of the same scheme $Proj S$ , with their own overlap identifications supplied by the same formulas of [F3] applied to the degree-one elements of $S$ ; on the intersection of a chart of the first family and a chart of the second, $D_{+} (g_{i}) \cap D_{+} (h_{k} t) = D_{+} (g_{i} h_{k} t)$ , the transition is again induced by the canonical localisation of $S$ , so the two cover structures are compatible. No bijection between the two chart families is asserted: the charts are indexed by different generating sets and need not correspond individually.

6.1F3F5step 5.1∎

The formulas allow zero divisors and empty charts: step 3.1 never cancels $g_{i}$ in $S$ , and if $f_{i}$ is nilpotent then $g_{i} = f_{i} t$ is nilpotent, so $D_{+} (g_{i}) = \emptyset$ and $B_{i} = S_{(g_{i})} = 0$ by [F3] and [F5]; the same holds for the overlap formula in the degenerate cases. This completes the proof.

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Blowups restrict to open subschemes of the base

Statement

Assume the Axiom of Choice as inherited from the relative Proj construction. Let $j ⁣ : U \to X$ be an open subscheme of a scheme $X$ , let $I$ be a quasi-coherent ideal sheaf on $X$ and let $I ∣_{U}$ be its restriction. Then there is a canonical isomorphism of $U$ -schemes ${Bl}_{I ∣_{U}} U \to {Bl}_{I} X \times_{X} U$ , equivalently an isomorphism of the open subscheme $π^{- 1} (U)$ of ${Bl}_{I} X$ with ${Bl}_{I ∣_{U}} U$ over $U$ ; these isomorphisms are compatible with inclusions of opens.

Facts & Assumptions

Given: A scheme $X$ , an open subscheme $j ⁣ : U \to X$ (Open immersions of schemes), a quasi-coherent ideal sheaf $I \subseteq O_{X}$ (Quasi-coherent ideal sheaves) with restriction $I ∣_{U} = j^{*} I$ , and the blowup $π ⁣ : {Bl}_{I} X \to X$ of (Blowup of a scheme along an ideal sheaf), whose Rees algebra is $R (I) = ⨁_{n \geq 0} I^{n}$ .

[F1]

Relative Proj commutes with arbitrary base change: For a morphism $g ⁣ : S^{'} \to S$ and a quasi-coherent graded $O_{S}$ -algebra $A$ , with $A^{'} = g^{*} A$ graded by $(A^{'})_{d} = g^{*} A_{d}$ , there is a canonical isomorphism of $S^{'}$ -schemes ${Proj}_{S} A \times_{S} S^{'} ≅ {Proj}_{S^{'}} A^{'}$ , natural in $S^{'} \to S$ , compatible with the relative twists. No flatness and no finite-generation hypothesis is required.

[F2]

Rees algebra sheaf of a finite type ideal: For a quasi-coherent ideal sheaf $I$ on a scheme $X$ , the Rees algebra sheaf is $R (I) = ⨁_{n \geq 0} I^{n}$ with degree- $n$ piece $I^{n}$ and multiplication induced by multiplication in $O_{X}$ ; the construction is local on $X$ and on an affine chart $Spec A$ with $I = \tilde{I}$ it restricts to the sheaf associated to $⨁_{n \geq 0} I^{n}$ .

[F3]

Scheme pullback preserves quasi-coherence: Pullback of a quasi-coherent module along a morphism of schemes is quasi-coherent, and on affine opens with $f (U) \subseteq V$ , $U = Spec B$ , $V = Spec A$ and $F ∣_{V} = \tilde{M}$ one has $f^{*} F ∣_{U} ≅ \tilde{(B \otimes_{A} M)}$ .

[F4]

Base change of immersions: Open immersions remain open immersions after arbitrary base change.

[F5]

Base change of objects, morphisms and properties: The base change of $f ⁣ : X \to S$ along $h ⁣ : S^{'} \to S$ is $X \times_{S} S^{'}$ with second projection as structure map, and the formulas preserve identities and composition.

[F6]

Blowup of a scheme along an ideal sheaf: For a scheme $X$ and a quasi-coherent ideal sheaf of finite type with zero scheme $Z$ , the blowup is ${Bl}_{I} X = {Proj}_{X} R (I)$ with structural morphism to $X$ and relative twists.

Proof

1.1F2F3

The pullback along $j$ of the Rees algebra is the Rees algebra of the restricted ideal: $j^{*} R (I) ≅ R (I ∣_{U})$ as graded $O_{U}$ -algebras. Indeed, restriction to the open subscheme $U$ is exact and commutes with tensor products, so $j^{*} (I^{n}) ≅ (j^{*} I)^{n} = (I ∣_{U})^{n}$ for every $n \geq 0$ , and these identifications are compatible with the multiplications inherited from $O_{X}$ and $O_{U}$ ; the graded pieces of both sides are quasi-coherent by [F3], and $I ∣_{U}$ is again quasi-coherent (of finite type when $I$ is).

2.1F1F6step 1.1

Applying [F1] to the morphism $j ⁣ : U \to X$ and the graded algebra $A = R (I)$ gives a canonical isomorphism of $U$ -schemes ${Bl}_{I} X \times_{X} U = {Proj}_{X} R (I) \times_{X} U ≅ {Proj}_{U} (j^{*} R (I)) ≅ {Proj}_{U} R (I ∣_{U}) = {Bl}_{I ∣_{U}} U$ , where the last equality is the definition of the blowup of $U$ along $I ∣_{U}$ ; the isomorphism is compatible with the relative twists.

3.1F4F5step 2.1

The first projection ${Bl}_{I} X \times_{X} U \to {Bl}_{I} X$ is the base change of the open immersion $j$ along $π$ , hence an open immersion by [F4], and its underlying image is the open subset $π^{- 1} (U)$ . Identifying the fibre product with this open subscheme via that open immersion turns the isomorphism of step 2.1 into an isomorphism $π^{- 1} (U) \to {Bl}_{I ∣_{U}} U$ over $U$ .

4.1F1F5step 3.1∎

The isomorphisms are compatible with inclusions of opens: for open subschemes $U^{'} \subseteq U \subseteq X$ one has $({Bl}_{I} X \times_{X} U) \times_{U} U^{'} = {Bl}_{I} X \times_{X} U^{'}$ by [F5], and the isomorphism of [F1] is natural in the base morphism, so the identifications for $U$ and for $U^{'}$ restrict to one another; the same naturality makes the passage to $π^{- 1} (U)$ of step 3.1 compatible with the inclusion $π^{- 1} (U^{'}) \subseteq π^{- 1} (U)$ .

Remarks

The statement is written for a quasi-coherent ideal sheaf; the finite type hypothesis of Blowup of a scheme along an ideal sheaf is not needed for either side of the comparison and is preserved under restriction when it is imposed.
The identifications are canonical: on the overlaps of two open subschemes the two blowups agree because both restrict the same graded algebra $R (I)$ .

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The blowup is independent of chosen ideal generators

Statement

Assume the Axiom of Choice, inherited from the relative Proj construction used to define the blowup (The Axiom of Choice). Let $X$ be a scheme and let $I$ be a quasi-coherent ideal sheaf of finite type on $X$ (Quasi-coherent ideal sheaves). For two finite families of local generators of $I$ on an open cover of $X$ , the corresponding collections of standard affine charts and overlap identifications of the blowup ${Bl}_{I} X$ of Blowup of a scheme along an ideal sheaf glue to canonically isomorphic $X$ -schemes; on a common chart the canonical isomorphism is the identity on the common affine blowup algebra. In particular ${Bl}_{I} X$ , as a relative Proj, does not depend on any chosen finite generating set, and the affine blowup presentations $A [I / a]$ for $a \in I$ are canonically identified with the standard charts.

Facts & Assumptions

Given: A scheme $X$ with a quasi-coherent ideal sheaf $I$ of finite type, its Rees algebra sheaf $R (I) = ⨁_{n \geq 0} I^{n}$ (Rees algebra sheaf of a finite type ideal), the blowup ${Bl}_{I} X = {Proj}_{X} R (I)$ (Blowup of a scheme along an ideal sheaf), and for an affine open $U = Spec A \subseteq X$ with $I = Γ (U, I)$ and $a \in I$ the affine blowup algebra $A [I / a] = (R (I))_{(a)}$ , the degree-zero part of the localisation of $R (I)$ at the degree-one element $a t$ (Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains).

[F1]

Blowup of a scheme along an ideal sheaf: The blowup is the relative Proj ${Bl}_{I} X = {Proj}_{X} R (I)$ of the Rees algebra sheaf, with structural morphism to $X$ ; the Rees algebra and its graded pieces are intrinsic to $I$ , with no auxiliary generating data.

[F2]

Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: For $a \in I$ the affine blowup algebra $A [I / a]$ has $I A [I / a] = a A [I / a]$ with $a$ a nonzerodivisor, equals $A_{a}$ after inverting $a$ , and is independent of the generating set and of the representative used for the homogeneous localisation, up to canonical $A$ -algebra isomorphism.

[F3]

Affine blowup standard charts and overlaps: For $I = (f_{0}, \dots, f_{r})$ the standard opens $U_{i} = D_{+} (f_{i} t) = Spec A [I / f_{i}]$ cover ${Bl}_{I} Spec A$ ; their overlaps are $U_{i} \cap U_{j} = D (u_{i j})$ for $u_{i j} = (f_{j} t) / (f_{i} t)$ in $B_{i} = A [I / f_{i}]$ , with canonical identifications $(B_{i})_{u_{i j}} = (B_{j})_{u_{j i}}$ , $u_{i j} \mapsto u_{j i}^{- 1}$ , satisfying the identity and cocycle conditions and preserving the structure maps to $Spec A$ . Different finite generating families give compatible chart covers of the same canonical blowup.

[F4]

Blowups restrict to open subschemes of the base: For an open subscheme $j ⁣ : U ↪ X$ there is a canonical isomorphism ${Bl}_{I ∣_{U}} U \to {Bl}_{I} X \times_{X} U$ , compatible with inclusions of opens.

Proof

1.1F2

The Rees algebra $R (I) = ⨁_{n \geq 0} I^{n} t^{n}$ and the affine blowup algebra $A [I / a] = (R (I))_{(a)}$ for $a \in I$ depend only on $I$ and $a$ : by [F2] the affine blowup algebra is independent of the chosen generating set and of the representative of the homogeneous localisation, up to canonical isomorphism.

1.2F3

Let $U = Spec A$ be an affine open and let $f_{0}, \dots, f_{r}$ generate $I = Γ (U, I)$ . By [F3] the standard opens $U_{i} = D_{+} (f_{i} t) = Spec A [I / f_{i}]$ cover ${Bl}_{I} U$ , with overlaps $U_{i} \cap U_{j} = D (u_{i j})$ and the canonical identifications $(B_{i})_{u_{i j}} = (B_{j})_{u_{j i}}$ of chart rings given by the ratios of the degree-one elements $f_{i} t, f_{j} t$ of $R (I)$ .

2.1F1F3step 1.2

Now let $g_{0}, \dots, g_{s}$ be a second finite family generating the same ideal $I$ . Both families present open covers of the single scheme ${Proj}_{U} R (I) = {Bl}_{I} U$ by [F1]: a standard chart $D_{+} (h t)$ is the basic open of the degree-one element $h t \in R (I)_{1}$ , so the charts of the two families are open subschemes of the same relative Proj, and every overlap $D_{+} (f_{i} t) \cap D_{+} (g_{j} t)$ is the basic open of the degree-zero ratio of the two degree-one elements inside this Proj, identified with the corresponding localised chart ring as in [F3].

2.2F2step 1.1

If a chart occurs in both families, that is $f_{i} = g_{j} = h$ for some $h \in I$ , the two chart rings are both the affine blowup algebra $A [I / h]$ of [F2], and the identification is the identity of this common algebra, well defined independently of the family by step 1.1.

3.1F1F4step 2.1step 2.2∎

The chartwise identifications of steps 2.1 and 2.2 are the restrictions of the identity of the single scheme ${Bl}_{I} U = {Proj}_{U} R (I)$ to the members and pairwise overlaps of the two covers, so they satisfy the identity and cocycle conditions automatically, and glue to an isomorphism of presentations of ${Bl}_{I} U$ ; over an affine cover of $X$ these isomorphisms are compatible on overlaps by [F4], so they glue to a canonical isomorphism of $X$ -schemes between the presentations of ${Bl}_{I} X$ built from the two generating families. In particular ${Bl}_{I} X$ does not depend on a chosen finite generating set, and the affine blowup presentations $A [I / a]$ , $a \in I$ , are precisely the standard charts $D_{+} (a t)$ of the canonical blowup.

Remarks

No bijection between the two chart families is produced, and none is needed: the two covers are compared inside the same relative Proj through their pairwise overlaps, as in [F3].
The statement is used in practice to read off the standard charts $A [I / a]$ for any convenient $a \in I$ without changing the blowup; the fractional rescaling invariance of Invariance of the blowup under invertible (fractional) rescaling of the ideal is a different statement, comparing blowups of different ideals.

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The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier

Statement

Assume the Axiom of Choice, inherited from the relative Proj construction (The Axiom of Choice). Let $I$ be a quasi-coherent ideal sheaf of finite type on a scheme $X$ (Quasi-coherent ideal sheaves), let $π ⁣ : {Bl}_{I} X \to X$ be its blowup and let $E = π^{- 1} (Z)$ be the exceptional subscheme, with the convention that $O (1)$ is the positive relative twist of the Rees algebra $R (I)$ and $O (- 1) : = O (1)^{\lor}$ . Then:

$O (1)$ is invertible;
the natural degree-one map $π^{*} I \to O (1)$ is surjective, its image is the inverse image ideal $I O_{Bl}$ , and the induced map $I O_{Bl} \to O (1)$ of invertible sheaves is an isomorphism;
$I O_{Bl}$ is invertible and $E = V (I O_{Bl})$ is an effective Cartier divisor on ${Bl}_{I} X$ , with $O_{Bl} (- E) = I O_{Bl} = O (1)$ and $O_{Bl} (E) = O (- 1)$ .

Facts & Assumptions

Given: A scheme $X$ , a quasi-coherent ideal sheaf $I$ of finite type, the Rees algebra sheaf $R (I) = ⨁_{n \geq 0} I^{n}$ (Rees algebra sheaf of a finite type ideal), the blowup $π ⁣ : {Bl}_{I} X = {Proj}_{X} R (I) \to X$ with relative twists $O (n)$ (Blowup of a scheme along an ideal sheaf, Relative Proj of a graded quasi-coherent algebra), and the exceptional subscheme $E = π^{- 1} (Z)$ with ideal sheaf the inverse image ideal $I O_{Bl}$ (Exceptional subscheme of a blowup).

[F1]

Affine blowup standard charts and overlaps and Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: For $a \in I = Γ (U, I)$ on an affine open $U = Spec A$ , the chart $Spec A [I / a]$ of ${Bl}_{I} U$ has $I A [I / a] = a A [I / a]$ with $a$ a nonzerodivisor of $A [I / a]$ , and the charts over a finite generating family cover the blowup.

[F2]

Invertible twists for degree-one generated rings: For a commutative graded ring $S$ generated over $S_{0}$ by $S_{1}$ , all twists $O_{Proj S} (n)$ are invertible, with $O (m) \otimes O (n) ≅ O (m + n)$ ; the empty Proj is allowed.

[F3]

Twisting sheaf on Proj and Relative Proj of a graded quasi-coherent algebra: The relative twist $O (1)$ of a relative Proj is the sheaf whose sections over the chart $D_{+} (f)$ are the degree-one part $S (1)_{(f)}$ of the localised graded algebra; it is the sheafification of the degree-one part, and on an affine base it restricts to the absolute twist $O_{Proj} (1)$ .

[F4]

Invertible sheaves and Effective cartier divisor: A sheaf is invertible when it is locally free of rank one; an effective Cartier divisor on a scheme is given by local equations that are regular sections, i.e. nonzerodivisors on the stalks.

[F5]

Invertible sheaf of cartier divisor and Effective Cartier divisors give a short exact sequence: For an effective Cartier divisor $D$ the sheaf $O_{X} (- D)$ is the ideal sheaf $I_{D} \subseteq O_{X}$ , and there is a short exact sequence $0 \to O_{X} (- D) \to O_{X} \to i_{*} O_{D} \to 0$ .

Proof

1.1F2F3given

The Rees algebra is generated in degree one. Thus its positive twist is invertible on each affine base, and these restrictions give an invertible sheaf $O (1)$ on the relative Proj. There are two natural maps from $π^{*} I$ : the structural multiplication map $ψ$ to $O_{Bl}$ , whose image is $I O_{Bl}$ , and the degree-one map $γ$ to $O (1)$ .

2.1F1F3step 1.1

On a standard chart $B = A [I / a]$ , $O (1)$ is free with frame $a t$ . For $b \in I$ and $c \in B$ , the maps are $γ (b \otimes c) = c (b / a) (a t)$ and $ψ (b \otimes c) = c b = a c (b / a)$ . The first is surjective since $γ (a \otimes 1) = a t$ . Since $a$ is a nonzerodivisor in $B$ , these formulas give $\ker γ = \ker ψ$ , including for sums of tensors. Consequently $γ$ factors uniquely through the image $I B = a B$ of $ψ$ and induces an isomorphism $I B \to B (a t)$ taking $a$ to $a t$ . This does not assert that $I \otimes_{A} B$ is free or torsion-free.

3.1F1F4F5step 2.1∎

The equality of the kernels is local and therefore global. The induced isomorphisms are restrictions of this unique factorization, so agree on overlaps. Hence $I O_{Bl} ≅ O (1)$ canonically. The ideal cuts out $E$ and is locally generated by the nonzerodivisor $a$ , so $E$ is effective Cartier. Its ideal is $O (- E)$ , and dualizing the isomorphism gives $O (E) ≅ O (- 1)$ . Empty charts and the empty blowup satisfy the same assertions.

Remarks

Assertion 2 identifies the inverse image ideal with the twist $O (1)$ as an invertible sheaf, which is what makes $E$ Cartier even when the centre $Z$ is neither reduced nor Cartier in $X$ .
Combining assertion 3 with Effective Cartier divisors give a short exact sequence gives the short exact sequence $0 \to O (1) \to O_{Bl} \to i_{*} O_{E} \to 0$ on the blowup, a form used in cohomological computations.

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Universal property of an affine blowup chart

Statement

Let $φ ⁣ : A \to B$ be a ring map, $I \subseteq A$ an ideal and $a \in I$ ; suppose the image $b = φ (a)$ is a nonzerodivisor in $B$ and $I B = b B$ . Then there is a unique $A$ -algebra homomorphism $A [I / a] \to B$ sending $x / a^{n}$ to the unique $y \in B$ with $x = b^{n} y$ ( $x \in I^{n}$ ); equivalently, $Spec B \to Spec A [I / a]$ is the unique $A$ -morphism into the chart $Spec A [I / a]$ along which the image of $a$ generates $I O_{Spec B}$ . The chart $A [I / a]$ itself satisfies the hypothesis with $b$ the image of $a$ .

Facts & Assumptions

Given: A commutative ring $A$ , an ideal $I \subseteq A$ , an element $a \in I$ , a ring map $φ ⁣ : A \to B$ whose image $b = φ (a)$ is a nonzerodivisor in $B$ and satisfies $I B = b B$ , and the affine blowup algebra $A [I / a]$ of (Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains) built from the Rees algebra $R (I) = ⨁_{n \geq 0} I^{n} t^{n}$ of (The Rees algebra of an ideal and the Rees module of a filtered module).

[F1]

Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: For a commutative ring $A$ , an ideal $I \subseteq A$ and $a \in I$ , the affine blowup algebra $A [I / a] : = (R (I))_{(a)}$ is the degree-zero part of the localisation of the Rees algebra $R (I)$ at the multiplicative set generated by $a t$ ; the image of $a$ in $A [I / a]$ is a nonzerodivisor and $I A [I / a] = a A [I / a]$ .

[F2]

The Rees algebra of an ideal and the Rees module of a filtered module: For a commutative ring $R$ and an ideal $I \subset R$ , the Rees algebra is the graded subring $R (I) = ⨁_{n \geq 0} I^{n} t^{n} \subset R [t]$ ; equivalently, it is the graded ring whose degree- $n$ piece is $I^{n}$ .

[F3]

Multiplicative subsets and the localisation $S^{- 1} R$ as equivalence classes of fractions: For a commutative ring $R$ and a multiplicative subset $S \subseteq R$ , the localisation $S^{- 1} R$ has elements written $r / s$ with $r / s = r^{'} / s^{'}$ if and only if $u (r s^{'} - r^{'} s) = 0$ for some $u \in S$ ; the operations are $r / s + r^{'} / s^{'} = (r s^{'} + r^{'} s) / (s s^{'})$ and $(r / s) (r^{'} / s^{'}) = r r^{'} / (s s^{'})$ ; every $s \in S$ maps to a unit.

Proof

1.1F1F2F3given

Put $β = φ (a)$ . By the fraction description of the affine blowup algebra, every element of $C = A [I / a]$ has the form $r / a^{n}$ , $r \in I^{n}$ , and $r / a^{n} = s / a^{m}$ precisely when $a^{k} (a^{m} r - a^{n} s) = 0$ for some $k \geq 0$ . Since $I^{n} B = (I B)^{n} = β^{n} B$ , there is a unique $y \in B$ with $φ (r) = β^{n} y$ : existence follows from this ideal equality and uniqueness from the nonzerodivisor hypothesis on $β$ .

2.1step 1.1

Define $ψ (r / a^{n}) = y$ . If $r / a^{n} = s / a^{m}$ , write $φ (r) = β^{n} y$ and $φ (s) = β^{m} z$ . Applying $φ$ to the equality criterion gives $β^{k + m + n} (y - z) = 0$ , hence $y = z$ . Thus $ψ$ is well defined. The numerator of the sum is $a^{m} r + a^{n} s$ , whose image is $β^{m + n} (y + z)$ ; the product numerator $r s$ has image $β^{m + n} y z$ . Uniqueness of division by $β^{m + n}$ proves additivity and multiplicativity. Degree-zero fractions show that $ψ$ restricts to $φ$ on $A$ and sends $1$ to $1$ .

3.1F1step 2.1∎

Any $A$ -algebra map $χ : C \to B$ satisfies $β^{n} χ (r / a^{n}) = φ (r)$ , because $a^{n} (r / a^{n}) = r$ in $C$ . Cancellation of $β^{n}$ forces $χ (r / a^{n}) = ψ (r / a^{n})$ for every fraction. Hence the map is unique among all $A$ -algebra maps. The chart itself has $I C = a C$ with $a$ a nonzerodivisor, and the affine scheme/ring correspondence gives the stated geometric formulation.

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Universal property of the blowup

Statement

Assume the Axiom of Choice. Let $I$ be a quasi-coherent ideal sheaf of finite type on $X$ with zero scheme $Z$ , and let $π ⁣ : {Bl}_{I} X \to X$ be the blowup. For every $X$ -scheme $f ⁣ : Y \to X$ such that the inverse image $f^{- 1} (Z)$ is an effective Cartier divisor on $Y$ , there is a unique $X$ -morphism $Y \to {Bl}_{I} X$ . Equivalently, ${Bl}_{I} X$ is the final object of the category of $X$ -schemes in which the inverse image of $Z$ is an effective Cartier divisor.

Facts & Assumptions

Given: The Axiom of Choice, a quasi-coherent ideal sheaf $I$ of finite type on $X$ with zero scheme $Z$ , the blowup $π ⁣ : {Bl}_{I} X \to X$ , and an $X$ -scheme $f ⁣ : Y \to X$ such that $f^{- 1} (Z)$ is an effective Cartier divisor on $Y$ .

[A1]

Choice. The Axiom of Choice is assumed, as in the statement; the cited suppliers used below are stated under it.

[F1]

Universal property of an affine blowup chart: Let $φ ⁣ : A \to B$ be a ring map, $I \subseteq A$ an ideal and $a \in I$ , and suppose the image $b = φ (a)$ is a nonzerodivisor in $B$ with $I B = b B$ . Then there is a unique $A$ -algebra homomorphism $A [I / a] \to B$ sending $x / a^{n}$ to the unique $y \in B$ with $x = b^{n} y$ ; equivalently, $Spec B \to Spec A [I / a]$ is the unique $A$ -morphism into the chart along which the image of $a$ generates $I O_{Spec B}$ .

[F2]

Affine blowup standard charts and overlaps: If $I = (f_{0}, \dots, f_{r}) \subseteq A$ and $B_{i} = A [I / f_{i}]$ , the standard opens $U_{i} = Spec B_{i}$ cover ${Bl}_{I} Spec A$ , with transition maps sending $u_{i j} = (f_{j} t) / (f_{i} t)$ to $u_{j i}^{- 1} = f_{j} / f_{i}$ on the overlaps.

[F3]

Blowup of a scheme along an ideal sheaf: ${Bl}_{I} X = {Proj}_{X} R (I)$ with structural morphism $π$ , and the blowup is local on the base: over an affine open $Spec A$ with $I = (f_{0}, \dots, f_{r})$ it is covered by the charts $Spec A [I / f_{i}]$ .

[F4]

Scheme-theoretic inverse images of subschemes: For $f ⁣ : Y \to X$ and the closed subscheme $Z = V (I)$ , the scheme-theoretic inverse image is $Y \times_{X} Z$ , and the inverse-image ideal is $Im (f^{*} I \to O_{Y})$ .

[F5]

Effective cartier divisor: A Cartier divisor is effective when it has a local-equation representation by regular sections $f_{i} \in O_{X} (U_{i})$ ; the local principal ideals $f_{i} O_{U_{i}}$ glue to an ideal sheaf, and a unit equation represents the empty divisor.

[F6]

Morphisms of schemes are local on compatible open covers: Compatible morphisms on an open cover glue uniquely, and two morphisms out of $Y$ are equal if their restrictions to an open cover are equal.

[F7]

Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: The chart algebra $A [I / a]$ is the degree-zero part of the localization of $R (I)$ at $a$ , with $I A [I / a] = a A [I / a]$ , and $a$ a nonzerodivisor; for $I = (a_{0}, \dots, a_{r})$ , $a = a_{0}$ , the chart receives a surjection $A [x_{1}, \dots, x_{r}] / (a x_{i} - a_{i}) \to A [I / a]$ .

[F8]

The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier: The inverse-image center ideal on the blowup is invertible and locally generated by a nonzerodivisor; its zero scheme is an effective Cartier divisor.

Proof

1.1F1F3F4F5

Work over an affine $U = Spec A$ with $I = (f_{0}, \dots, f_{r})$ . Cover its inverse image in $Y$ by affines $V = Spec B$ on which $I B = β B$ with $β$ a nonzerodivisor. Write $b_{i} = φ (f_{i}) = u_{i} β$ . A relation $β = \sum c_{i} b_{i}$ and cancellation of $β$ show $1 = \sum c_{i} u_{i}$ , so the $D (u_{i})$ cover $V$ . On each $D (u_{i})$ , $b_{i}$ is a nonzerodivisor generating the inverse-image ideal, and the affine chart property gives a map to chart $i$ , sending $f_{l} / f_{i}$ to $u_{l} / u_{i}$ .

2.1F1F2F6F7step 1.1

We first prove uniqueness for any two lifts on such a $V$ . At a point $y \in D (u_{i})$ , any lift $h$ has image in some chart $j$ ; shrink around $y$ so it lands in that chart. There the pulled-back ideal is generated by $b_{j}$ , since $I A [I / f_{j}] = f_{j} A [I / f_{j}]$ . Since $b_{i}$ also generates $I B$ near $y$ , write $b_{i} = v b_{j}$ and $b_{j} = w b_{i}$ . Cancellation of the regular element $b_{i}$ gives $v w = 1$ . Thus the pullback of the ratio $f_{i} / f_{j}$ is a unit. A local ring map then puts $h (y)$ in the ratio open of chart $j$ , which is its intersection with chart $i$ . This holds at every $y \in D (u_{i})$ , so $h ∣_{D (u_{i})}$ factors through chart $i$ . The unique chart map of [F1] therefore determines any lift. Equality on this open cover proves local uniqueness.

3.1F6step 1.1step 2.1

The maps constructed in step 1.1 agree on intersections by this local uniqueness, after refining intersections by affines on which the pulled-back ideal has a regular generator. The same argument compares constructions from different base affines and different local equations. They consequently glue to an $X$ -morphism $Y \to {Bl}_{I} X$ . Any two global lifts coincide on these local covers by step 2.1, hence coincide globally.

4.1F8step 3.1∎

The blowup itself belongs to the specified category: its inverse image of $Z$ is effective Cartier by [F8]. Every object has exactly one morphism to it by step 3.1. This is precisely finality in that category.

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Uniqueness of the blowup

Statement

Assume the Axiom of Choice. Let $I$ be a quasi-coherent ideal sheaf of finite type with zero scheme $Z$ . If $π^{'} ⁣ : Y \to X$ is an $X$ -scheme such that $(π^{'})^{- 1} (Z)$ is an effective Cartier divisor and $Y$ carries the universal property of ${Bl}_{I} X$ (every $X$ -scheme in which the inverse image of $Z$ is an effective Cartier divisor maps uniquely to $Y$ over $X$ ), then there is a unique $X$ -isomorphism $Y \to {Bl}_{I} X$ . In particular any two models of the blowup are uniquely isomorphic over $X$ .

Facts & Assumptions

Given: A quasi-coherent ideal sheaf $I$ of finite type on $X$ with zero scheme $Z$ , the blowup ${Bl}_{I} X$ , and an $X$ -scheme $π^{'} ⁣ : Y \to X$ whose inverse image of $Z$ is an effective Cartier divisor and which carries the same universal property.

[A1]

Choice. The Axiom of Choice is assumed as inherited from the blowup and Proj constructions used by the cited items.

[F1]

Universal property of the blowup: For every $X$ -scheme $f ⁣ : T \to X$ in which the inverse image of $Z$ is an effective Cartier divisor there is a unique $X$ -morphism $T \to {Bl}_{I} X$ ; equivalently ${Bl}_{I} X$ is final among such $X$ -schemes.

[F2]

The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier: The inverse image ideal $I O_{Bl}$ is invertible and $E = V (I O_{Bl})$ is an effective Cartier divisor on ${Bl}_{I} X$ ; in particular the blowup is itself an $X$ -scheme in which the inverse image of $Z$ is an effective Cartier divisor.

[F3]

Blowup of a scheme along an ideal sheaf: The blowup is the relative Proj of the Rees algebra with its structural morphism to $X$ . The identification of its exceptional subscheme with the inverse image of $Z$ used here is supplied by [F2].

Proof

1.1F2F3

By [F2] the blowup $π ⁣ : {Bl}_{I} X \to X$ is an object of the category of $X$ -schemes in which the inverse image of $Z$ is an effective Cartier divisor, and by hypothesis $Y$ is such an object as well.

2.1F1step 1.1

Applying the universal property of the blowup [F1] to the $X$ -scheme $Y$ gives a unique $X$ -morphism $u ⁣ : Y \to {Bl}_{I} X$ with $π \circ u = π^{'}$ ; applying the universal property carried by $Y$ to the $X$ -scheme ${Bl}_{I} X$ gives a unique $X$ -morphism $v ⁣ : {Bl}_{I} X \to Y$ with $π^{'} \circ v = π$ .

3.1F1step 2.1∎

The composite $v \circ u ⁣ : Y \to Y$ is an $X$ -morphism with $π^{'} \circ (v \circ u) = π^{'}$ , and so is ${id}_{Y}$ ; since by hypothesis there is at most one $X$ -morphism from the admissible $X$ -scheme $Y$ to $Y$ , namely the map required by the universal property, we get $v \circ u = {id}_{Y}$ ; symmetrically $u \circ v = {id}_{{Bl}_{I} X}$ because $u \circ v$ and the identity are both $X$ -morphisms from ${Bl}_{I} X$ to itself and [F1] gives a unique one. Hence $u$ is an $X$ -isomorphism, and it is the unique one: any $X$ -isomorphism $Y \to {Bl}_{I} X$ is an $X$ -morphism between admissible objects and therefore equals $u$ by the uniqueness clause of [F1]; in particular any two models of the blowup are uniquely isomorphic over $X$ .

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The blowup is an isomorphism off the center

Statement

Let $I$ be a quasi-coherent ideal sheaf of finite type with zero scheme $Z$ and let $π ⁣ : {Bl}_{I} X \to X$ be the blowup. Then the restriction $π ⁣ : π^{- 1} (X ∖ Z) \to X ∖ Z$ is an isomorphism of schemes, with inverse characterized by the universal property applied to the identity of $X ∖ Z$ (where the inverse image of $Z$ is empty) and to the open immersion $π^{- 1} (X ∖ Z) ↪ X$ . Consequently $E = π^{- 1} (Z)$ is the complement of this open subscheme.

Facts & Assumptions

Given: A quasi-coherent ideal sheaf $I$ of finite type with zero scheme $Z$ , and the blowup $π ⁣ : {Bl}_{I} X \to X$ .

[A1]

Choice. The Axiom of Choice is assumed as inherited from the blowup and Proj constructions used by the cited items.

[F1]

Affine blowup standard charts and overlaps: For $I = (f_{0}, \dots, f_{r}) \subseteq A$ , the standard opens $Spec A [I / f_{i}]$ cover ${Bl}_{I} Spec A$ , with transition functions $u_{i j} \mapsto u_{j i}^{- 1}$ .

[F2]

Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: For $a \in I$ the affine blowup algebra satisfies $I A [I / a] = a A [I / a]$ and $(A [I / a])_{a} = A_{a}$ , the latter being the ordinary localization of $A$ at $a$ .

[F3]

Blowup of a scheme along an ideal sheaf: ${Bl}_{I} X = {Proj}_{X} R (I)$ with its structural morphism to $X$ ; the affine chart cover is supplied by [F1].

[F4]

Universal property of the blowup: For every $X$ -scheme $f ⁣ : Y \to X$ whose inverse image of $Z$ is an effective Cartier divisor there is a unique $X$ -morphism $Y \to {Bl}_{I} X$ .

[F5]

Effective cartier divisor: A unit equation represents the zero Cartier divisor, the empty effective divisor, and the empty scheme has only this effective divisor.

[F6]

Exceptional subscheme of a blowup: $E = π^{- 1} (Z) = Z \times_{X} {Bl}_{I} X$ with ideal sheaf $I O_{Bl}$ , and $E$ is set-theoretically the preimage of $Z$ .

Proof

1.1F1F2F3

Cover $X$ by affine opens $U = Spec A$ with $I ∣_{U} = (f_{0}, \dots, f_{r})$ ; by [F1] and [F3] the preimage $π^{- 1} (U)$ is covered by the charts $Spec A [I / f_{i}]$ , and by [F2] the chart ring localizes to $A_{f_{i}}$ after inverting $f_{i}$ , and this chart contains the entire inverse image of $D (f_{i})$ . Indeed, in every chart $j$ one has $f_{i} = f_{j} (f_{i} / f_{j})$ ; wherever $f_{i}$ is invertible, both $f_{j}$ and the ratio $f_{i} / f_{j}$ are invertible. The ratio-overlap formula of [F1] puts this open of chart $j$ in chart $i$ . Thus the restriction of $π$ over the principal open $D (f_{i})$ is an isomorphism $D (f_{i}) \to D (f_{i})$ : it is the structural map $Spec A [I / f_{i}] \to Spec A$ followed by localization, and $(A [I / f_{i}])_{f_{i}} = A_{f_{i}}$ .

2.1step 1.1

These local inverses glue. Let $O$ and $O'$ be any two base opens of the form $D (f_{i})$ in step 1.1, possibly in different affine base neighborhoods. The structural map on the whole inverse image $π^{- 1} (O)$ is an isomorphism onto $O$ . Restricting it to $O \cap O'$ gives an isomorphism $π^{- 1} (O \cap O') \to O \cap O'$ . Both local inverse maps restricted to this intersection land in that inverse image and are inverses of this same isomorphism, so they agree. Thus they glue to $σ : X ∖ Z \to W : = π^{- 1} (X ∖ Z)$ with $π \circ σ = id$ . For two charts in one affine base, only the restriction of their chart overlap over $D (f_{i} f_{j})$ is identified with $D (f_{i} f_{j})$ ; the whole ratio overlap can also contain points over $Z$ .

3.1step 2.1algebra

The morphism $σ$ is an inverse for $π ∣_{W}$ . Since $X ∖ Z$ is covered by the opens $D (f_{i})$ of step 1.1, and over each such open the restriction of $σ$ is the inverse of the restriction of $π$ (step 1.1), the composite $σ \circ π ∣_{W}$ agrees with ${id}_{W}$ after restriction to the cover of $W$ by the opens $π^{- 1} (D (f_{i})) \cap Spec A [I / f_{i}]$ , on each of which $π$ is an isomorphism; hence $σ \circ π ∣_{W} = {id}_{W}$ and $π ∣_{W} \circ σ = {id}_{X ∖ Z}$ , so $π ∣_{W}$ is an isomorphism.

4.1F4F5step 3.1

The inverse is characterized by the universal property: the inverse image of $Z$ under the identity $X ∖ Z \to X$ is empty, hence the zero Cartier divisor, which is effective by [F5]; so [F4] gives a unique $X$ -morphism $τ ⁣ : X ∖ Z \to {Bl}_{I} X$ lifting the identity, and $τ$ is an inverse of $π$ over $X ∖ Z$ ; by uniqueness of the inverse of the isomorphism $π ∣_{W}$ of step 3.1, $τ = σ$ . In particular $σ$ is the unique morphism over $X$ from $X ∖ Z$ into the blowup.

5.1F6step 3.1∎

Finally $E$ is the complement of $W$ : by [F6], $E = π^{- 1} (Z)$ is set-theoretically the preimage of $Z$ , so its underlying set is the complement of the underlying set of $π^{- 1} (X ∖ Z) = W$ , i.e. $E$ is the complement of the open subscheme $W$ in the blowup.

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Blowups of finite type ideals are locally H-projective, and proper

Statement

$π$ is locally H-projective on $X$ : for every affine open $U = Spec A$ with $I ∣_{U} = (f_{0}, \dots, f_{r})$ , the pullback of the graded surjection $A [x_{0}, \dots, x_{r}] \to R (I ∣_{U})$ , $x_{i} \mapsto f_{i} t$ , exhibits $π^{- 1} (U)$ as a closed subscheme of $P_{U}^{r}$ .
If $I$ is generated as an $O_{X}$ -module by finitely many global sections $f_{0}, \dots, f_{r}$ (Global generation by the evaluation map), then the same surjection $O_{X} [x_{0}, \dots, x_{r}] \to R (I)$ makes ${Bl}_{I} X$ a closed subscheme of $P_{X}^{r}$ over $X$ ; in particular $π$ is H-projective (Projective morphisms before Proj).
In all cases $π$ is proper, since properness is local on the base and each $π^{- 1} (U) \to U$ is H-projective hence proper.

Facts & Assumptions

Given: A scheme $X$ , a quasi-coherent ideal sheaf $I$ of finite type, the blowup $π ⁣ : {Bl}_{I} X = {Proj}_{X} R (I) \to X$ (Blowup of a scheme along an ideal sheaf), and for an affine open $U = Spec A \subseteq X$ the restricted ideal $I = Γ (U, I)$ with affine blowup algebras $A [I / a]$ .

[F1]

Closed subschemes of projective space and saturated ideals: For a commutative ring $A$ and a homogeneous ideal $J \subseteq A [x_{0}, \dots, x_{r}]$ one has $V_{+} (J) = Proj (A [x_{0}, \dots, x_{r}] / J)$ as a closed subscheme of $P_{A}^{r}$ , and on the chart $D_{+} (x_{i})$ it is $Spec (A [x_{0}, \dots, x_{r}]_{(x_{i})} / J_{(x_{i})})$ ; every closed subscheme of $P_{A}^{r}$ arises from a unique saturated homogeneous ideal.

[F2]

Affine-local graded algebras glue their Proj charts: The relative Proj of a quasi-coherent graded $O_{X}$ -algebra is obtained by gluing the spectra $Proj Γ (U, A)$ over affine opens $U \subseteq X$ , with canonical restriction and cocycle identifications.

[F3]

Affine blowup standard charts and overlaps and Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: For $I = (f_{0}, \dots, f_{r})$ the standard opens $D_{+} (f_{i} t) = Spec A [I / f_{i}]$ cover ${Bl}_{I} Spec A$ , and the chart presentations are independent of the chosen generating family; for $U \subseteq X$ affine there is a canonical identification $π^{- 1} (U) ≅ {Bl}_{I ∣_{U}} U$ (Blowups restrict to open subschemes of the base).

[F4]

Global generation by the evaluation map: A quasi-coherent sheaf $I$ is generated by global sections $f_{0}, \dots, f_{r} \in Γ (X, I)$ exactly when the evaluation morphism $O_{X}^{r + 1} \to I$ , $(g_{0}, \dots, g_{r}) \mapsto \sum_{i} g_{i} f_{i}$ , is surjective.

[F5]

Projective morphisms are proper and Properness is local on the target: An H-projective morphism is proper, and properness is local on an open cover of the base.

[F6]

Closed immersions of schemes and Quasi-coherent module on a scheme: A morphism $i$ is a closed immersion when it is a homeomorphism onto a closed subset and $O \to i_{*} O$ is surjective; both conditions are local on the target, and kernels of morphisms of quasi-coherent modules are quasi-coherent.

Proof

1.1F1F3

Let $U = Spec A \subseteq X$ be affine and let $f_{0}, \dots, f_{r} \in A$ generate $I = Γ (U, I)$ . The graded $A$ -algebra homomorphism $φ ⁣ : A [x_{0}, \dots, x_{r}] \to R (I)$ , $x_{i} \mapsto f_{i} t$ , is surjective, because in degree $n$ the images $f^{α} t^{n}$ of the monomials $x^{α}$ of degree $n$ generate $I^{n} t^{n}$ ; hence by [F1] it presents $Proj R (I) = {Bl}_{I} U ≅ π^{- 1} (U)$ as the closed subscheme $V_{+} (\ker φ) ↪ P_{U}^{r}$ over $U$ , which is assertion 1.

1.2F1F4

If $I$ is generated by global sections $f_{0}, \dots, f_{r}$ , then the evaluation morphism $O_{X}^{r + 1} \to I$ is surjective by [F4], and consequently the induced morphism of graded $O_{X}$ -algebras $ψ ⁣ : O_{X} [x_{0}, \dots, x_{r}] \to R (I)$ , $x_{i} \mapsto f_{i} t$ , is surjective in every degree, because in degree $n$ its image is the subsheaf generated by the products of $n$ of the global sections, which is $I^{n}$ by hypothesis.

2.1F2step 1.1step 1.2

Assume the global generation of step 1.2. On every affine open $U \subseteq X$ the restriction of $ψ$ is the surjection $φ$ of step 1.1 for the restricted generators, so by step 1.1 the morphisms $π^{- 1} (U) \to P_{U}^{r} \subseteq P_{X}^{r}$ are closed immersions over $U$ ; these affine-local morphisms agree on overlaps because they are induced by the restrictions of the single graded morphism $ψ$ and the identifications of [F2] are canonical, so they glue to a morphism ${Bl}_{I} X \to P_{X}^{r}$ over $X$ .

3.1F6step 2.1

The glued morphism of step 2.1 is a closed immersion: surjectivity of the structure map of sheaves is checked on stalks, and a subset of $P_{X}^{r}$ whose traces on the members of an open cover are closed is closed, so both conditions of [F6] are affine-local and hold because each restriction is a closed immersion; hence ${Bl}_{I} X$ is a closed subscheme of $P_{X}^{r}$ over $X$ , and $π$ is H-projective by Projective morphisms before Proj, which is assertion 2.

4.1F5step 1.1∎

For every affine open $U \subseteq X$ the restriction $π^{- 1} (U) \to U$ is a closed subscheme of $P_{U}^{r}$ over $U$ by step 1.1, hence H-projective and therefore proper by [F5]; since properness is local on an open cover of the base by [F5], the morphism $π$ itself is proper, which is assertion 3, and it holds whether or not $I$ is globally generated.

Remarks

Assertion 1 holds for an arbitrary finite type ideal sheaf; assertion 2 needs the stronger hypothesis that the ideal is generated by finitely many global sections, and it is this case that produces a globally defined closed immersion into relative projective space.
The properness in assertion 3 is the only part used in the sequel for valuative arguments and for the direct image computations on an affine base; the local H-projectivity is used to read off charts.

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Blowing up a nonzero ideal on an integral scheme is birational

Statement

Assume the Axiom of Choice, inherited from the blowup construction (The Axiom of Choice). Let $X$ be an integral scheme (Integral schemes) and let $I$ be a nonzero quasi-coherent ideal sheaf of finite type. Then ${Bl}_{I} X$ is integral and $π ⁣ : {Bl}_{I} X \to X$ is birational: $π$ is an isomorphism over the nonempty dense open $X ∖ Z$ , and the generic point of ${Bl}_{I} X$ maps to the generic point of $X$ . If moreover $X$ is normal and every irreducible component of $Z$ has codimension at least two, the blowup is an isomorphism in codimension one, i.e. over the complement of a closed subset of codimension at least two.

Facts & Assumptions

Given: An integral scheme $X$ , a nonzero quasi-coherent ideal sheaf $I$ of finite type with zero scheme $Z = V (I)$ , the blowup $π ⁣ : {Bl}_{I} X \to X$ (Blowup of a scheme along an ideal sheaf), and the generic points $η_{X}$ of $X$ and $η_{Bl}$ of ${Bl}_{I} X$ (Generic points of irreducible closed subsets).

[F1]

Integrality and reducedness of blowups from the Rees charts: For an integral $X$ and a nonzero ideal sheaf $I$ of finite type, the blowup ${Bl}_{I} X$ is integral; in particular it is nonempty, reduced and irreducible, with a unique generic point.

[F2]

The blowup is an isomorphism off the center: The restriction $π ⁣ : π^{- 1} (X ∖ Z) \to X ∖ Z$ is an isomorphism of schemes, and $E = π^{- 1} (Z)$ is the complement of this open subscheme.

[F3]

Birational morphisms of integral finite-type schemes: For integral $k$ -schemes of finite type, a morphism $f$ is birational when it carries the generic point of the source to the generic point of the target and the induced map on local rings at the generic points is an isomorphism; equivalently $f$ identifies the function fields.

[F4]

Integral schemes and The reduction of a scheme: An integral scheme is reduced, so its nilradical ideal is zero; hence a nonzero ideal sheaf $I$ has $V (I) \neq X$ , and $X ∖ Z$ is a nonempty open subset of the irreducible space $X$ , therefore dense.

Proof

1.1F1F2F3F4

The blowup is integral by [F1], and $W = π^{- 1} (X ∖ Z)$ is isomorphic to the nonempty dense open $X ∖ Z$ by [F2, F4]. The generic point of an integral scheme belongs to every nonempty open; it is also the generic point of that open. Hence the generic point of the blowup belongs to $W$ and maps to the generic point of $X ∖ Z$ , namely the generic point of $X$ . The open isomorphism identifies their local rings. This proves the concrete birational assertion for arbitrary integral $X$ , and the function-field formulation when [F3] applies.

2.1F2step 1.1∎

Under the codimension assumption, a point in $Z$ is a specialization of the generic point of an irreducible component of $Z$ . Codimension cannot decrease under specialization: locally, the corresponding prime contains that component's prime, and every chain below the latter is also a chain below the former. Thus no point of codimension at most one belongs to $Z$ . The isomorphism over $X ∖ Z$ is therefore an isomorphism in codimension one, in exactly the sense stated. Normality is not needed for this implication.

Remarks

Normality of $X$ is not needed for the direction proved here; it is the standard hypothesis in the converse statements comparing a birational morphism with a blowup, which are not claimed on this page.
The birationality statement for an arbitrary integral base is the concrete one: isomorphism over a nonempty dense open with the generic point carried to the generic point; the function-field formulation of Birational morphisms of integral finite-type schemes applies over a field.

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Flat base change for blowups, and failure without flatness

Statement

Assume the Axiom of Choice as inherited from the relative Proj construction. Let $g ⁣ : X^{'} \to X$ be a flat morphism of schemes and $I$ a quasi-coherent ideal sheaf of finite type on $X$ . Then there is a canonical isomorphism of $X^{'}$ -schemes ${Bl}_{g^{- 1} I} X^{'} \to {Bl}_{I} X \times_{X} X^{'}$ , where $g^{- 1} I$ is the inverse image ideal sheaf, compatible with the structural morphisms and the relative twists. Without flatness the natural comparison map need not be an isomorphism: the powers $(g^{- 1} I)^{n}$ can differ from $g^{*} (I^{n})$ by torsion (compare the companion counterexample).

Facts & Assumptions

Given: A morphism of schemes $g ⁣ : X^{'} \to X$ , a quasi-coherent ideal sheaf $I \subseteq O_{X}$ of finite type (Quasi-coherent ideal sheaves) with Rees algebra sheaf $R (I) = ⨁_{n \geq 0} I^{n}$ (Rees algebra sheaf of a finite type ideal), the inverse image ideal sheaf $g^{- 1} I = Im (g^{*} I \to O_{X^{'}})$ , the blowups ${Bl}_{I} X$ and ${Bl}_{g^{- 1} I} X^{'}$ (Blowup of a scheme along an ideal sheaf), and the base change $X \times_{X} X^{'}$ of (Base change of objects, morphisms and properties).

[F1]

Flat and faithfully flat modules and ring homomorphisms: A module $M$ over a commutative ring $R$ is flat if $- \otimes_{R} M$ preserves exact sequences; a ring map $A \to B$ is flat when $B$ is flat as an $A$ -module. Flatness of a morphism of schemes is the corresponding local condition.

[F2]

Scheme pullback preserves quasi-coherence: Pullback of a quasi-coherent module is quasi-coherent, and on affine opens with $f (U) \subseteq V$ , $U = Spec B$ , $V = Spec A$ and $F ∣_{V} = \tilde{M}$ , one has $f^{*} F ∣_{U} ≅ \tilde{(B \otimes_{A} M)}$ .

[F3]

Tensor product preserves quasi-coherence: The tensor product of quasi-coherent $O_{X}$ -modules is quasi-coherent, and on an affine open $Spec A$ with $F = \tilde{M}$ , $G = \tilde{N}$ it restricts to $\tilde{(M \otimes_{A} N)}$ .

[F4]

Rees algebra sheaf of a finite type ideal: The Rees algebra sheaf is $R (I) = ⨁_{n \geq 0} I^{n}$ with degree- $n$ piece $I^{n}$ , multiplication induced by multiplication in $O_{X}$ ; it is a quasi-coherent graded $O_{X}$ -algebra.

[F5]

Blowup of a scheme along an ideal sheaf: For a scheme $X$ and a quasi-coherent ideal sheaf of finite type, ${Bl}_{I} X = {Proj}_{X} R (I)$ with structural morphism to $X$ and relative twists.

[F6]

Relative Proj commutes with arbitrary base change: For $g ⁣ : S^{'} \to S$ and a quasi-coherent graded $O_{S}$ -algebra $A$ with $A^{'} = g^{*} A$ , there is a canonical isomorphism of $S^{'}$ -schemes ${Proj}_{S} A \times_{S} S^{'} ≅ {Proj}_{S^{'}} A^{'}$ , natural in $S^{'} \to S$ , compatible with the relative twists; no flatness is required.

[F7]

Affine blowup standard charts and overlaps and Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: The blowup of $(x, y) \subset k [x, y]$ has charts $k [x, T]$ with $y = x T$ and $k [y, U]$ with $x = y U$ , with inverse ratio transition. The blowup of a principal regular ideal $(x)$ in $k [x]$ is the identity, since its sole chart is $k [x] [(x) / x] = k [x]$ .

Proof

1.1F1F2F3

Flat pullback commutes with the ideal powers and with the inverse image ideal: the natural maps $g^{*} I \to g^{- 1} I$ and $g^{*} (I^{n}) \to (g^{- 1} I)^{n}$ are isomorphisms. Affine-locally over $U = Spec A \subseteq X$ with $I ∣_{U} = \tilde{I}$ and $U^{'} = Spec B \subseteq X^{'}$ mapping into $U$ , flatness of $g$ says $B$ is a flat $A$ -module; the sequence $0 \to I^{n} \to A \to A / I^{n} \to 0$ then stays exact after $- \otimes_{A} B$ , so $B \otimes_{A} I^{n} \to B$ is injective, and its image is the ideal $I^{n} B = (I B)^{n}$ ; the pullback sheaf $g^{*} (I^{n})$ restricts to $\tilde{(B \otimes_{A} I^{n})}$ by [F2] and $(g^{- 1} I)^{n}$ restricts to $\tilde{(I B)^{n}}$ by [F2] and [F3], so the comparison is an isomorphism, and taking $n = 1$ identifies $g^{*} I$ with its image $g^{- 1} I$ in $O_{X^{'}}$ .

2.1F3F4step 1.1

Consequently $g^{*} R (I) ≅ R (g^{- 1} I)$ as quasi-coherent graded $O_{X^{'}}$ -algebras: by step 1.1 the degree- $n$ pieces are both $(g^{- 1} I)^{n}$ , and the pullback of the multiplication $I^{m} \otimes I^{n} \to I^{m + n}$ is the multiplication of the inverse image ideal, so the graded algebra structures agree; all pieces are quasi-coherent by [F2], [F3] and [F4].

3.1F5F6step 2.1

Applying [F6] to the morphism $g ⁣ : X^{'} \to X$ and the graded algebra $A = R (I)$ gives a canonical isomorphism of $X^{'}$ -schemes ${Bl}_{I} X \times_{X} X^{'} = {Proj}_{X} R (I) \times_{X} X^{'} ≅ {Proj}_{X^{'}} g^{*} R (I)$ , and step 2.1 identifies the target with ${Proj}_{X^{'}} R (g^{- 1} I) = {Bl}_{g^{- 1} I} X^{'}$ by [F5]; the inverse of this composite is the canonical isomorphism of the statement. The comparison is compatible with the structural morphisms because both sides are the relative Proj of the pulled-back graded algebra over $X^{'}$ , and with the relative twists by the corresponding clause of [F6].

4.1F1F7step 3.1∎

The comparison need not be an isomorphism without flatness. Take $A = k [x, y]$ , $I = (x, y)$ and $B = A / (y) = k [x]$ . Then $I B = (x)$ is principal regular and its blowup is $Spec B$ . Base changing the two charts of the original blowup gives $k [x, T] / (x T)$ and $k [U]$ , respectively, with inverse ratio gluing. The fiber of this base-changed blowup over $x = 0$ is the two affine lines glued by $U = T^{- 1}$ , hence $P_{k}^{1}$ , whereas the fiber of ${Bl}_{(x)} Spec B$ is $Spec k$ . Thus the comparison is not an isomorphism. The difference already appears in degree two: $B \otimes_{A} I^{2} = I^{2} / y I^{2}$ contains the nonzero class of $x y$ , since $x \notin I^{2}$ and cancellation of $y$ in $A$ shows $x y \notin y I^{2}$ . This class maps to zero in $(I B)^{2}$ , and is killed by $x$ since $x^{2} y \in y I^{2}$ . Hence the flatness hypothesis cannot be dropped, and the stated torsion caveat is proved within this item.

Remarks

The failure of flatness is not a defect of the relative Proj construction but of the identification of the pulled-back Rees algebra with the Rees algebra of the pulled-back ideal: Nonflat base change of a blowup can fail ↗ computes the torsion kernel in degree two and shows that the two sides of the comparison are not isomorphic.
The theorem applies in particular to open immersions and to flat morphisms of finite type over a field, and no finite presentation of $g$ is assumed.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Strict transform of a closed subscheme

Definition

Assume the Axiom of Choice, inherited from the relative Proj construction used by the blowup (The Axiom of Choice). Let $X$ be a scheme, let $I$ be a quasi-coherent ideal sheaf of finite type (Quasi-coherent ideal sheaves), let $π ⁣ : {Bl}_{I} X \to X$ be the blowup of Blowup of a scheme along an ideal sheaf with exceptional subscheme $E = π^{- 1} (Z)$ (Exceptional subscheme of a blowup), and let $Y ↪ X$ be a closed subscheme (Closed immersions of schemes) with scheme-theoretic inverse image $Y \times_{X} {Bl}_{I} X$ . Write $U : = (Y \times_{X} {Bl}_{I} X) ∖ E$ for the open subscheme obtained by deleting $E$ from this inverse image, with its open immersion $j ⁣ : U \to Y \times_{X} {Bl}_{I} X$ .

The strict transform (or proper transform) of $Y$ is the scheme-theoretic closure of $U$ in $Y \times_{X} {Bl}_{I} X$ , that is, the scheme-theoretic image $Y^{'} : = \overline{U}$ of $j$ (Scheme-theoretic image). This image exists for all the data above: the kernel ideal sheaf $K = \ker (O_{Y \times_{X} Bl} \to j_{*} O_{U})$ is quasi-coherent by the chart calculation below. The corresponding closed subscheme exists by Quasi-coherent ideals and closed subschemes, complete route and is the smallest closed subscheme through which $j$ factors. Its structural map $Y^{'} \to Y$ is the restriction of the projection.

Chartwise description. The strict transform is determined by its restriction to the standard affine charts of the blowup. Let $U_{0} = Spec A \subseteq X$ be an affine open and let $a \in Γ (U_{0}, I)$ ; on the chart $Spec A [I / a] \subseteq {Bl}_{I} X$ the exceptional subscheme is cut out by $a$ by Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains and Exceptional subscheme of a blowup, the inverse image of $Y$ is cut out by the ideal $J A [I / a]$ , where $J = Γ (U_{0}, I_{Y})$ , and the strict transform meets the chart in the closed subscheme defined by the saturation $(J A [I / a] : a^{\infty}) = {f \in A [I / a] : a^{n} f \in J A [I / a] for some n \geq 0},$ the largest ideal of the chart ring containing $J A [I / a]$ whose localisation at $a$ equals the localisation of $J A [I / a]$ . To verify both existence and this description, put $C = A [I / a]$ and $H = J C$ . The inverse image of $Y$ on this chart is $Spec (C / H)$ , and its intersection with $U$ is $D (a)$ (A principal localization identifies its spectrum with a distinguished open). The kernel of $C \to (C / H)_{a}$ is precisely $(H : a^{\infty})$ : a class becomes zero after localization exactly when some power of $a$ annihilates it. For every $g \in C$ , localization of this kernel at $g$ is the kernel of $C_{g} \to (C / H)_{a g}$ , by Localisation of modules is exact. Thus $K$ on $Spec (C / H)$ is the associated sheaf of $(H : a^{\infty}) / H$ , so it is quasi-coherent without any Noetherian hypothesis. These kernels agree on chart overlaps, since they all consist of sections whose restriction to $U$ is zero; equivalently the ratio transition of Affine blowup standard charts and overlaps makes $a$ and $b$ unit multiples. Consequently the chart subschemes glue to the closed subscheme cut out by $K$ . Its ideal restricts to zero on $U$ , so $j$ factors through it. Any other closed subscheme through which $j$ factors has ideal contained in $K$ , and therefore contains this one. This proves that the glued saturation construction is the scheme-theoretic closure in all cases, including an empty deleted open, whose closure is empty.

Reducedness. If $Y$ is reduced, then $Y^{'}$ is reduced. Indeed, on a chart as above the ring of the strict transform is $A [I / a] / (J A [I / a] : a^{\infty})$ , and the saturation is by construction the kernel of the localisation $A [I / a] / (J A [I / a]) \to A [I / a] [1 / a] / (J)$ , so this ring embeds into $(A [I / a] / (J)) [1 / a] = (A / J) [1 / a]$ , which is reduced when $Y$ is; a subring of a reduced ring is reduced, and reducedness is local, so $Y^{'}$ is reduced.

Iteration. The construction applies verbatim to a blowup of ${Bl}_{I} X$ along any quasi-coherent ideal sheaf of finite type on it: for a closed subscheme $W ↪ {Bl}_{I} X$ , its inverse image under a further blowup and the deletion of that blowup's exceptional subscheme define the strict transform of $W$ , and the chartwise saturation description is unchanged. In particular a strict transform of a strict transform may be formed along a further blowup.

Remarks

The strict transform depends on the scheme structure of the center, not just its underlying closed set. Multiplication of its ideal by an invertible ideal can preserve the blowup canonically while changing the center, exceptional locus and deleted open; it need not preserve strict transforms. For example, on $A_{k}^{2}$ the ideals $O$ and $(x)$ both have identity blowup. The strict transform of $V (x)$ is $V (x)$ for the first center and empty for the second.
The chartwise saturation description is the one used in computations: the strict transform of a hypersurface with local equation $f$ in the chart of $a$ is cut out by the saturation $(f : a^{\infty})$ , which removes the components supported inside the exceptional divisor; no closure operation is visible beyond this saturation.
No reducedness, regularity or normality of $X$ or $Y$ is assumed; the reducedness conclusion above is a statement about the strict transform, not about $Y \times_{X} {Bl}_{I} X$ , which need not be reduced even for reduced $Y$ .

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Total transform of a Cartier divisor

Definition

Assume the Axiom of Choice (The Axiom of Choice) as inherited from the blowup construction, which is a relative Proj. Let $π ⁣ : {Bl}_{I} X \to X$ be the blowup of a scheme $X$ along a quasi-coherent ideal sheaf $I$ of finite type (Blowup of a scheme along an ideal sheaf), and let $D$ be a Cartier divisor on $X$ (Cartier divisor) whose pullback $π^{*} D$ along $π$ is defined (Pullback of a Cartier divisor). The total transform of $D$ under $π$ is the pullback Cartier divisor

$π^{*} D .$

Its associated invertible sheaf is the pullback of the invertible sheaf of $D$ : there is a canonical isomorphism $O_{Bl} (π^{*} D) ≅ π^{*} O_{X} (D)$ of $O_{Bl}$ -modules (Pullback of a Cartier divisor computes the pullback of its line bundle).

The pullback is defined for every effective Cartier divisor. Indeed, on a standard chart $A [I / a] \subseteq A_{a}$ , a nonzerodivisor $f \in A$ remains a nonzerodivisor after localization and on this subalgebra. Thus every effective local equation pulls back to a regular equation, without a dominance assumption. For integral $X$ and nonzero $I$ , the chart embeddings in the function field similarly pull back nonzero rational local equations, so every Cartier divisor has a pullback. If $I = 0$ , the blowup is empty and these assertions hold vacuously.

For a reduced curve $D$ on a regular surface, blowing up a closed point with two-dimensional regular local ring gives the formula $π^{*} D = D^{'} + m E,$ where $m$ is the order of its local equation at that point. This formula is proved in Total transform equals strict transform plus multiplicity times the exceptional divisor ↗; it is not part of the definition for arbitrary centers. For example, on $A_{k}^{2}$ , the blowup of $(x^{2})$ is the identity, its exceptional Cartier divisor is $2 V (x)$ , and the strict transform of $V (x)$ is empty. No integer $m$ expresses $V (x)$ as $m \cdot 2 V (x)$ .

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Total transform equals strict transform plus multiplicity times the exceptional divisor

Statement

Assume the Axiom of Choice. Let $S$ be a regular surface over a field $k$ and let $C \subseteq S$ be a reduced curve (an effective Cartier divisor) with a closed point $p$ such that $\dim O_{S, p} = 2$ , at which the multiplicity $m = {mult}_{p} (C)$ of a local equation is finite and positive. Let $π ⁣ : S^{'} \to S$ be the blowup of the point $p$ with exceptional curve $E$ and let $C^{'}$ be the strict transform of $C$ . Then $π^{*} C = C^{'} + m E$ as effective Cartier divisors on $S^{'}$ ; equivalently, the strict transform is defined by dividing a local equation of the total transform by the $m$ -th power of an exceptional equation on each chart, and $C^{'}$ meets $E$ in the $0$ -cycle of degree $m$ cut out by the degree- $m$ leading form of a local equation of $C$ at $p$ (its degree over $κ (p)$ is $m$ ; its degree over $k$ is $m [κ (p) : k]$ when this residue degree is finite).

Facts & Assumptions

Given: A regular surface $S$ over $k$ , a reduced curve $C \subseteq S$ that is an effective Cartier divisor, a closed point $p \in S$ with $\dim O_{S, p} = 2$ at which a local equation $f$ of $C$ has finite positive multiplicity, the blowup $π ⁣ : S^{'} \to S$ of $p$ , the exceptional curve $E$ , and the strict transform $C^{'}$ of $C$ .

[A1]

Choice. The Axiom of Choice is assumed as inherited from the blowup and associated-graded constructions used below. (The Axiom of Choice).

[F1]

Total transform of a Cartier divisor: The total transform of an effective Cartier divisor is its effective Cartier pullback, with associated line bundle the pulled-back line bundle.

[F2]

Strict transform of a closed subscheme: The strict transform of a closed subscheme is the scheme-theoretic closure of the inverse image minus $E$ ; when the ideal of $E$ is invertible on a chart, it is the closed subscheme defined by the saturation of the inverse-image ideal by the ideal of $E$ .

[F3]

The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier: The inverse image ideal of the blowup is invertible, and $E = V (I O_{Bl})$ is an effective Cartier divisor cut locally by a generator of that ideal.

[F4]

Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: For a domain $A$ and $a \neq 0$ , the affine blowup algebra $A [I / a]$ is a domain with $I A [I / a] = a A [I / a]$ .

[F5]

associated graded ring of a regular local ring: At the regular local ring $A = O_{S, p}$ of dimension two with regular parameters $x, y$ , the associated graded ring is ${gr}_{m} A = κ (p) [X, Y]$ with $X, Y$ the initial classes of $x, y$ .

[F6]

Cartier divisor local equation equivalence: Two effective Cartier divisors agree where their local equations differ by a unit; effective Cartier data are exactly principal ideals generated by regular sections.

[F7]

Affine blowup standard charts and overlaps: The blowup at $p$ has the two charts $Spec A [y / x]$ and $Spec A [x / y]$ , glued by inverting the ratio.

[F8]

Regular centers have projective-bundle exceptional divisors: At a closed point with two-dimensional regular local ring, the exceptional curve is $P_{κ (p)}^{1}$ . The quotient chart presentations are established directly in step 1.1.

[F9]

regular local rings are domains and cohen macaulay: The regular local ring $A$ is a domain; its regular parameters $x, y$ form a regular sequence, so $x$ is a nonzerodivisor and $y$ is a nonzerodivisor modulo $x$ .

Proof

1.1A1F4F5F7F9

At $p$ put $A = O_{S, p}$ , $m = (x, y)$ and $κ = κ (p)$ . The equation $f \in m^{m} ∖ m^{m + 1}$ has nonzero leading form $f_{m} (X, Y)$ in $κ [X, Y]$ . Write the finite ideal-power expression $f = \sum_{i + j = m} c_{i j} x^{i} y^{j}$ with $c_{i j} \in A$ . By [F9], $A$ is a domain and $x, y$ form a regular sequence. The chart is $B_{x} = A [T] / (x T - y) = A [y / x]$ : if $x h = (x T - y) q$ , reduction modulo $x$ and regularity of $y$ modulo $x$ give $q = x q_{1}$ , and cancellation gives $h = (x T - y) q_{1}$ ; hence the incidence quotient has no $x$ -power torsion and [F4, F7] identify it with the chart. In this chart, the ideal-power expression gives $f = x^{m} g$ , where $g = \sum c_{i j} T^{j}$ and $g m o d x = f_{m} (1, T) \neq 0$ . The ring $B_{x}$ is a domain and $B_{x} / x B_{x} = κ [T]$ , so $x$ is prime and does not divide $g$ .

2.1F1F2F3F6F8step 1.1

If $x h = g q$ in $B_{x}$ , primality of $x$ forces $q = x q_{1}$ , and cancellation gives $h = g q_{1}$ . Therefore $(g) : x = (g)$ , and iteration gives $(x^{m} g) : x^{\infty} = (g)$ . The strict-transform chart is thus $V (g)$ . Since $g$ is a nonzero element of a domain, it is a Cartier equation, and the factorization $f = x^{m} g$ gives the total-transform identity there. In the second chart $B_{y} = A [U] / (y U - x)$ the identical argument gives $f = y^{m} h$ , with $h m o d y = f_{m} (U, 1) \neq 0$ . On the overlap $y = x T$ , so cancellation of $x^{m}$ gives $g = T^{m} h$ ; the equations differ by a unit and glue. Off the exceptional curve the blowup is the identity, as follows by inverting the chart denominators. Hence globally $C^{'}$ is effective Cartier and $π^{*} C = C^{'} + m E$ .

3.1F5F8step 2.1∎

On $E ≅ P_{κ}^{1}$ , these equations cut the homogeneous divisor of the nonzero degree- $m$ form $f_{m}$ . To include the entire projective line, set $d = \deg f_{m} (1, T) \leq m$ . Its zeros in this affine chart have total degree $d$ , since $κ [T] / (f_{m} (1, T))$ has dimension $d$ (zero if $d = 0$ ); this counts local lengths times residue degrees. At the omitted point, $U = 0$ , the identity $f_{m} (U, 1) = U^{m} f_{m} (1, 1 / U)$ gives order $m - d$ . Thus the complete zero-cycle degree over $κ$ is $d + (m - d) = m$ . When $[κ : k]$ is finite, each residue degree over $k$ is that degree times its degree over $κ$ , giving $m [κ : k]$ . No splitting or separability assumption is used.

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The exceptional divisor is the projectivized normal cone

Statement

Assume the Axiom of Choice, inherited from the relative Proj construction (The Axiom of Choice). Let $Z = V (I)$ be a closed subscheme of a scheme $X$ cut out by a quasi-coherent ideal sheaf $I$ of finite type (Quasi-coherent ideal sheaves) and let $E = π^{- 1} (Z)$ be the exceptional subscheme of the blowup $π ⁣ : {Bl}_{I} X \to X$ (Exceptional subscheme of a blowup). Then there is a canonical isomorphism of $Z$ -schemes $E ⟶ {Proj}_{Z} ({gr}_{I} O_{X}) = {Proj}_{Z} (⨁_{n \geq 0} I^{n} / I^{n + 1}),$ the projectivized normal cone of $Z$ in $X$ . On an affine chart $Spec A$ with $I = (f_{0}, \dots, f_{r})$ , the fibre of $E$ over a point $z$ of $Z$ is $Proj ({gr}_{I} (A) \otimes_{A / I} κ (z))$ , the projectivized fibre of the normal cone (for a closed point center this is its projectivized tangent cone), and the closed immersion $E ↪ {Bl}_{I} X$ identifies $E$ with the divisor $V (a)$ in the chart $Spec A [I / a]$ .

Facts & Assumptions

Given: A scheme $X$ , a quasi-coherent ideal sheaf $I$ of finite type with zero scheme $Z = V (I)$ , the blowup $π ⁣ : {Bl}_{I} X = {Proj}_{X} R (I) \to X$ (Blowup of a scheme along an ideal sheaf), the exceptional subscheme $E = π^{- 1} (Z) = Z \times_{X} {Bl}_{I} X$ with ideal sheaf $I O_{Bl}$ (Exceptional subscheme of a blowup), and the associated graded sheaf ${gr}_{I} O_{X} = ⨁_{n \geq 0} I^{n} / I^{n + 1}$ (The associated graded ring and associated graded module of an ideal-adic filtration).

[F1]

Exceptional subscheme of a blowup: The exceptional subscheme is the scheme-theoretic inverse image $E = Z \times_{X} {Bl}_{I} X$ , with ideal sheaf the inverse image ideal $I O_{Bl}$ ; it is a closed subscheme of the blowup mapping to $Z$ .

[F2]

Relative Proj commutes with arbitrary base change: For a morphism $S^{'} \to S$ and a quasi-coherent graded $O_{S}$ -algebra $A$ , there is a canonical isomorphism ${Proj}_{S} A \times_{S} S^{'} ≅ {Proj}_{S^{'}} (A \otimes_{O_{S}} O_{S^{'}})$ , natural in the base and compatible with graded quotients; no flatness is needed.

[F3]

Rees algebra sheaf of a finite type ideal and The associated graded ring and associated graded module of an ideal-adic filtration: $R (I) = ⨁_{n \geq 0} I^{n}$ , and the degree- $n$ piece of $R (I) / I R (I)$ is $I^{n} / I^{n + 1}$ , so $R (I) / I R (I) ≅ {gr}_{I} O_{X}$ as quasi-coherent graded $O_{X}$ -algebras; the identifications are compatible with restriction to open subschemes and with localisation.

[F4]

Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains and Affine blowup standard charts and overlaps: On the chart $Spec A [I / a] \subseteq {Bl}_{I} Spec A$ , for $a \in I$ , one has $I A [I / a] = a A [I / a]$ with $a$ a nonzerodivisor, and the charts over a generating family cover the blowup.

[F5]

Blowups restrict to open subschemes of the base: The blowup of $U \subseteq X$ is the restriction of the blowup of $X$ , canonically over $U$ .

Proof

1.1F1F2F5

Let $U = Spec A \subseteq X$ be affine with $I ∣_{U} = \tilde{I}$ . Restricting the fibre product of [F1] to $U$ gives $E \times_{X} U = Spec (A / I) \times_{Spec A} {Bl}_{I} U$ , and ${Bl}_{I} U = {Proj}_{A} R (I)$ ; applying [F2] to the morphism $Spec (A / I) \to Spec A$ and the graded algebra $R (I)$ yields a canonical isomorphism ${Proj}_{A} R (I) \times_{Spec A} Spec (A / I) ≅ {Proj}_{A / I} (R (I) \otimes_{A} A / I)$ .

1.2F1F4

In the standard chart $Spec A [I / a]$ , $a \in I$ , the ideal of $E$ is $I O_{Bl} ∣_{chart} = I A [I / a] = a A [I / a]$ by [F1] and [F4], so $E$ meets the chart in $V (a)$ , and the charts over a generating family of $I$ cover the blowup by [F4].

2.1F3step 1.1

By [F3] the graded $A / I$ -algebra $R (I) \otimes_{A} A / I = R (I) / I R (I)$ has degree- $n$ piece $I^{n} / I^{n + 1}$ , so it is canonically ${gr}_{I} (A)$ ; combining with step 1.1 gives a canonical isomorphism $E \times_{X} U ≅ {Proj}_{A / I} {gr}_{I} (A)$ over $Spec (A / I)$ .

3.1F2F3F5step 2.1

The isomorphisms of step 2.1 are canonical and compatible with restriction to smaller affine opens: for $A \to A_{f}$ both $Bl$ and the associated graded construction localise, $I^{n} A_{f} / I^{n + 1} A_{f} = (I A_{f})^{n} / (I A_{f})^{n + 1}$ , and the identifications of [F2] are natural; hence they glue over an affine cover of $X$ to a canonical isomorphism of $Z$ -schemes $E \to {Proj}_{Z} ({gr}_{I} O_{X})$ , the projectivized normal cone.

4.1F2step 3.1

For a point $z \in Z$ , applying [F2] to the morphism $Spec κ (z) \to Z$ and the graded $O_{Z}$ -algebra ${gr}_{I} O_{X}$ identifies the fibre of ${Proj}_{Z} ({gr}_{I} O_{X})$ over $z$ , and hence the fibre of $E$ over $z$ by step 3.1, with ${Proj}_{κ (z)} ({gr}_{I} (A) \otimes_{A / I} κ (z))$ ; for a closed point center, where $I = m$ is maximal, ${gr}_{m} (A) = ⨁_{n \geq 0} m^{n} / m^{n + 1}$ is the associated graded ring of the local ring at the centre, so its Proj is the projectivized tangent cone.

5.1step 2.1step 1.2step 4.1∎

Steps 2.1, 1.2 and 4.1 prove all the assertions: $E ≅ {Proj}_{Z} ({gr}_{I} O_{X})$ over $Z$ , the fibre description over points of $Z$ , and the identification of $E$ with the divisor $V (a)$ in each standard chart.

Remarks

The computation is the reason the exceptional divisor of a point blowup is a projective space: for a reduced point the associated graded of a regular local ring is a polynomial ring, so the projectivized tangent cone is projective space over the residue field.
No regularity, Noetherianity or reducedness of $Z$ is assumed; the identification with the projectivized normal cone is purely a statement about the Rees algebra and its quotient by $I$ .

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Regular centers have projective-bundle exceptional divisors

Statement

Assume the Axiom of Choice. Let $i ⁣ : Z \to X$ be a regular immersion (so the conormal sheaf $I / I^{2}$ is locally free, with $I$ the ideal sheaf of $Z$ ), and let $E$ be the exceptional divisor of the blowup of $X$ along $Z$ . Then $E \to Z$ is canonically isomorphic to the projective bundle $P_{Z} (I / I^{2}) = {Proj}_{Z} Sym (I / I^{2})$ in the quotient convention. In particular, if $p$ is a closed point of a regular surface $S$ over a field $k$ with $\dim O_{S, p} = 2$ (automatic for finite-type pure-dimensional surfaces), then $I / I^{2} = m_{p} / m_{p}^{2}$ is free of rank two over $κ (p)$ and $E$ is isomorphic to the projective line $P_{κ (p)}^{1}$ over $κ (p)$ .

Facts & Assumptions

Given: The Axiom of Choice, a regular immersion $i ⁣ : Z \to X$ with ideal sheaf $I$ , the blowup of $X$ along $Z$ with exceptional divisor $E$ , and, for the second claim, a closed point $p$ of a regular surface $S$ over a field $k$ with two-dimensional local ring.

[A1]

Regular-immersion hypothesis, local form. The immersion $i$ is regular: every point of $Z$ has an affine open neighbourhood in $X$ $Spec A$ on which $Z$ is cut out by an $A$ -regular sequence $f_{1}, \dots, f_{c} \in A$ , and the conormal sheaf $I / I^{2}$ is locally free there, with the classes of $f_{1}, \dots, f_{c}$ as a basis over $A / J$ , $J = (f_{1}, \dots, f_{c})$ ; the empty sequence $c = 0$ is allowed.

[A2]

Choice. The Axiom of Choice is assumed, as in the statement, and the cited suppliers used below are stated under it.

[F1]

The exceptional divisor is the projectivized normal cone: Assume the Axiom of Choice. Let $Z = V (I)$ be a closed subscheme of $X$ cut out by a quasi-coherent ideal sheaf $I$ of finite type and let $E = π^{- 1} (Z)$ be the exceptional subscheme of the blowup. Then there is a canonical isomorphism of $Z$ -schemes $E \to {Proj}_{Z} ({gr}_{I} O_{X}) = {Proj}_{Z} (⨁_{n \geq 0} I^{n} / I^{n + 1})$ , the projectivized normal cone of $Z$ in $X$ .

[F2]

Associated graded algebra of an ideal generated by a regular sequence: Let $R$ be a commutative ring and $f_{1}, \dots, f_{c}$ an $R$ -regular sequence, $J = (f_{1}, \dots, f_{c})$ . The canonical graded homomorphism $(R / J) [X_{1}, \dots, X_{c}] \to {gr}_{J} R$ , $X_{i} \mapsto f_{i}$ modulo $J^{2}$ , is an isomorphism. In particular $J / J^{2}$ is free on the classes of $f_{i}$ and ${Sym}_{R / J} (J / J^{2}) = {gr}_{J} R$ . No Noetherian or domain hypothesis is required.

[F3]

associated graded ring of a regular local ring: Assume the Axiom of Choice. If $(R, m, k)$ is regular local of dimension $d$ , any cotangent basis induces a graded isomorphism $k [X_{1}, \dots, X_{d}] ≅ {gr}_{m} R$ .

[F4]

embedding dimension and regular local ring: For a nonzero commutative Noetherian local ring $(R, m, k)$ , $edim R = \dim_{k} (m / m^{2})$ . The ring is regular local when $edim R = \dim R$ .

[F5]

embedding dimension is minimal maximal ideal generator number: Assume the Axiom of Choice. For a nonzero Noetherian local ring $(R, m, k)$ , $edim R$ is the least number of generators of $m$ .

[F6]

Projective bundle in the quotient convention: Let $E$ be a finite locally free $O_{S}$ -module of locally constant rank $r \geq 0$ . The projective bundle of $E$ over $S$ is the relative Proj $P_{S} (E) = {Proj}_{S} Sym (E) \to S$ , with the quotient convention: over an $S$ -scheme $g ⁣ : T \to S$ , an $S$ -morphism $T \to P_{S} (E)$ is the same as an isomorphism class of surjections $g^{*} E \to L$ with $L$ invertible on $T$ .

[F7]

regular local rings are domains and cohen macaulay, one dimensional regular local rings are dvrs: Regular parameters form a regular sequence; a one-dimensional regular local ring is a DVR.

Proof

1.1A1F2

Let $Spec A \subseteq X$ be an affine chart on which $I$ is generated by an $A$ -regular sequence $f_{1}, \dots, f_{c}$ and put $J = (f_{1}, \dots, f_{c})$ , so that $J / J^{2}$ is free on the classes of the $f_{i}$ with ${Sym}_{A / J} (J / J^{2}) ≅ A / J [X_{1}, \dots, X_{c}]$ . By [F2] the canonical graded homomorphism $A / J [X_{1}, \dots, X_{c}] \to {gr}_{J} A$ , $X_{i} \mapsto f_{i} + J^{2}$ , is an isomorphism, so it is a canonical isomorphism ${Sym}_{A / J} (J / J^{2}) \to {gr}_{J} A$ ; for the empty sequence this reads $Sym (0) = A / J = {gr}_{0} A$ .

2.1A2F1F6step 1.1

Over this chart [F1] identifies the exceptional divisor $E$ with ${Proj}_{Z} ({gr}_{I} O_{X})$ , that is, with ${Proj}_{Spec A / J} ({gr}_{J} A)$ , canonically in $Z$ ; combining with step 1.1 gives a canonical isomorphism $E ≅ {Proj}_{A / J} Sym (J / J^{2}) = P_{Spec A / J} (J / J^{2})$ over the chart, the projective bundle in the quotient convention recorded in [F6]. In the degenerate case $J = 0$ both sides of this display are empty and the identification is the empty isomorphism.

3.1F1F6step 2.1

These neighborhoods cover $Z$ . Over $X ∖ Z$ the inverse-image exceptional subscheme is empty, so no regular-sequence presentation of the unit ideal is required there. The chartwise isomorphisms of step 2.1 are canonical: on an overlap of two charts each is induced by the canonical graded multiplication map $Sym (I / I^{2}) \to {gr}_{I} O_{X}$ , together with the canonical isomorphism of [F1], which involves no choices; hence they agree on overlaps and glue to a single canonical isomorphism of $Z$ -schemes $E \to P_{Z} (I / I^{2}) = {Proj}_{Z} Sym (I / I^{2})$ , the projective bundle of [F6] in the quotient convention.

4.1F4F7step 3.1

For the second claim let $p$ have the stated two-dimensional regular local ring on $S$ and let $I$ be the ideal sheaf of $Z = {p}$ , so that $I / I^{2} = m_{p} / m_{p}^{2}$ . The point immersion is regular: its regular parameters form a regular sequence at $p$ , and this property and generation of the point ideal extend to a neighborhood by killing the finite coherent quotients and multiplication kernels whose stalks vanish at $p$ . Thus step 3.1 gives a canonical isomorphism $E \to P_{Z} (m_{p} / m_{p}^{2})$ of schemes over $Spec κ (p)$ . By hypothesis $O_{S, p}$ is a regular local ring of dimension two with residue field $κ (p)$ , so [F4] gives $edim O_{S, p} = \dim_{κ (p)} (m_{p} / m_{p}^{2}) = \dim O_{S, p} = 2$ ; thus $m_{p} / m_{p}^{2}$ is a free $κ (p)$ -module of rank two.

5.1A2F3F5step 4.1

Let $u, v$ be a $κ (p)$ -basis of $m_{p} / m_{p}^{2}$ ; it has exactly two elements by step 4.1, and by [F5] the lifted elements $u, v$ minimally generate $m_{p}$ , so their initial classes generate ${gr}_{m_{p}} O_{S, p}$ in degree one. Applying [F3] to the two-dimensional regular local ring $O_{S, p}$ with this cotangent basis gives a graded $κ (p)$ -algebra isomorphism $κ (p) [X, Y] \to {gr}_{m_{p}} O_{S, p}$ with $X \mapsto u$ , $Y \mapsto v$ .

6.1F6step 3.1step 5.1∎

Finally $P_{Z} (m_{p} / m_{p}^{2}) = {Proj}_{κ (p)} Sym (m_{p} / m_{p}^{2})$ for the free rank-two module $m_{p} / m_{p}^{2}$ is by [F6] exactly ${Proj}_{κ (p)} κ (p) [X, Y] = P_{κ (p)}^{1}$ , and step 5.1 identifies this graded algebra with ${gr}_{m_{p}} O_{S, p}$ ; hence, with the canonical isomorphism $E \to P_{Z} (I / I^{2})$ of step 3.1, the exceptional divisor satisfies $E ≅ P_{κ (p)}^{1}$ and $I / I^{2}$ is free of rank two over $κ (p)$ , as claimed.

Remarks

For a general Noetherian regular surface a closed point can instead have local dimension one. Its point ideal is then locally Cartier (a DVR parameter near the point and the unit ideal elsewhere), and its principal regular chart algebra is $A [(a) / a] = A$ , so its blowup is the identity and the exceptional fiber is $P_{κ (p)}^{0} = Spec κ (p)$ . No degree $- 1$ on a point is asserted. The main regular-immersion statement covers this rank-one case as well.

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Blowing up an effective Cartier divisor does nothing

Statement

Assume the Axiom of Choice, inherited from the blowup construction (The Axiom of Choice). Let $I$ be a quasi-coherent ideal sheaf of finite type on a scheme $X$ that is invertible as an $O_{X}$ -module (Invertible sheaves) — equivalently, $Z = V (I)$ is an effective Cartier divisor on $X$ (Effective cartier divisor, Cartier divisor). Then the blowup $π ⁣ : {Bl}_{I} X \to X$ of Blowup of a scheme along an ideal sheaf is an isomorphism. If $I$ is generated by a single nonzerodivisor $f$ on an affine open $U = Spec A$ , then the blowup is $Spec A [I / f] = Spec A$ over $U$ .

Facts & Assumptions

Given: A scheme $X$ , a quasi-coherent ideal sheaf $I$ of finite type with zero scheme $Z$ , and the blowup $π ⁣ : {Bl}_{I} X \to X$ .

[F1]

Effective cartier divisor and Cartier divisor: An effective Cartier divisor on $X$ is given by an open cover ${U_{i}}$ with regular sections $f_{i} \in Γ (U_{i}, O_{X})$ , regular meaning that multiplication by the germ $(f_{i})_{x}$ is injective for every $x \in U_{i}$ , and the local principal ideals $f_{i} O_{U_{i}}$ glue to an ideal sheaf $I_{D}$ ; conversely an invertible ideal sheaf is locally generated by one element and that generator is a nonzerodivisor, because the map $O_{U_{i}} \to I ∣_{U_{i}}$ , $1 \mapsto f_{i}$ , is an isomorphism.

[F2]

The sheaf of a Cartier divisor is invertible: For a Cartier divisor $D$ the sheaf $O_{X} (D)$ is invertible and locally freely generated by $f_{i}^{- 1}$ ; in particular the ideal sheaf of an effective Cartier divisor is an invertible $O_{X}$ -module.

[F3]

Affine blowup standard charts and overlaps and Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: For $I = (f_{0}, \dots, f_{r})$ on an affine $U = Spec A$ the standard charts are $Spec A [I / f_{i}]$ and they cover ${Bl}_{I} U$ ; for a principal ideal $I = (f)$ with $f$ a nonzerodivisor the affine blowup algebra is $A [I / f] = A$ .

[F4]

Universal property of the blowup: For every $X$ -scheme $Y \to X$ whose inverse image of $Z$ is an effective Cartier divisor there is a unique $X$ -morphism $Y \to {Bl}_{I} X$ ; equivalently ${Bl}_{I} X$ is final among such $X$ -schemes, and Uniqueness of the blowup makes the resulting identifications unique.

Proof

1.1F1F2

If $I$ is invertible then it is locally generated by a single element $f_{i}$ on an open cover ${U_{i}}$ and the generator is a nonzerodivisor by [F1], so the zero scheme $Z$ is an effective Cartier divisor with local equations $f_{i}$ ; conversely if $Z = V (I)$ is an effective Cartier divisor with local equations $f_{i}$ , then $I ∣_{U_{i}} = (f_{i})$ and [F2] exhibits $I$ as invertible.

2.1F3step 1.1

Suppose $I$ is invertible. On each affine open $U = Spec A$ with $I ∣_{U} = (f)$ and $f$ a nonzerodivisor, the single standard chart of the blowup is $Spec A [I / f] = Spec A = U$ by [F3], and it covers ${Bl}_{I} U = π^{- 1} (U)$ ; hence $π^{- 1} (U) \to U$ is an isomorphism.

3.1F4step 2.1

The restriction $π^{- 1} (U) \to U$ is an isomorphism for the members of an affine open cover of $X$ by step 2.1, and being an isomorphism is local on the target, so $π ⁣ : {Bl}_{I} X \to X$ is an isomorphism; its inverse is characterized by the universal property [F4] applied to the identity of $X$ , whose inverse image of $Z$ is the effective Cartier divisor $Z$ itself, so the inverse is the unique $X$ -morphism $X \to {Bl}_{I} X$ supplied there, and it is unique by Uniqueness of the blowup.

3.2F3step 2.1

In the affine case $X = Spec A$ and $I = \tilde{I}$ with $I = (f)$ for a nonzerodivisor $f$ , step 2.1 with $U = X$ gives ${Bl}_{I} Spec A = Spec A [I / f] = Spec A$ , as claimed.

4.1step 1.1step 3.1step 3.2∎

Steps 1.1, 3.1 and 3.2 prove the statement: invertible centers are exactly effective Cartier divisors, and the blowup along such an ideal sheaf is an isomorphism, computed on an affine chart as $Spec A [I / f] = Spec A$ when $I$ is generated by one nonzerodivisor.

Remarks

This is the case in which a blowup changes nothing at all: it is an isomorphism precisely when the centre is already Cartier, so nontrivial blowups require a centre that fails to be Cartier in a neighbourhood of itself.
Combining the statement with Invariance of the blowup under invertible (fractional) rescaling of the ideal shows that ${Bl}_{I} X$ depends only on the class of $I$ modulo invertible rescaling, a class that is trivial exactly in the Cartier case.

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Blowing up a rational point of a smooth surface

Statement

Assume the Axiom of Choice. Let $S$ be a smooth surface over a field $k$ and let $p \in S (k)$ be a $k$ -rational point. Then the blowup $S^{'} = {Bl}_{p} S$ is smooth over $k$ , with exceptional curve $E ≅ P_{k}^{1}$ and $O_{E} (E) ≅ O_{P_{k}^{1}} (- 1)$ (Point blowups of regular surfaces stay regular, with rational exceptional fibre over the residue field). Choose a sufficiently small affine neighbourhood $U = Spec R$ of $p$ and functions $x, y \in R$ generating the ideal of $p$ on $U$ and giving regular parameters at $p$ ; such a choice exists. Over $U$ the blowup is the incidence subscheme $V (x v - y u) \subseteq U \times_{k} P_{k}^{1},$ with homogeneous coordinates $(u : v)$ on the second factor, its two charts are $Spec R [T] / (x T - y) = Spec R [y / x]$ and $Spec R [U_{1}] / (y U_{1} - x) = Spec R [x / y]$ , and the overlap inverts $T$ and $U_{1}$ with $T U_{1} = 1$ . After base change to $Spec O_{S, p}$ , replace $R$ by $O_{S, p}$ in these formulas. The charts over $R$ are smooth surfaces over $k$ , and the local rings on $E$ have dimension one at its generic point and dimension two at its closed points. For $S = A_{k}^{2}$ with coordinates $x, y$ and $p = 0$ the charts are the affine planes $Spec k [x, T]$ and $Spec k [y, U_{1}]$ , and the incidence subscheme lies in $A_{k}^{2} \times_{k} P_{k}^{1}$ .

Facts & Assumptions

Given: A field $k$ , a smooth surface $S$ over $k$ , a $k$ -rational point $p$ , the local ring $A = O_{S, p}$ , regular parameters $\overset{ˉ}{x}, \overset{ˉ}{y} \in A$ , an affine neighbourhood $U = Spec R$ of $p$ , lifts $x, y \in R$ of $\overset{ˉ}{x}, \overset{ˉ}{y}$ , the blowup $π ⁣ : S^{'} \to S$ of $p$ , and the Axiom of Choice, inherited from the Proj and gluing constructions (The Axiom of Choice).

[F1]

Smooth morphism of schemes: $S \to Spec k$ is smooth, hence flat, locally of finite presentation, and geometrically regular on the fibres; the fibre over the unique point of $Spec k$ is $S$ itself, so every local ring of $S$ is regular. Smoothness is local on the source.

[F2]

embedding dimension and regular local ring, regular local rings are domains and cohen macaulay, regular local quotient by parameter is regular and localisations of regular local rings are regular: $A$ is a regular local ring of dimension two, $2 = \dim A = edim A$ ; regular local rings are domains and Cohen-Macaulay, their regular systems of parameters are regular sequences in any order, and $A / (\overset{ˉ}{x})$ is a regular local ring of dimension one, hence a domain.

[F3]

localisation and polynomial extension of regular rings: Localizations and finite polynomial extensions of a regular Noetherian ring are regular, and regularity is tested at maximal ideals.

[F4]

Affine blowup standard charts and overlaps and Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: For an ideal $I = (f_{0}, \dots, f_{r})$ the standard charts $Spec A [I / f_{i}]$ cover ${Bl}_{I} Spec A$ , and the homomorphism $A [x_{1}, \dots, x_{r}] / (a x_{i} - a_{i}) \to A [I / a]$ is surjective with kernel the $a$ -power torsion; the image of $a$ is a nonzerodivisor and $(A [I / a])_{a} = A_{a}$ .

[F5]

Blowups restrict to open subschemes of the base: On the open subscheme $U$ the blowup of the point is the blowup of $U$ along the restriction of the ideal sheaf of $p$ ; if $(x, y)$ is the ideal of $p$ on $U$ , this is ${Bl}_{(x, y)} U$ .

[F6]

Gluing affine schemes along compatible open isomorphisms: Affine schemes with open subschemes and isomorphisms on overlaps satisfying the cocycle condition glue to a scheme, uniquely up to unique isomorphism respecting the charts.

[F7]

Standard opens of Proj and Projective space is Proj of a polynomial ring: On $P_{k}^{1} = Proj k [u, v]$ the standard opens $D_{+} (u)$ and $D_{+} (v)$ are the affine lines $Spec k [T]$ , $T = v / u$ , and $Spec k [U_{1}]$ , $U_{1} = u / v$ , glued by $T U_{1} = 1$ .

[F8]

Point blowups of regular surfaces stay regular, with rational exceptional fibre over the residue field: ${Bl}_{p} S$ is regular of pure dimension two, $E$ is an effective Cartier divisor isomorphic to $P_{κ (p)}^{1} = P_{k}^{1}$ with $O_{E} (E) = O (- 1)$ , the base change to $Spec A$ has charts $Spec A [T] / (x T - y)$ and $Spec A [U_{1}] / (y U_{1} - x)$ glued by $T U_{1} = 1$ , the local rings on $E$ have dimension one at the generic point and two at closed points, and if $S$ is smooth over $k$ and $p$ is $k$ -rational then ${Bl}_{p} S$ is smooth over $k$ .

[F9]

Pushforward and vanishing for an affine point blowup: For a ring $A$ and $I = (x, y)$ generated by a regular sequence, the two standard charts cover ${Bl}_{I} Spec A$ and, over the affine base, the structure-sheaf pushforward is the structure sheaf of the base with all higher direct images vanishing.

[F10]

The blowup of the plane at the origin as an incidence scheme: Over $k [x, y]$ the blowup of the origin is $V (x v - y u) \subseteq A_{k}^{2} \times_{k} P_{k}^{1}$ with charts $Spec k [x, T]$ , $y = x T$ , and $Spec k [y, U_{1}]$ , $x = y U_{1}$ , glued by $T U_{1} = 1$ .

[F11]

The blowup is an isomorphism off the center: The blowup is an isomorphism over the complement of the centre, so the descriptions over the open neighbourhood $U$ glue to the global blowup.

Proof

1.1F1F2F5

By [F1] the local ring $A = O_{S, p}$ is regular, and by [F2] it has dimension and embedding dimension two, so there are regular parameters $\overset{ˉ}{x}, \overset{ˉ}{y}$ ; lifting them along $R \to R_{m_{p}} = A$ and clearing denominators gives $x, y \in R$ with these images, and the failure loci of the conditions below are closed subsets of the affine scheme $U$ not containing $p$ , so $U$ may be shrunk while keeping $p$ . Arrange that (i) $U$ is connected and $R$ is a domain: every local ring of the smooth surface $S$ is a domain by [F1] and [F2], so a connected affine open neighbourhood of $p$ has domain ring; (ii) $(x, y)$ generates the ideal of $p$ on $U$ , which holds at $p$ because the images generate $m_{p}$ and the locus where the two coherent ideals differ is closed and avoids $p$ ; (iii) $(x, y)$ and $(y, x)$ are regular sequences in $R$ , namely $x$ and $y$ are nonzerodivisors and each is a nonzerodivisor modulo the other: this holds at $p$ because $\overset{ˉ}{x}, \overset{ˉ}{y}$ is a regular system of parameters in the Cohen-Macaulay ring $A$ by [F2], and each failure is the support of the kernel of multiplication on a coherent module, a closed subset avoiding $p$ . Thus a sufficiently small affine neighbourhood and functions as in the statement exist.

2.1F4F6F7step 1.1algebra

Let $Z = V (x v - y u) \subseteq U \times_{k} P_{k}^{1}$ . By [F7] the two charts of the projective factor give $Z \cap {u \neq 0} = Spec R [T] / (x T - y)$ with $T = v / u$ , and $Z \cap {v \neq 0} = Spec R [U_{1}] / (y U_{1} - x)$ with $U_{1} = u / v$ ; on the overlap both $T$ and $U_{1}$ are invertible and $T U_{1} = 1$ , so $Z$ is obtained by gluing these two affine charts along $R [T, T^{- 1}] / (x T - y)$ . On the other hand, by [F4] the standard charts of ${Bl}_{(x, y)} U$ are $Spec R [(x, y) / x]$ and $Spec R [(x, y) / y]$ with overlap $R [(x, y) / x] [x / y]$ . Since $(x, y)$ is a regular sequence in the domain $R$ by step 1.1, the homomorphism $R [T] / (x T - y) \to R [(x, y) / x]$ , $T \mapsto y / x$ , is an isomorphism: it is surjective with kernel the $x$ -power torsion by [F4], and a coefficient comparison in a relation $x g = (x T - y) h$ , using that $y$ is a nonzerodivisor modulo $x$ , shows $g \in (x T - y)$ , so no nonzero torsion exists; symmetrically $R [U_{1}] / (y U_{1} - x) ≅ R [(x, y) / y]$ via $U_{1} \mapsto x / y$ . These identifications carry $T \mapsto y / x$ and $U_{1} \mapsto x / y$ , matching the ratio identifications of the blowup charts, so by [F6] they glue to an isomorphism $Z \to {Bl}_{(x, y)} U$ over $U$ , canonical because both sides are determined by the same chart data.

3.1F5F8F9step 2.1

By [F5] the restriction of the blowup of $S$ at $p$ to the open $U$ is ${Bl}_{(x, y)} U$ , so step 2.1 identifies it with the incidence subscheme $V (x v - y u)$ and gives the two charts $Spec R [T] / (x T - y) = Spec R [y / x]$ and $Spec R [U_{1}] / (y U_{1} - x) = Spec R [x / y]$ with $T U_{1} = 1$ , the descriptions displayed in the statement. Base change to $Spec A$ replaces $R$ by $A$ : the formulas $Spec A [T] / (x T - y)$ and $Spec A [U_{1}] / (y U_{1} - x)$ are exactly the local charts of [F8], and over this affine base the structure-sheaf pushforward is $A$ with vanishing higher direct images by [F9].

4.1F1F2F3F8F10F11step 3.1

Smoothness and the local structure of $E$ . Since $S$ is smooth over $k$ and $p$ is $k$ -rational, [F8] gives that ${Bl}_{p} S$ is smooth over $k$ with exceptional curve $E ≅ P_{k}^{1}$ and $O_{E} (E) = O (- 1)$ , and that the local rings on $E$ have dimension one at its generic point and two at closed points. The two charts of step 3.1 cover $π^{- 1} (U)$ and are open subschemes of ${Bl}_{p} S$ ; smoothness is local on the source by [F1], so each chart is a smooth surface over $k$ . The centre is a single point, so by [F11] the blowup is an isomorphism away from $p$ , and the chart descriptions of steps 2.1 and 3.1 glue to the global blowup. In the model $S = A_{k}^{2}$ , $p = 0$ with the coordinate functions $x, y$ , [F10] gives literally $Spec k [x, T]$ and $Spec k [y, U_{1}]$ inside $A_{k}^{2} \times_{k} P_{k}^{1}$ .

5.1step 1.1step 2.1step 3.1step 4.1∎

Steps 1.1-4.1 prove the statement: a sufficiently small affine neighbourhood $U = Spec R$ with regular parameters $x, y$ generating the ideal of $p$ exists, over $U$ the blowup is the incidence subscheme $V (x v - y u) \subseteq U \times_{k} P_{k}^{1}$ with charts $Spec R [T] / (x T - y) = Spec R [y / x]$ and $Spec R [U_{1}] / (y U_{1} - x) = Spec R [x / y]$ glued by $T U_{1} = 1$ , the base change to $Spec O_{S, p}$ is obtained by replacing $R$ by $O_{S, p}$ , the charts are smooth surfaces over $k$ , the local rings on $E$ have the asserted dimensions, and the plane model has the two affine-plane charts inside $A_{k}^{2} \times_{k} P_{k}^{1}$ .

Remarks

The quotient chart description only requires the indicated regular sequence. Regularity at the point gives the exceptional projective line and its normal twist; smoothness of $S$ and rationality of $p$ give absolute smoothness of the blowup over $k$ .
For a general closed point the local charts are over $O_{S, p}$ . The exceptional curve is over $κ (p)$ ; the whole blowup need not have a $κ (p)$ -algebra structure.

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Pushforward and vanishing for an affine point blowup

Statement

Assume the Axiom of Choice, inherited from the Proj construction and from the cohomology suppliers (The Axiom of Choice). Let $A$ be a commutative ring with $1$ , let $I = (x, y) \subseteq A$ be an ideal generated by a regular sequence $(x, y)$ (for example $A$ a regular local ring of dimension two with parameters $x, y$ ), let $π ⁣ : {Bl}_{I} Spec A \to Spec A$ be the blowup of Blowup of a scheme along an ideal sheaf and let $E$ be its exceptional subscheme. Then the two standard charts $X_{0} = D_{+} (x t)$ and $X_{1} = D_{+} (y t)$ cover ${Bl}_{I} Spec A$ , the ordered Čech complex of this two-element affine cover computes the cohomology of $O_{{Bl}_{I} Spec A}$ (Cech cohomology computes quasi-coherent cohomology on a separated scheme), and $H^{0} ({Bl}_{I} Spec A, O) = A, H^{1} ({Bl}_{I} Spec A, O) = 0, H^{q} ({Bl}_{I} Spec A, O) = 0 (q \geq 2) .$ Consequently $π_{*} O_{{Bl}_{I} Spec A} = O_{Spec A}$ and $R^{q} π_{*} O_{{Bl}_{I} Spec A} = 0$ for all $q > 0$ .

Facts & Assumptions

Given: A commutative ring $A$ , an ideal $I = (x, y) \subseteq A$ such that $x$ is a nonzerodivisor of $A$ and $y$ is a nonzerodivisor of $A / (x)$ , the Rees algebra $R (I) = ⨁_{n \geq 0} I^{n} t^{n}$ (Rees algebra sheaf of a finite type ideal), the blowup $π ⁣ : {Bl}_{I} Spec A = Proj R (I) \to Spec A$ (Blowup of a scheme along an ideal sheaf), and the relative projective line $P_{A}^{1} = Proj A [u, v]$ with twisting sheaves $O (d)$ (Twisting sheaf on Proj).

[F1]

Affine blowup standard charts and overlaps: For $I = (f_{0}, \dots, f_{r})$ and $B_{i} = A [I / f_{i}]$ the standard opens $D_{+} (f_{i} t) = Spec B_{i}$ cover ${Bl}_{I} Spec A$ , with overlap identifications given by the ratios $(f_{j} t) / (f_{i} t)$ .

[F2]

Closed subschemes of projective space and saturated ideals: A homogeneous ideal $a \subseteq A [u, v]$ determines a closed subscheme $V_{+} (a) ↪ P_{A}^{1}$ , equal to $Proj (A [u, v] / a)$ under the canonical closed immersion, and $V_{+} (a) \cap D_{+} (u) = Spec (A [u, v]_{(u)} / a_{(u)})$ .

[F3]

Hypersurface cohomology sequence: For $f \in A [u, v]$ homogeneous of degree $d > 0$ with every dehomogenisation $f / u^{d}$ , $f / v^{d}$ a nonzerodivisor of the corresponding chart ring, the multiplication map $O_{P_{A}^{1}} (- d) \overset{\cdot f}{\to} O_{P_{A}^{1}}$ and the structure map of the closed immersion $i ⁣ : V_{+} (f) ↪ P_{A}^{1}$ form a short exact sequence $0 \to O_{P_{A}^{1}} (- d) \overset{\cdot f}{\to} O_{P_{A}^{1}} \overset{i^{♯}}{\to} i_{*} O_{V_{+} (f)} \to 0,$ and the induced long exact sequence of sheaf cohomology computes the cohomology of $V_{+} (f)$ from that of the twists $O_{P_{A}^{1}} (d)$ .

[F4]

Cohomology of O(d) on projective space: On $P_{A}^{1}$ one has $H^{0} (O) = A$ , $H^{1} (O) = 0$ , $H^{q} (O) = 0$ for $q \geq 2$ , and $H^{0} (O (- 1)) = 0$ , $H^{1} (O (- 1)) = 0$ ; indeed $H^{q} (P_{A}^{1}, O (d)) = 0$ unless $q = 0$ or $q = 1$ , with $H^{0}$ the degree- $d$ part of $A [u, v]$ for $d \geq 0$ and $H^{1}$ the free $A$ -module on the Laurent monomials $u^{e_{0}} v^{e_{1}}$ with $e_{0}, e_{1} < 0$ and $e_{0} + e_{1} = d$ , which is zero for $d = - 1$ .

[F5]

Cech cohomology computes quasi-coherent cohomology on a separated scheme: For a quasi-compact separated scheme $X$ , a finite affine open cover and a quasi-coherent $O_{X}$ -module $F$ , the canonical comparison map ${\overset{ˇ}{H}}^{q} (U, F) \to H^{q} (X, F)$ is an isomorphism for every $q \geq 0$ .

[F6]

Čech complex for a two-open cover: For a cover by two open sets $U_{0}, U_{1}$ , the ordered Čech complex has $C^{0} = F (U_{0}) \oplus F (U_{1})$ , $C^{1} = F (U_{0} \cap U_{1})$ and $C^{p} = 0$ for $p \geq 2$ , with $δ^{0} (s_{0}, s_{1}) = s_{1} ∣_{U_{0} \cap U_{1}} - s_{0} ∣_{U_{0} \cap U_{1}}$ ; hence ${\overset{ˇ}{H}}^{0} = \ker δ^{0}$ , ${\overset{ˇ}{H}}^{1} = coker δ^{0}$ and ${\overset{ˇ}{H}}^{p} = 0$ for $p \geq 2$ .

[F7]

Higher direct images localize over an affine base: For a quasi-compact separated morphism $f ⁣ : X \to S$ and a quasi-coherent $O_{X}$ -module $F$ , each $R^{q} f_{*} F$ is quasi-coherent (Higher direct image of a sheaf), and for every affine open $V = Spec A \subseteq S$ there is a canonical isomorphism $(R^{q} f_{*} F) ∣_{V} ≅ \tilde{H^{q} (f^{- 1} V, F)}$ .

[F8]

Closed immersions of schemes, Quasi-compact and quasi-separated schemes, Separated morphism of schemes: A closed subscheme of a quasi-compact scheme is quasi-compact, and a closed subscheme of a separated scheme is separated; the relative projective line $P_{A}^{1}$ is quasi-compact and separated over $Spec A$ .

Proof

1.1F1

The graded $A$ -algebra homomorphism $A [u, v] \to R (I)$ with $u \mapsto x t$ , $v \mapsto y t$ is surjective because $I^{n} = (x, y)^{n}$ is generated by the monomials $x^{n - i} y^{i}$ , and its kernel is $(x v - y u)$ : for a homogeneous $F = \sum_{i = 0}^{n} c_{i} u^{n - i} v^{i}$ with $F (x, y) = 0$ , reduction modulo $x$ gives $c_{n} y^{n} \equiv 0 (m o d x)$ , so ${\overset{ˉ}{c}}_{n} = 0$ because $y$ is a nonzerodivisor modulo $x$ , say $c_{n} = x d$ ; then $F - d v^{n - 1} (x v - y u)$ has zero $v^{n}$ -coefficient, hence equals $u G$ for a homogeneous $G$ of degree $n - 1$ , and evaluating at $(x, y)$ gives $x G (x, y) = 0$ , so $G (x, y) = 0$ because $x$ is a nonzerodivisor, and induction on $n$ down to degree $0$ yields $F \in (x v - y u)$ , while conversely $x v - y u \mapsto x \cdot y t - y \cdot x t = 0$ ; hence $R (I) ≅ A [u, v] / (x v - y u)$ as graded $A$ -algebras.

1.2F1F2

The element $f : = x v - y u \in A [u, v]$ is a nonzerodivisor: as a polynomial in $v$ over $A [u]$ its leading coefficient is the nonzerodivisor $x$ , so $f g = 0$ forces the top $v$ -coefficient of $g$ to vanish and descending induction gives $g = 0$ ; the dehomogenisation $f / u = x (v / u) - y$ is a nonzerodivisor of $A [v / u]$ for the same reason, and $y U - x$ is a nonzerodivisor of $A [U]$ , since comparing coefficients of $U^{i}$ in $(y U - x) g = 0$ gives $- x g_{0} = 0$ and $y g_{i - 1} = x g_{i}$ for $i \geq 1$ , so $g_{0} = 0$ and induction gives all $g_{i} = 0$ ; hence by [F2] the blowup is identified with the closed subscheme ${Bl}_{I} Spec A = Proj (A [u, v] / (x v - y u)) = V_{+} (x v - y u) \subseteq P_{A}^{1}$ , whose standard charts $D_{+} (u) \cap V_{+} (f) = Spec (A [v / u] / (x (v / u) - y)) = X_{0}$ and $D_{+} (v) \cap V_{+} (f) = Spec (A [u / v] / (y (u / v) - x)) = X_{1}$ are affine and cover it, matching the standard blowup charts of [F1] under the identification of $u, v$ with $x t, y t$ .

1.3F4

On $P_{A}^{1}$ the groups $H^{0} (O) = A$ , $H^{1} (O) = 0$ , $H^{q} (O) = 0$ for $q \geq 2$ and $H^{0} (O (- 1)) = H^{1} (O (- 1)) = 0$ hold by [F4] with $n = 1$ .

2.1F3step 1.2

Applying [F3] to the homogeneous degree-one element $f = x v - y u$ , whose chart dehomogenisations are nonzerodivisors by step 1.2, gives a short exact sequence of $O_{P_{A}^{1}}$ -modules $0 \to O_{P_{A}^{1}} (- 1) \overset{\cdot f}{\to} O_{P_{A}^{1}} \to i_{*} O_{{Bl}_{I} Spec A} \to 0$ , where $i$ is the closed immersion of step 1.2.

3.1F4step 2.1step 1.3

The long exact cohomology sequence of step 2.1, with $H^{q} (P_{A}^{1}, i_{*} O_{Bl}) ≅ H^{q} ({Bl}_{I} Spec A, O)$ for the closed immersion $i$ and the groups of step 1.3, gives $H^{0} ({Bl}_{I} Spec A, O) ≅ coker (H^{0} (O (- 1)) \to H^{0} (O)) = A$ , then $H^{1} ({Bl}_{I} Spec A, O) = 0$ because it is squeezed between $H^{1} (O) = 0$ and $H^{2} (O (- 1)) = 0$ , and for $q \geq 2$ the group $H^{q} ({Bl}_{I} Spec A, O)$ is squeezed between $H^{q} (O) = 0$ and $H^{q + 1} (O (- 1)) = 0$ .

4.1F5F6F8step 1.2step 3.1

By [F8] the blowup ${Bl}_{I} Spec A$ is a closed subscheme of the quasi-compact separated $P_{A}^{1}$ , hence quasi-compact and separated, so the two-element affine cover of step 1.2 has Čech cohomology computing the sheaf cohomology of $O$ by [F5] and, by [F6], a Čech complex concentrated in degrees $0$ and $1$ with ${\overset{ˇ}{H}}^{0} = \ker δ^{0}$ and ${\overset{ˇ}{H}}^{1} = coker δ^{0}$ , so its Čech groups are $A$ , $0$ and $0$ in degrees $0$ , $1$ and $\geq 2$ , consistently with step 3.1.

5.1F7F8step 3.1∎

The structural morphism $π$ is quasi-compact and separated, being the composite of the closed immersion of step 1.2 with the quasi-compact separated structure morphism of $P_{A}^{1}$ from [F8], and $O_{Bl}$ is quasi-coherent, so [F7] with $V = Spec A$ identifies $R^{q} π_{*} O$ with the sheaf associated to the $A$ -module $H^{q} ({Bl}_{I} Spec A, O)$ , which is $A$ in degree $0$ and $0$ for $q > 0$ by step 3.1; hence $π_{*} O = \tilde{A} = O_{Spec A}$ and $R^{q} π_{*} O = 0$ for every $q > 0$ .

Remarks

Regularity of the sequence $(x, y)$ enters twice: to identify the Rees algebra with the incidence algebra $A [u, v] / (x v - y u)$ (step 1.1) and to make the dehomogenisations of $x v - y u$ nonzerodivisors (step 1.2). For a general pair of generators the map $A [u, v] \to R (I)$ has a nontrivial kernel in general, and the blowup need not be a hypersurface in $P_{A}^{1}$ .
The theorem is stated for an affine base precisely so that the direct image computation reduces to the two cohomology modules $A$ and $0$ ; over a nonaffine base the same proof applies over each affine open, and the sheaf statement is the affine-local one of [F7].
The two-chart computation never inverts $x$ or $y$ in $A$ : the overlap identification inverts the ratio $v / u$ only, consistent with [F1].

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Pushforward and vanishing for point blowups on a surface

Statement

Assume the Axiom of Choice. Let $S$ be a regular surface over a field $k$ (more generally a locally Noetherian scheme of dimension two whose local rings at the center are regular of dimension two) and let $p$ be a closed point with residue field $κ (p)$ . Let $π ⁣ : S^{'} \to S$ be the blowup of $p$ with exceptional curve $E$ . Then $π_{*} O_{S^{'}} = O_{S}$ and $R^{q} π_{*} O_{S^{'}} = 0$ for every $q > 0$ . Moreover the same conclusions hold after composing finitely many point blowups.

Facts & Assumptions

Given: The Axiom of Choice, a scheme $S$ as in the statement, a closed point $p \in S$ with residue field $κ (p)$ , the blowup $π ⁣ : S^{'} \to S$ of $p$ , and its exceptional curve $E$ .

[A1]

Choice. The Axiom of Choice is assumed, as in the statement; the cited local computation and its proof are choice-carrying, and no additional choices are made below.

[F1]

Pushforward and vanishing for an affine point blowup: For the affine-local presentation of a point blowup on a surface, the structure-sheaf pushforward is the structure sheaf of the base and all higher direct images of the structure sheaf vanish.

[F3]

Blowups restrict to open subschemes of the base: For an open $U \subseteq S$ , $π^{- 1} (U)$ is canonically the blowup of $U$ at the restricted center.

[F4]

Higher direct image of a sheaf: $R^{q} π_{*} F$ is the sheaf associated to $U \mapsto H^{q} (π^{- 1} (U), F)$ ; for $q = 0$ it is the ordinary pushforward, and a morphism of sheaves is an isomorphism on the base if and only if it is so on an open cover.

[F5]

The blowup is an isomorphism off the center: The blowup restricts to an isomorphism away from its center.

[F6]

Cohomology comparison when higher direct images vanish: If $R^{q} g_{*} G = 0$ for $q > 0$ , the natural maps $H^{n} (T, g_{*} G) \to H^{n} (T', G)$ are isomorphisms for every $n$ .

[F7]

Affine acyclicity of quasi-coherent sheaves: A quasi-coherent module on an affine scheme has zero higher cohomology.

[F8]

one dimensional regular local rings are dvrs and regular local rings are domains and cohen macaulay: A regular local ring is a domain, and in dimension one it is a DVR with principal maximal ideal.

[F9]

Blowing up an effective Cartier divisor does nothing: The blowup of an effective Cartier center is the identity.

Proof

1.1F3F4F5F8F9given

The calculation is local on the base, by the sheafification description of higher direct images and the locality of the blowup. In the regular-surface alternative, the local dimension at a closed point is positive: if it were zero, the regular local domain would be a field and the closed point would also be the generic point of its ambient irreducible component, forcing that component to be a point, contrary to the pure dimension two convention for a surface. If its dimension is one, its maximal ideal has a regular generator by [F8]. Lift this generator to an affine Noetherian neighborhood; shrinking kills the finite quotient of the point ideal by that generator and the finite kernel of multiplication by it, just as below. The point center is then effective Cartier on that neighborhood, so its blowup is the identity there by [F9], and it is also the identity off the center by [F5]. Thus the asserted pushforward and vanishing follow in this case. The more general alternative in the statement already assumes local dimension two at the center. It remains to treat local dimension two. Near $p$ , choose an affine Noetherian neighborhood $Spec R$ . Lift regular parameters of $R_{m_{p}}$ to functions $x, y$ after inverting denominators not vanishing at $p$ . The ideal of $p$ is finite; its quotient by $(x, y)$ has zero stalk at $p$ , so shrinking kills this finite module. Likewise the kernels of multiplication by $x$ on $R$ and by $y$ on $R / (x)$ are finite modules with zero stalk at $p$ , and another shrinking kills them. Thus $(x, y)$ is a regular sequence generating the point ideal on this affine neighborhood, with nonzero quotient $κ (p)$ .

2.1F1F3F4F5step 1.1

The affine regular-sequence calculation applies on this neighborhood and gives the asserted direct images. On the complement of $p$ the blowup is the identity, with the same direct images. These local results give $π_{*} O = O$ and $R^{q} π_{*} O = 0$ globally. The argument uses only local Noetherianity and the two-dimensional regular local ring at the center.

3.1F4F6F7step 2.1∎

For a finite composition of the point blowups just considered, write it as $f \circ g$ , where $g$ is the last step. Assume by induction the conclusions for $f$ . For every affine open $V$ in the original base, apply the vanishing-direct-image comparison to $g$ restricted over $f^{- 1} (V)$ . Step 2.1 makes its higher direct images zero and its degree-zero image the structure sheaf. Hence $H^{q} (g^{- 1} f^{- 1} V, O) = H^{q} (f^{- 1} V, O)$ . A second comparison for $f$ over $V$ , followed by affine acyclicity, identifies the latter with $Γ (V, O)$ for $q = 0$ and zero for $q > 0$ . Sheafifying these identifications proves the same direct-image conclusions for the composition, completing the induction.

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Projection formula for invertible twists

Statement

Assume the Axiom of Choice. Let $f ⁣ : X \to Y$ be a morphism of schemes, let $F$ be a quasi-coherent $O_{X}$ -module and let $L$ be an invertible $O_{Y}$ -module. Then the natural map

$R^{q} f_{*} (F) \otimes_{O_{Y}} L ⟶ R^{q} f_{*} (F \otimes_{O_{X}} f^{*} L)$

is an isomorphism for every $q \geq 0$ . In particular, if $f$ is a $k$ -morphism, $X$ and $Y$ are proper over a field $k$ and $F$ is coherent, then the Euler characteristics satisfy $χ (X, f^{*} L) = χ (Y, L)$ whenever $f_{*} O_{X} = O_{Y}$ and $R^{q} f_{*} O_{X} = 0$ for $q > 0$ .

Facts & Assumptions

Given: A morphism $f ⁣ : X \to Y$ of schemes, a quasi-coherent $O_{X}$ -module $F$ and an invertible $O_{Y}$ -module $L$ ; the Axiom of Choice is inherited from the cohomology and adjunction suppliers cited below (The Axiom of Choice).

[F1]

Higher direct image of a sheaf: For a morphism of ringed spaces $f$ and an $O_{X}$ -module $G$ , the higher direct images are $R^{q} f_{*} G = H^{q} (f_{*} I (G)_{d e l})$ for a fixed functorial injective resolution datum, with $R^{0} f_{*} G = f_{*} G$ canonically and $R^{q} f_{*} = 0$ for $q < 0$ ; the functor $f_{*}$ is left exact and additive.

[F2]

Pullback of modules is left adjoint to pushforward: For a morphism of ringed spaces $f$ , the inverse image functor $f^{*}$ on modules is left adjoint to the direct image functor $f_{*}$ , with unit $η ⁣ : i d \to f_{*} f^{*}$ and counit $ε ⁣ : f^{*} f_{*} \to i d$ .

[F3]

Invertible sheaves and Locally free sheaves of finite rank: An invertible sheaf is locally free of rank one; its dual is an inverse for tensor product, and its pullback is invertible.

[F5]

Cohomology comparison when higher direct images vanish: If the higher direct images of a module vanish, its cohomology equals the cohomology of its degree-zero direct image, naturally in every degree.

[F7]

Euler characteristic of a coherent sheaf: For a scheme proper over a field $k$ and a coherent module, the Euler characteristic is the finite alternating sum of the $k$ -dimensions of the cohomology groups.

[F8]

Finite-dimensional coherent cohomology over a field: For a scheme $X$ proper over a field $k$ and a coherent $O_{X}$ -module $F$ , each $H^{q} (X, F)$ is finite-dimensional over $k$ and only finitely many of the groups are nonzero.

[F9]

Coherent module sheaves: On a locally Noetherian scheme, a finite-type quasi-coherent module is coherent. Schemes proper over a field are of finite type by Proper morphisms, and their affine coordinate rings are Noetherian by Every algebra of finite type over a principal ideal domain is a Noetherian ring (a field is a principal ideal domain).

Proof

1.1F3

Tensoring with an invertible sheaf $L$ is an exact autoequivalence, with inverse tensoring with $L^{\lor}$ : exactness is checked in local trivializations. It preserves injectives, since $Hom (M, I \otimes L) ≅ Hom (M \otimes L^{\lor}, I)$ is exact in $M$ when $I$ is injective. The same statements hold on $X$ for $f^{*} L$ .

1.2F2F3

For any module $G$ on $X$ , adjunction gives the natural map $ν_{G} : f_{*} G \otimes L \to f_{*} (G \otimes f^{*} L)$ , adjoint to the counit map $f^{*} (f_{*} G) \otimes f^{*} L \to G \otimes f^{*} L$ . On every open trivializing $L$ this is the identity under the trivializations, hence it is an isomorphism. This ordinary direct-image argument requires no quasi-compactness or separatedness of $f$ .

2.1F1step 1.1step 1.2

Take an injective resolution $I^{∙}$ of $F$ . By step 1.1, $I^{∙} \otimes f^{*} L$ is an injective resolution of $F \otimes f^{*} L$ . Naturality of $ν$ gives an isomorphism of complexes $f_{*} I^{∙} \otimes L ≅ f_{*} (I^{∙} \otimes f^{*} L)$ . Exact tensor with $L$ commutes with taking cohomology, so the resulting isomorphism is precisely $R^{q} f_{*} F \otimes L ≅ R^{q} f_{*} (F \otimes f^{*} L)$ for every $q$ .

3.1F3F5F7F8F9step 2.1∎

If $f_{*} O_{X} = O_{Y}$ and the higher direct images of $O_{X}$ vanish, applying step 2.1 to $O_{X}$ gives $f_{*} f^{*} L = L$ and $R^{q} f_{*} f^{*} L = 0$ for $q > 0$ . For the $k$ -morphism in the final assertion, the vanishing-direct-image comparison gives $k$ -linear isomorphisms $H^{n} (X, f^{*} L) ≅ H^{n} (Y, L)$ . The proper schemes of the final assertion are locally Noetherian by [F9]; the line bundles are finite-type quasi-coherent modules by [F3], hence coherent by [F9], and their cohomology is finite-dimensional and vanishes in sufficiently high degree. Taking the finite alternating sums proves the Euler-characteristic identity.

Remarks

The formula uses invertibility to obtain an exact tensor autoequivalence. The Euler-characteristic clause uses the specified direct-image vanishing and requires no flatness of $f$ .

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Euler characteristic of line bundles on a projective line over a finite field extension

Statement

Assume the Axiom of Choice. Let $k$ be a field and let $κ$ be a finite extension of $k$ of degree $r = [κ : k]$ . Let $E$ be a scheme isomorphic to $P_{κ}^{1}$ over $κ$ , and let $M$ be an invertible sheaf on $E$ . Then $H^{q} (E, M) = 0$ for $q \geq 2$ , and writing $d$ for the degree of $M$ over $κ$ one has

$\dim_{k} H^{0} (E, M) - \dim_{k} H^{1} (E, M) = r (1 + d),$

so the $k$ -Euler characteristic of $M$ is $χ_{k} (E, M) = r (1 + d)$ . In particular a line bundle of degree $- j$ on $E$ has Euler characteristic $r (1 - j)$ .

Facts & Assumptions

Given: A field $k$ , a finite field extension $κ / k$ of degree $r = [κ : k]$ , a $κ$ -scheme $E$ isomorphic to $P_{κ}^{1}$ over $κ$ , and an invertible sheaf $M$ on $E$ ; the Axiom of Choice is inherited from the Picard, cohomology and Euler-characteristic suppliers cited below (The Axiom of Choice).

[F1]

Sheaf cohomology as right derived global sections: For a scheme $X$ and an $O_{X}$ -module $F$ , the cohomology groups $H^{q} (X, F)$ are the right derived functors of the global-section functor $Γ (X, -)$ , so an isomorphism of $O_{X}$ -modules induces an isomorphism $H^{q} (X, F) ≅ H^{q} (X, F^{'})$ , functorially in $q$ .

[F2]

The Picard group of the projective line: For every field $K$ , the degree homomorphism induces an isomorphism $Pic (P_{K}^{1}) \to Z$ under which $O_{P_{K}^{1}} (d)$ corresponds to $d$ ; every invertible sheaf on $P_{K}^{1}$ is isomorphic to $O_{P_{K}^{1}} (d)$ for a unique integer $d$ .

[F3]

Cohomology of O(d) on projective space: For $n = 1$ over a field $K$ , $H^{0} (O (d))$ has basis the degree- $d$ ordinary monomials when $d \geq 0$ and is zero otherwise; $H^{1} (O (d))$ has basis the Laurent monomials $x^{e} y^{f}$ with $e, f < 0$ and $e + f = d$ ; all higher groups vanish.

[F4]

Euler characteristic of a coherent sheaf: For a scheme $X$ proper over a field $k$ and a coherent $O_{X}$ -module $F$ , all groups $H^{q} (X, F)$ are finite-dimensional over $k$ and only finitely many are nonzero, and the Euler characteristic is the alternating sum $χ (X, F) = \sum_{q \geq 0} (- 1)^{q} \dim_{k} H^{q} (X, F)$ .

[F5]

Finite-dimensional projective space is proper over every base: For a scheme $S$ and $n \geq 0$ the structure morphism $P_{S}^{n} \to S$ is proper.

[F6]

Coherent module sheaves: On a locally Noetherian scheme, a finite locally free sheaf is coherent; in particular an invertible sheaf on a locally Noetherian scheme is coherent.

Proof

1.1F1

Fix a $κ$ -isomorphism $φ ⁣ : E \to P_{κ}^{1}$ and set $N : = φ_{*} M$ , so that $N$ is an invertible sheaf on $P_{κ}^{1}$ and $Γ (E, M) = Γ (P_{κ}^{1}, N)$ by definition of the direct image. The direct image along an isomorphism is an exact equivalence of module categories with inverse $φ^{*}$ , so it preserves the global-section functors and their right derived functors; hence $φ$ induces $κ$ -linear isomorphisms $H^{q} (E, M) ≅ H^{q} (P_{κ}^{1}, N)$ for all $q \geq 0$ .

2.1F2step 1.1

By [F2] applied to the field $κ$ , the invertible sheaf $N$ on $P_{κ}^{1}$ is isomorphic to $O_{P_{κ}^{1}} (d)$ for a unique integer $d$ , which we take as the definition of the degree $d$ of $M$ over $κ$ .

3.1F1F3step 1.1step 2.1

By [F3] with $A = κ$ and $n = 1$ the groups $H^{q} (P_{κ}^{1}, O (d))$ vanish unless $q = 0$ or $q = 1$ ; the same description gives $H^{0} ≅ κ [x, y]_{d}$ , of dimension $h^{0} = d + 1$ for $d \geq 0$ and $0$ for $d < 0$ , and $H^{1}$ free on the Laurent monomials $x^{e} y^{f}$ with $e, f < 0$ and $e + f = d$ , of dimension $h^{1} = - d - 1$ for $d \leq - 2$ and $0$ otherwise, so $h^{0} - h^{1} = 1 + d$ . Since cohomology depends only on the isomorphism class of the sheaf, [F1] gives $\dim_{κ} H^{q} (P_{κ}^{1}, N) = h^{q}$ ; combined with step 1.1 this yields $H^{q} (E, M) = 0$ for $q \geq 2$ and $\dim_{κ} H^{0} (E, M) - \dim_{κ} H^{1} (E, M) = 1 + d$ .

4.1step 3.1algebra

The structure morphism $E \to Spec κ$ makes $κ \to Γ (E, O_{E})$ a ring homomorphism, so each $H^{q} (E, M)$ is a $κ$ -vector space; for a $κ$ -vector space $V$ of dimension $h$ one has $\dim_{k} V = r h$ , because if $v_{1}, \dots, v_{h}$ is a $κ$ -basis of $V$ and $μ_{1}, \dots, μ_{r}$ is a $k$ -basis of $κ$ , then the products $μ_{j} v_{i}$ span $V$ over $k$ and are $k$ -independent. Applying this to the groups of step 3.1 gives $\dim_{k} H^{0} (E, M) - \dim_{k} H^{1} (E, M) = r (h^{0} - h^{1}) = r (1 + d)$ .

5.1F4F5F6step 3.1step 4.1

The scheme $E$ is proper over $κ$ because it is $κ$ -isomorphic to $P_{κ}^{1}$ and $P_{S}^{1} \to S$ is proper for every $S$ [F5], and $M$ is coherent on the locally Noetherian scheme $E$ because it is invertible [F6]; thus the Euler characteristic of [F4], with base field $κ$ , is the alternating sum over the finitely many nonzero cohomology groups. By step 3.1 only the terms $q = 0, 1$ occur, and passing to $k$ -dimensions as in step 4.1 gives the $k$ -Euler characteristic $χ_{k} (E, M) = \dim_{k} H^{0} (E, M) - \dim_{k} H^{1} (E, M) = r (1 + d)$ .

6.1step 2.1step 5.1∎

If $φ^{'}$ is another $κ$ -isomorphism with associated integer $d^{'}$ , then step 5.1 applied to both gives $r (1 + d) = χ_{k} (E, M) = r (1 + d^{'})$ , and $r \geq 1$ because $κ / k$ is a finite extension, so $d = d^{'}$ ; thus the degree of $M$ over $κ$ is well defined. For $d = - j$ the formula reads $χ_{k} (E, M) = r (1 - j)$ , which is the final assertion.

Remarks

The extension $κ / k$ need not be separable or Galois: the proof never decomposes $κ \otimes_{k} κ$ , using only that $H^{q} (E, M)$ is a $κ$ -vector space and that $r = [κ : k]$ is the $k$ -dimension of $κ$ . The case $d = - 1$ gives $χ_{k} = 0$ , and $d = 0$ gives $χ_{k} = r$ , the $k$ -dimension of the structure sheaf's cohomology.

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Point blowups of regular surfaces stay regular, with rational exceptional fibre over the residue field

Statement

Assume the Axiom of Choice. Let $S$ be a regular finite-type $k$ -scheme of pure dimension two, let $p$ be a closed point, put $κ = κ (p)$ and $r = [κ : k]$ , and let $π ⁣ : S^{'} = {Bl}_{p} S \to S$ be the blowup of $S$ at $p$ with exceptional subscheme $E$ . Then $S^{'}$ is regular of pure dimension two, $E$ is an effective Cartier divisor canonically isomorphic to $P_{κ} (m_{p} / m_{p}^{2})$ , hence isomorphic to $P_{κ}^{1}$ after choosing regular parameters, and $O_{E} (E) = O_{P_{κ}^{1}} (- 1)$ . For $A = O_{S, p}$ and regular parameters $x, y$ , the base change to $Spec A$ has charts $Spec A [T] / (x T - y) = Spec A [y / x]$ and $Spec A [U] / (y U - x) = Spec A [x / y]$ , glued by inverting $T$ and $U$ with $U = T^{- 1}$ . Their local rings at the generic point of $E$ have dimension one and at its closed points dimension two. If $S$ is smooth over $k$ and $p$ is $k$ -rational, $S^{'}$ is smooth over $k$ ; literal affine-plane charts occur in the model $S = A_{k}^{2}$ , $p = 0$ . No smoothness over an imperfect $k$ is asserted for a general inseparable closed point. Regularity is a property of these local rings and does not require a $κ$ -algebra structure on $S^{'}$ .

Facts & Assumptions

Given: The Axiom of Choice, a regular finite-type $k$ -scheme $S$ of pure dimension two, a closed point $p \in S$ with residue field $κ = κ (p)$ , the blowup $S^{'} = {Bl}_{p} S$ with exceptional subscheme $E$ , and regular parameters $x, y$ of $A = O_{S, p}$ .

[A1]

Choice. The Axiom of Choice is assumed, as in the statement; the cited suppliers used below are stated under it.

[F1]

Affine blowup standard charts and overlaps: Let $A$ be a ring, $I = (f_{0}, \dots, f_{r})$ , $S = R (I)$ and $B_{i} = A [I / f_{i}]$ . The standard opens $U_{i} = D_{+} (f_{i} t) = Spec B_{i}$ cover ${Bl}_{I} Spec A$ , and on overlaps the identifications are $D (u_{i j})$ in $U_{i}$ with $u_{i j} = (f_{j} t) / (f_{i} t)$ , sending $u_{i j}$ to $u_{j i}^{- 1}$ and preserving the structure maps to $Spec A$ .

[F2]

Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: For a ring $A$ , an ideal $I$ and $a \in I$ , the affine blowup algebra $A [I / a] : = (R (I))_{(a)}$ satisfies: the image of $a$ is a nonzerodivisor, $I A [I / a] = a A [I / a]$ , and $(A [I / a])_{a} = A_{a}$ . If $I = (a_{0}, \dots, a_{r})$ and $a = a_{0}$ , then $A [x_{1}, \dots, x_{r}] / (a x_{i} - a_{i}) \to A [I / a]$ , $x_{i} \mapsto a_{i} / a$ , is surjective; if $A$ is a domain and $a \neq 0$ , then $A [I / a]$ is a domain.

[F3]

Flat base change for blowups, and failure without flatness: For a flat base change $X^{'} \to X$ , the blowup of $X$ along a quasi-coherent ideal sheaf of finite type base-changes to the blowup of $X^{'}$ along the pulled-back ideal; in particular the base change of ${Bl}_{(x, y)} Spec A$ to $Spec A$ over $S$ is the blowup of $Spec A$ at its closed point.

[F5]

Maximal ideals of an affine domain have full height: Let $k$ be a field, $B$ a finite-type $k$ -domain and $m \subseteq B$ maximal. Then $ht (m) = \dim B$ .

[F6]

regular local rings are domains and cohen macaulay: A regular local ring $R$ of dimension $d$ is a domain and Cohen-Macaulay. For every regular system $(x_{1}, \dots, x_{d})$ , the tuple is $R$ -regular and $R / (x_{1}, \dots, x_{c})$ is regular local of dimension $d - c$ for all $0 \leq c \leq d$ .

[F7]

localisation and polynomial extension of regular rings: Localizations and finite polynomial extensions of a commutative regular Noetherian ring are regular.

[F8]

localisations of regular local rings are regular: Every prime localization $R_{p}$ of a regular local ring $R$ is regular, and $edim R_{p} = ht p$ .

[F9]

regular local quotient by parameter is regular: Let $(R, m, k)$ be regular local of dimension $d$ and $x \in m ∖ m^{2}$ . Then $R / (x)$ is regular local of dimension and embedding dimension $d - 1$ .

[F10]

embedding dimension and regular local ring: For a nonzero commutative Noetherian local ring $(R, m, k)$ , $edim R = \dim_{k} (m / m^{2})$ ; $R$ is regular local when $edim R = \dim R$ .

[F11]

dimension at most embedding dimension: Every nonzero commutative Noetherian local ring $R$ satisfies $\dim R \leq edim R < \infty$ .

[F12]

associated graded ring of a regular local ring: If $(R, m, k)$ is regular local of dimension $d$ , any cotangent basis induces a graded isomorphism $k [X_{1}, \dots, X_{d}] ≅ {gr}_{m} R$ .

[F13]

The exceptional divisor is the projectivized normal cone: For $Z = V (I)$ cut out by a quasi-coherent ideal sheaf $I$ of finite type, there is a canonical isomorphism of $Z$ -schemes $E \to {Proj}_{Z} ({gr}_{I} O_{X})$ from the exceptional subscheme of the blowup.

[F14]

The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier: For the blowup of $I$ with exceptional subscheme $E = V (I O_{Bl})$ : $O (1)$ is invertible, the natural map $π^{*} I \to O (1)$ is surjective with image $I O_{Bl}$ , so $I O_{Bl}$ is invertible and $E$ is an effective Cartier divisor with $O_{Bl} (- E) = I O_{Bl} = O (1)$ and $O_{Bl} (E) = O (- 1)$ .

[F15]

Smooth morphism of schemes: A morphism $f ⁣ : X \to S$ is smooth at $x$ when it is locally of finite presentation at $x$ , flat at $x$ , and the scheme-theoretic fibre $X_{f (x)}$ is geometrically regular at $x$ (regular after every field extension of $κ (f (x))$ ); $f$ is smooth when this holds everywhere.

[F16]

Smoothness survives base change and composition: Smooth morphisms are stable under arbitrary base change.

[F17]

Every vector space has a basis: Assuming the Axiom of Choice, every vector space over a field has a basis.

[F18]

Under the stated choice boundary, free modules are projective and hence flat: Every free module over a commutative ring is flat.

[F19]

Affine-domain dimension equals transcendence degree: For any field $k$ and any finite-type $k$ -domain $B$ , $\dim B = {trdeg}_{k} Frac (B)$ .

[F20]

Projective bundle in the quotient convention: For a finite locally free sheaf $V$ , $P (V) = Proj (Sym V)$ , with its standard positive twist.

Proof

1.1A1F5F6F10

The component through $p$ is open: regular local rings are domains, so distinct irreducible components of the Noetherian regular scheme cannot meet. Choose a domain affine neighborhood of $p$ in that component. Its dimension is two, and the maximal-ideal height theorem gives $\dim A = 2$ for $A = O_{S, p}$ . Choose regular parameters $x, y$ . They form a regular sequence; $A / (x)$ and $A / (y)$ are one-dimensional regular local domains.

2.1F1F2F3step 1.1

Flat localization of the base identifies the part over $Spec A$ with the blowup of $(x, y)$ . On the $x$ -chart, put $C = A [T] / (x T - y)$ . If $x g = (x T - y) h$ , reduction modulo $x$ gives $\overset{ˉ}{y} \overset{ˉ}{h} = 0$ in the domain $(A / (x)) [T]$ , so $h = x h_{1}$ ; cancellation of $x$ gives $g = (x T - y) h_{1}$ . Hence $C$ has no $x$ -power torsion, and the chart algebra theorem identifies $C = A [y / x]$ . It has $C_{x} = A_{x}$ , exceptional ideal $x C$ and quotient $C / x C = κ [T]$ . The second chart is $A [U] / (y U - x)$ by the same argument, with overlap $U = T^{- 1}$ . Localization of the base does not change local rings at points over $p$ .

3.1F7F8F9F10F11step 2.1

Outside $V (x)$ the local rings of $C$ are prime localizations of $A$ , hence regular. A prime of $A [T]$ lying over a point of $V (x)$ in $C$ is either $Q = m A [T]$ or $Q = (m, h)$ , where $\overset{ˉ}{h}$ is a monic irreducible polynomial over $κ$ . The ambient local ring $A [T]_{Q}$ is regular. Its maximal ideal is generated respectively by $x, y$ or by $x, y, h$ . The prime chains $(0) ⊊ (x) ⊊ m A [T]$ and, in the second case, their extension by $Q$ , together with the embedding-dimension bound, give dimensions two and three. These generators therefore form a cotangent basis. The class of $x T - y$ is $\overset{ˉ}{T} \overset{ˉ}{x} - \overset{ˉ}{y}$ , which is nonzero because the coefficient of $\overset{ˉ}{y}$ is $- 1$ , even when $\overset{ˉ}{T} = 0$ . Quotienting by this parameter gives regular local rings of dimension one at the generic exceptional point and two at its closed points. The same proof works in the $y$ -chart. These computations also show the local chart rings have dimension two, without asserting $\dim A_{x} = 2$ .

4.1F1F2F6F19step 3.1

Away from $p$ the structural morphism is an isomorphism: on the complement of the exceptional ideal in each standard chart its denominator is inverted and the chart becomes the corresponding base principal open, compatibly with the ratio transitions. Thus all local rings of $S^{'}$ are regular. Its charts over finite-type affine bases are finitely generated algebras, so $S^{'}$ is finite type over $k$ . Its irreducible components are disjoint and open, as for $S$ . No component has generic point in $E$ , since the local rings computed there have positive dimension, while a component's generic local ring has dimension zero. Every component consequently meets the unchanged open $S ∖ {p}$ and shares the function field of a two-dimensional component of $S$ . By the affine-domain dimension formula every nonempty affine open in it has dimension two. This gives dimension two for the component itself: any finite strict chain of irreducible closed subsets remains strict after intersecting an affine open meeting its smallest member, since such an open contains every member's generic point. Therefore $S^{'}$ is pure of dimension two.

4.2F12F13F14F20step 3.1

The exceptional subscheme is canonically ${Proj}_{κ} {gr}_{m} A$ . The multiplication map ${Sym}_{κ} (m / m^{2}) \to {gr}_{m} A$ is an isomorphism: choose any cotangent basis and apply the associated-graded theorem. Thus $E$ is canonically $P_{κ} (m / m^{2})$ ; the chosen basis $x, y$ identifies it with $P_{κ}^{1}$ . The center ideal is $O (- E) ≅ O (1)$ , so $E$ is effective Cartier and its normal line bundle is the restricted negative twist, $O_{P_{κ}^{1}} (- 1)$ . The projective-line coordinate identification depends on the chosen basis.

5.1F3F15F16F17F18step 4.1step 4.2∎

If $S$ is smooth over $k$ and $p$ is rational, then for every field extension $K / k$ , $S_{K}$ is smooth, regular and pure of dimension two, and $p_{K}$ is a rational closed point. Steps 1.1–4.2 apply over $K$ . Flat base change identifies $(S^{'})_{K}$ with this point blowup, so it is regular for every $K$ . The finite-type $k$ -algebras of the charts are finitely presented, and they are flat over $k$ because vector spaces are free. Hence the geometric-regularity definition proves smoothness. In the affine-plane model the quotients eliminate $y$ or $x$ , giving literal affine planes. For general $p$ only regularity is asserted; only $E$ , not the whole blowup, carries the indicated residue-field structure.

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The normal bundle of the exceptional curve is O(-1)

Statement

Assume the Axiom of Choice. Let $p$ be a closed point of a regular surface $S$ over a field $k$ , assume $\dim O_{S, p} = 2$ , and let $π ⁣ : S^{'} \to S$ be the blowup of $p$ and $E$ its exceptional curve. Then $E$ is isomorphic to the projective line over $κ (p)$ , and the restriction to $E$ of the invertible sheaf $O_{S^{'}} (E)$ is the dual tautological bundle $O_{P_{κ (p)}^{1}} (- 1)$ ; equivalently $O_{E} (E)$ has degree $- 1$ and $O_{E} (- E) = O (1)$ has degree $1$ . For $p$ $k$ -rational, $O_{E} (E) = O_{P_{k}^{1}} (- 1)$ and this twist index is an isomorphism invariant of $E$ .

Facts & Assumptions

Given: A regular surface $S$ over $k$ , a closed point $p \in S$ with two-dimensional local ring, the blowup $π ⁣ : S^{'} \to S$ of $p$ with exceptional curve $E$ and $A = O_{S, p}$ .

[A1]

Choice. The Axiom of Choice is assumed, as in the statement; the cited suppliers used below are stated under it (The Axiom of Choice).

[F1]

Regular centers have projective-bundle exceptional divisors: For a closed point $p$ of a regular surface $S$ over a field $k$ with two-dimensional local ring, the conormal sheaf $I / I^{2} = m_{p} / m_{p}^{2}$ is free of rank two over $κ (p)$ and the exceptional divisor is isomorphic to the projective line $P_{κ (p)}^{1}$ over $κ (p)$ ; the corollary identifies it with the projective bundle $P_{Z} (I / I^{2})$ in the quotient convention.

[F3]

Affine blowup standard charts and overlaps, Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains, Flat base change for blowups, and failure without flatness and regular local rings are domains and cohen macaulay: The localized point blowup has charts $A [(x, y) / x]$ and $A [(x, y) / y]$ with inverse ratio overlap, where $x, y$ are regular parameters and form a regular sequence in the domain $A$ .

[F4]

Projective bundle in the quotient convention: The projective bundle of a finite locally free module $E$ of rank $r$ over $S$ is the relative Proj $P_{S} (E) = {Proj}_{S} Sym (E) \to S$ , in the quotient convention in which an $S$ -morphism $T \to P_{S} (E)$ amounts to an isomorphism class of surjections $g^{*} E \to L$ with $L$ invertible on $T$ .

[F5]

The twist index on the projective line is an isomorphism invariant: For the twists of the relative projective line over a field, $O_{P_{k}^{1}} (n) ≅ O_{P_{k}^{1}} (m)$ if and only if $n = m$ ; hence the twist index attached to an invertible sheaf on $P_{k}^{1}$ is an isomorphism invariant.

[F6]

Twisting sheaf on Proj: For a commutative nonnegatively graded ring $S$ , the twisting sheaf on $Proj S$ is $O_{X} (n) = \tilde{S (n)}$ , the associated sheaf of the shifted graded module, with $Γ (D_{+} (f), O_{X} (n)) = S (n)_{(f)}$ on standard opens.

[F7]

The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier: For the blowup of a quasi-coherent ideal sheaf $I$ of finite type with exceptional subscheme $E = V (I O_{Bl})$ : $O (1)$ is invertible, the inverse-image ideal $I O_{Bl}$ is invertible and equals $O (1)$ , and the exceptional divisor is effective Cartier with $O_{Bl} (- E) = I O_{Bl} = O (1)$ and $O_{Bl} (E) = O (- 1)$ .

Proof

1.1A1F1F4F7

The inverse-image center ideal $I O_{S^{'}}$ is the invertible sheaf $O_{S^{'}} (- E) ≅ O (1)$ , and its dual is $O_{S^{'}} (E) ≅ O (- 1)$ . The exceptional curve is the projective bundle of the rank-two cotangent space, hence becomes $P_{κ (p)}^{1}$ on choosing a basis.

2.1F3F6F7step 1.1

Use regular parameters $x, y$ . If $x g = (x T - y) h$ , reduction modulo $x$ forces $h = x h_{1}$ , and cancellation gives $g = (x T - y) h_{1}$ ; hence the incidence quotient has no $x$ -power torsion and is the $x$ -chart. Symmetrically this proves the $y$ -chart presentation. Thus use the charts $A [T] / (x T - y)$ and $A [U] / (y U - x)$ , with $U = T^{- 1}$ . The exceptional ideal has frames $x$ and $y$ on them. On restriction to $E$ , these give frames $e_{0} = [x]$ and $e_{1} = [y]$ of $I_{E} / I_{E}^{2} = O_{E} (- E)$ ; they are not functions $x ∣_{E}$ or $y ∣_{E}$ , which are zero. Their transition is $e_{1} = T e_{0}$ , exactly the transition of the positive twist on $P_{κ (p)}^{1}$ . Thus $O_{E} (- E) ≅ O (1)$ and dualizing gives $O_{E} (E) ≅ O (- 1)$ .

3.1F5step 2.1∎

These twists have degrees $1$ and $- 1$ over $κ (p)$ , respectively. The uniqueness-of-twists lemma makes their indices isomorphism invariants. If $p$ is rational, $κ (p) = k$ and the same statements specialize to the asserted twists over $k$ ; no smoothness assumption on $S$ is needed for this specialization.

Remarks

The local-dimension assumption is automatic for closed points of finite-type pure two-dimensional surfaces. At a closed point with one-dimensional local ring the blowup is the identity and the exceptional fiber is a point; there is no exceptional-curve degree assertion in that case.

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The blowup of the plane at the origin as an incidence scheme

Statement

Assume the Axiom of Choice, inherited from the Proj constructions used below (The Axiom of Choice). Let $B$ be a commutative ring with $1$ , $X = Spec B [x, y] = A_{B}^{2}$ and $I = (x, y) \subseteq B [x, y]$ . With homogeneous coordinates $u, v$ on $P_{B}^{1}$ , the blowup ${Bl}_{I} X$ of Blowup of a scheme along an ideal sheaf is $V (x v - y u) \subseteq X \times_{B} P_{B}^{1}$ , and the projection is its structural morphism. Its two standard charts are $Spec B [x, T]$ with $y = x T$ and $Spec B [y, U]$ with $x = y U$ ; their overlap inverts $T$ and $U$ with $T U = 1$ . The exceptional subscheme is $V (x)$ and $V (y)$ respectively and is isomorphic to $P_{B}^{1}$ . In particular this holds over any field $B = k$ .

Facts & Assumptions

Given: A commutative ring $B$ , the polynomial ring $B [x, y]$ , the ideal $I = (x, y)$ , the Rees algebra $R (I) = ⨁_{n \geq 0} I^{n} t^{n}$ (Rees algebra sheaf of a finite type ideal), the blowup ${Bl}_{I} Spec B [x, y] = {Proj}_{B [x, y]} R (I)$ (Blowup of a scheme along an ideal sheaf), and the projective line $P_{B}^{1}$ with standard charts $U_{0} = Spec B [t_{0}]$ , $U_{1} = Spec B [t_{1}]$ glued by $t_{0} \mapsto 1 / t_{1}$ (Relative projective space from standard charts).

[F1]

Affine blowup standard charts and overlaps: For $I = (f_{0}, \dots, f_{r})$ the standard opens $D_{+} (f_{i} t) = Spec A [I / f_{i}]$ cover ${Bl}_{I} Spec A$ , with overlap identifications given by the ratios $(f_{j} t) / (f_{i} t)$ .

[F2]

Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: For $a \in I$ the affine blowup algebra $A [I / a] = (R (I))_{(a)}$ has $I A [I / a] = a A [I / a]$ with $a$ a nonzerodivisor and $(A [I / a])_{a} = A_{a}$ , and is independent of the chosen generating set; for $I = (a, a_{1}, \dots, a_{r})$ the surjection $A [x_{1}, \dots, x_{r}] / (a x_{i} - a_{i}) \to A [I / a]$ , $x_{i} \mapsto a_{i} / a$ , has kernel the $a$ -power torsion.

[F3]

Relative projective space from standard charts, Projective space is Proj of a polynomial ring: $P_{B}^{1}$ is the gluing of its two standard charts, and for every commutative ring $A$ there is a canonical isomorphism $Proj A [u, v] ≅ P_{A}^{1}$ , natural in $A$ ; in particular $P_{B [x, y]}^{1} ≅ X \times_{B} P_{B}^{1}$ over $Spec B$ .

[F4]

Closed subschemes of projective space and saturated ideals: A homogeneous element $f$ of $A [u, v]$ of degree $d > 0$ cuts out the closed subscheme $V_{+} (f) = Proj (A [u, v] / (f)) ↪ P_{A}^{1}$ (Closed immersions of schemes), whose intersection with the standard open $D_{+} (u)$ is $Spec (A [u, v]_{(u)} / (f / u^{d}))$ (Standard opens of Proj).

[F5]

Exceptional subscheme of a blowup: The exceptional subscheme is the scheme-theoretic inverse image of the center, cut out by its inverse-image ideal.

Proof

1.1F1

The graded $B [x, y]$ -algebra homomorphism $B [x, y] [u, v] \to R (I)$ with $u \mapsto x t$ , $v \mapsto y t$ is surjective, since $I^{n} = (x, y)^{n}$ is generated by the monomials $x^{n - i} y^{i}$ , and its kernel is $(x v - y u)$ : the degree-zero kernel is zero, and for a homogeneous $F = \sum_{i = 0}^{n} c_{i} u^{n - i} v^{i}$ with $n \geq 1$ and $F (x, y) = 0$ , reduction modulo $x$ gives $c_{n} y^{n} \equiv 0 (m o d x)$ , so $c_{n} = x d$ because $y$ is a nonzerodivisor modulo $x$ in the polynomial ring $B [x, y]$ ; then $F - d v^{n - 1} (x v - y u) = u G$ with $G$ homogeneous of degree $n - 1$ , and $0 = F (x, y) = x G (x, y)$ forces $G (x, y) = 0$ because $x$ is a nonzerodivisor, so induction on $n$ gives $F \in (x v - y u)$ ; conversely $x v - y u \mapsto 0$ , and thus $R (I) ≅ B [x, y] [u, v] / (x v - y u)$ as graded $B [x, y]$ -algebras.

2.1F1F3F4step 1.1

Consequently ${Bl}_{I} X = {Proj}_{B [x, y]} (B [x, y] [u, v] / (x v - y u))$ , and by [F4] with $A = B [x, y]$ this is the closed subscheme $V (x v - y u)$ of $P_{B [x, y]}^{1} ≅ X \times_{B} P_{B}^{1}$ of [F3]; under this identification the structural morphism of the blowup is the projection to $X$ , and the standard charts of [F1] are $D_{+} (u) \cap V (x v - y u)$ and $D_{+} (v) \cap V (x v - y u)$ .

3.1F4step 2.1

Computing the two charts by [F4]: on $D_{+} (u)$ the dehomogenised equation is $x (v / u) - y = 0$ in $B [x, y, v / u]$ , so the chart ring is $B [x, T]$ with $T = v / u$ and $y = x T$ ; on $D_{+} (v)$ it is the quotient of $B [x, y, u / v]$ by $y (u / v) - x$ , that is $B [y, U]$ with $U = u / v$ and $x = y U$ . On the overlap $D_{+} (u v)$ both $u$ and $v$ are invertible, so $T = v / u$ and $U = u / v$ are mutually inverse units, $T U = 1$ , and the two chart rings agree on the overlap as localisations $B [x, T]_{T}$ and $B [y, U]_{U}$ under $T \mapsto U^{- 1}$ .

4.1F2F3F5step 3.1

By [F5], the exceptional subscheme is the inverse image of the origin $V (x, y)$ ; by [F2] its ideal on the first chart is $I B [x, T] = (x, x T) = (x)$ , so $E \cap D_{+} (u) = V (x) = Spec B [T]$ , and on the second chart it is $I B [y, U] = (y U, y) = (y)$ , so $E \cap D_{+} (v) = V (y) = Spec B [U]$ . On the overlap the rings $B [T]_{T} = B [T, T^{- 1}]$ and $B [U]_{U} = B [U, U^{- 1}]$ are identified by $T \mapsto U^{- 1}$ , which is exactly the gluing datum of the standard charts $U_{0}, U_{1}$ of $P_{B}^{1}$ in [F3]; hence $E ≅ P_{B}^{1}$ .

5.1F3step 2.1step 3.1step 4.1∎

Assembling steps 2.1, 3.1 and 4.1: the blowup is $V (x v - y u) \subseteq X \times_{B} P_{B}^{1}$ with the projection as structural morphism, its charts are $Spec B [x, T]$ with $y = x T$ and $Spec B [y, U]$ with $x = y U$ glued by $T U = 1$ , and the exceptional subscheme is $P_{B}^{1}$ ; no hypothesis on $B$ beyond commutativity was used, so the statement specialises to any field $B = k$ .

Remarks

The proof uses only that $x$ is a nonzerodivisor of $B [x, y]$ and that $y$ is a nonzerodivisor modulo $x$ ; neither $B$ a domain nor $B$ a field is needed, and the zero ring gives the empty blowup on both sides.
The equation $x v - y u = 0$ is the incidence relation of the point $(x, y)$ and the line $[u : v]$ , which is why the blowup is described as the incidence scheme; the strict transform computations of this page use this explicit presentation in the two charts.

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Strict transforms of plane curves record tangent directions

Statement

Assume the Axiom of Choice, inherited from the blowup and Proj constructions (The Axiom of Choice). Let $k$ be a field and let $C = V (f) \subseteq A_{k}^{2}$ be a reduced plane curve through the origin with multiplicity $m = {mult}_{0} (C) \geq 1$ and leading form $f_{m}$ (the degree- $m$ part of $f$ ). Let $C^{'}$ be the strict transform of $C$ under the blowup of the origin and let $E = P_{k}^{1}$ be the exceptional curve. Then $C^{'} \cap E$ is the closed subscheme of $E$ cut out by the form $f_{m} (u, v)$ : its closed points correspond to the irreducible factors of $f_{m}$ , a factor of multiplicity $s$ contributes with multiplicity $s$ , and the underlying $0$ -cycle has total degree $m$ over $k$ . Over a field over which $f_{m}$ splits, these points are exactly the tangent directions of $C$ at the origin, with multiplicity. If $f_{m}$ is squarefree (in particular for a node, or for a cusp with reduced tangent cone) the strict transform meets $E$ transversally at each of these points.

Facts & Assumptions

Given: A field $k$ , a reduced plane curve $C = V (f) \subseteq A_{k}^{2} = Spec k [x, y]$ through the origin with $m = {mult}_{0} (f) \geq 1$ and leading form $f_{m}$ , the blowup $π ⁣ : S^{'} \to A_{k}^{2}$ of the origin with exceptional curve $E$ and its two standard charts, and the strict transform $C^{'}$ of $C$ .

[F1]

Multiplicity of a hypersurface equation at a rational point: The expansion of $f$ about the origin is $f = f_{m} + (terms of degree > m)$ with $f_{m} \neq 0$ homogeneous of degree $m$ ; equivalently $f$ has order $m$ in the local ring at the origin.

[F2]

The blowup of the plane at the origin as an incidence scheme: The blowup is $V (x v - y u) \subseteq A_{k}^{2} \times P_{k}^{1}$ with homogeneous coordinates $(u : v)$ on the second factor; its charts are $Spec k [x, s]$ with $y = x s$ and $E = V (x)$ , and $Spec k [t, y]$ with $x = y t$ and $E = V (y)$ , glued by inverting $s$ and $t$ with $s t = 1$ ; the exceptional curve is isomorphic to $P_{k}^{1}$ .

[F3]

Strict-transform equation by removing the maximal exceptional power: In the first chart $f (x, x s) = x^{m} g (x, s)$ with $g (0, s) = f_{m} (1, s) \neq 0$ , and $C^{'}$ is cut out there by $g = 0$ ; symmetrically $f (y t, y) = y^{m} h (t, y)$ with $h (t, 0) = f_{m} (t, 1) \neq 0$ , and $C^{'}$ is cut out in the second chart by $h = 0$ .

[F4]

Total transform equals strict transform plus multiplicity times the exceptional divisor: The total transform is $π^{*} C = C^{'} + m E$ , and $C^{'}$ meets $E$ in the $0$ -cycle of degree $m$ cut out by the degree- $m$ leading form of a local equation of $C$ at the origin; the degree over $k$ is $m [κ (0) : k] = m$ , because the origin is $k$ -rational.

[F5]

All initial forms define the tangent cone and The scheme-theoretic tangent cone at a point: For $I = (f)$ the initial ideal is $in (I) = (f_{m})$ , so the tangent cone of $C$ at the origin is ${Cone}_{0} (C) = Spec (k [u, v] / (f_{m})),$ and the points of its projectivization are the tangent directions of $C$ at the origin.

[F6]

The exceptional divisor is the projectivized normal cone and Effective cartier divisor: $E$ is an effective Cartier divisor on $S^{'}$ , and $E$ is the projectivized normal cone of the origin in the plane, here $P_{k}^{1} = Proj k [u, v]$ ; its standard charts are $Spec k [s]$ with $s = v / u$ and $Spec k [t]$ with $t = u / v$ (Standard opens of Proj, Projective space is Proj of a polynomial ring).

[F7]

Strict transform of a closed subscheme: $C^{'}$ is a reduced curve, cut out on the charts by the saturated ideals of [F3], and it has no component equal to $E$ because its components dominate components of $C$ while $E$ maps to the origin.

[F8]

Contact order of two regular components at a point: For two distinct reduced curves with no common component meeting at a closed point $q$ , the contact order $n_{q}$ is a finite length, and $n_{q} = 1$ if and only if the curves meet transversally at $q$ , that is, both are regular at $q$ with distinct tangent lines; the length is computed from local equations by $n_{q} = {length}_{O_{Y, q}} (O_{Y, q} / z O_{Y, q})$ .

Proof

1.1F1F2F3

Write $f = f_{m} + f_{m + 1} + \dots$ as in [F1]. By [F2] the two charts cover $S^{'}$ and meet in the locus $s t = 1$ , and by [F3] the strict transform is cut out in them by the equations $g = 0$ and $h = 0$ , where $g (0, s) = f_{m} (1, s)$ and $h (t, 0) = f_{m} (t, 1)$ ; thus $C^{'} \cap E$ is computed in the first chart by the pair of equations $x = 0$ , $g = 0$ and in the second by $y = 0$ , $h = 0$ .

1.2F2F3F6algebra

In the first chart $C^{'} \cap E$ is $Spec k [s] / (g (0, s)) = Spec k [s] / (f_{m} (1, s))$ , and in the second chart it is $Spec k [t] / (f_{m} (t, 1))$ . These are exactly the standard charts $D_{+} (u)$ and $D_{+} (v)$ of the closed subscheme $Z = Proj (k [U, V] / (f_{m})) \subseteq P_{k}^{1} = Proj k [U, V]$ : on $D_{+} (u)$ one has $s = v / u$ and the defining equation $f_{m} (1, s) = 0$ , and on $D_{+} (v)$ one has $t = u / v$ and $f_{m} (t, 1) = 0$ , with the overlap inverting $s$ and $t$ . Hence $C^{'} \cap E ≅ Z$ as closed subschemes of $E$ , the closed subscheme cut out by the form $f_{m} (u, v)$ .

2.1F2F5step 1.2algebra

By [F5] the ring $k [U, V] / (f_{m})$ is the tangent cone ring of $C$ at the origin, so $Z = Proj ({Cone}_{0} (C))$ is the projectivized tangent cone. Its closed points are the homogeneous prime ideals of $k [U, V]$ containing $f_{m}$ and not the irrelevant ideal, that is, the irreducible factors of $f_{m}$ ; writing $f_{m} = \prod_{i} p_{i}^{s_{i}}$ with $p_{i}$ irreducible homogeneous of degree $d_{i}$ , the point $q_{i}$ defined by $p_{i}$ has residue field of degree $d_{i}$ over $k$ . For a point $q_{i}$ lying in the first chart, that is $p_{i} \neq U$ , the local ring of $Z$ at $q_{i}$ is $k [s]_{(p_{i} (1, s))} / (f_{m} (1, s))$ , whose length as an $O_{Z, q_{i}}$ -module is the exponent $s_{i}$ ; the point at infinity is computed in the second chart with the roles of $U$ and $V$ exchanged. So a factor of multiplicity $s$ contributes to $C^{'} \cap E$ with multiplicity $s$ . Over a splitting field of $f_{m}$ the factors $p_{i}$ are linear forms and the points $q_{i}$ are exactly the tangent directions of $C$ at the origin, with these multiplicities.

3.1F4step 2.1

The underlying $0$ -cycle of $C^{'} \cap E$ has total degree $m$ over $k$ : by [F4] the intersection is the $0$ -cycle of degree $m$ cut out by the leading form, the origin being $k$ -rational. Equivalently, the degrees of the points $q_{i}$ weighted by the multiplicities $s_{i}$ add up to $m$ , matching the computation in the two charts of step 1.2.

3.2F7F8step 1.2step 2.1

Transversality in the squarefree case. Suppose $f_{m}$ is squarefree, so its irreducible factors occur with multiplicity one; this covers a node and a cusp with reduced tangent cone, where the leading form is a product of distinct linear or irreducible factors. Let $q$ be a closed point of $C^{'} \cap E$ lying in the first chart and let $p (s)$ be the corresponding irreducible factor of $f_{m} (1, s)$ , which is simple; the case of a point lying only in the second chart is symmetric. Write $g (x, s) = p (s) u (s) + x w (x, s)$ with $u (s)$ a unit at $p$ , which is possible because $g (0, s) = f_{m} (1, s) = p (s) u (s)$ and $p$ is simple. In the local ring $O_{S^{'}, q}$ with maximal ideal $m = (x, p (s))$ , the equation $g$ lies in $m ∖ m^{2}$ and the quotient $O_{C^{'}, q} = O_{S^{'}, q} / (g)$ has maximal ideal generated by $x$ , because $p (s) u (s) \equiv - x w (x, s)$ modulo $g$ and $u$ is a unit; hence $O_{C^{'}, q}$ is a regular one-dimensional local ring and $n_{q} (C^{'}, E) = {length}_{O_{C^{'}, q}} (O_{C^{'}, q} / x O_{C^{'}, q}) = 1$ . By [F8] contact order one is exactly transversality at $q$ , so $C^{'}$ and $E$ meet transversally at every point of $C^{'} \cap E$ when $f_{m}$ is squarefree.

4.1step 1.2step 2.1step 3.1step 3.2∎

Steps 1.2, 2.1, 3.1 and 3.2 prove all the assertions: $C^{'} \cap E$ is the closed subscheme of $E = P_{k}^{1}$ cut out by the form $f_{m} (u, v)$ , its closed points are the irreducible factors of $f_{m}$ with the corresponding multiplicities, the underlying $0$ -cycle has total degree $m$ over $k$ , the points are the tangent directions of $C$ at the origin with multiplicity over a splitting field, and for squarefree $f_{m}$ the intersection is transverse at every point.

Remarks

The theorem is the local input to the resolution algorithm on this page: a point of multiplicity $m \geq 2$ whose leading form is a product of $m$ distinct linear forms is replaced by $m$ points at which the strict transform meets the new exceptional curve transversally.
For a cusp $y^{2} = x^{3}$ the leading form $y^{2}$ is not squarefree, the strict transform meets $E$ at the single point $[1 : 0]$ with multiplicity two; its first strict transform has equation $s^{2} = x$ and is already regular, but tangent to $E$ ; this is why the resolution argument must be iterated rather than applied once, and A point blowup lowers pairwise contact order by one and separates transverse branches is the companion statement controlling the pairwise behaviour of regular branches.

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Strict-transform equation by removing the maximal exceptional power

Statement

Let $k$ be a field, let $0$ be the origin of $A_{k}^{2} = Spec k [x, y]$ and let $f \in k [x, y]$ be a reduced local equation of a curve through $0$ of multiplicity $m = {mult}_{0} (f) \geq 1$ (Multiplicity of a hypersurface equation at a rational point). In the chart with coordinates $(x, s)$ where $y = x s$ , the total transform equation is $f (x, x s) = x^{m} g (x, s)$ with $g (0, s)$ the leading form evaluated at $(1, s)$ , and the strict transform is defined by $g = 0$ ; symmetrically in the other chart. In particular the strict transform has multiplicity at most $m$ at any point of the exceptional curve $E$ , and its equation is obtained from the total transform by dividing by the largest power of the exceptional equation, which is exactly the $m$ -th power.

Facts & Assumptions

Given: The plane $A_{k}^{2} = Spec k [x, y]$ , the origin $0$ , a reduced local equation $f \in k [x, y]$ with $m = {mult}_{0} (f)$ (Multiplicity of a hypersurface equation at a rational point), the blowup $π ⁣ : S^{'} \to A_{k}^{2}$ of the origin with exceptional curve $E$ and its two standard charts $Spec k [x, s] = Spec k [x, y] [y / x]$ and $Spec k [u, y] = Spec k [x, y] [x / y]$ (The blowup of the plane at the origin as an incidence scheme), and the strict transform $C^{'}$ of the curve $C = V (f)$ (Strict transform of a closed subscheme).

[F1]

Multiplicity of a hypersurface equation at a rational point: Expanding $f (t_{1}, t_{2}) = \sum_{d \geq 0} f_{d} (t_{1}, t_{2})$ into homogeneous parts about the origin, the multiplicity $m$ is the least $d$ with $f_{d} \neq 0$ ; equivalently $f_{d} \in k [x, y]$ , $f_{d} \neq 0$ , and $f = f_{m} + (terms of degree > m)$ with $f_{m}$ the leading form.

[F2]

The blowup of the plane at the origin as an incidence scheme: The blowup of the origin is $V (x v - y u) \subseteq A_{k}^{2} \times P_{k}^{1}$ ; in the chart $Spec k [x, s]$ with $s = v / u$ one has $y = x s$ and $E = V (x)$ ; in the chart $Spec k [u, y]$ with $u = u / v$ one has $x = y u$ and $E = V (y)$ ; the overlap inverts $s$ and $u$ with $s u = 1$ .

[F3]

Affine blowup standard charts and overlaps and Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: On the affine chart cut by the element $x$ of the ideal $(x, y)$ , the affine blowup algebra is $k [x, y] [(x, y) / x] = k [x, y] [s] / (x s - y) = k [x, s]$ , with $(x, y) k [x, s] = x k [x, s]$ and $x$ a nonzerodivisor; the two charts cover the blowup.

[F4]

Total transform equals strict transform plus multiplicity times the exceptional divisor: For a reduced curve $C$ through the origin with multiplicity $m$ , $π^{*} C = C^{'} + m E$ as effective Cartier divisors (Effective cartier divisor), and the strict transform is obtained on each chart by dividing a local equation of the total transform by the $m$ -th power of an exceptional equation.

Proof

1.1F1

Write the homogeneous decomposition of $f$ about the origin as $f = f_{m} + f_{m + 1} + \dots$ , with $f_{m} \neq 0$ the leading form by [F1]. Substituting $y = x s$ gives $f (x, x s) = \sum_{d \geq m} x^{d} f_{d} (1, s) = x^{m} (f_{m} (1, s) + x f_{m + 1} (1, s) + x^{2} f_{m + 2} (1, s) + \dots) = x^{m} g (x, s)$ , where $g (x, s) : = \sum_{d \geq m} x^{d - m} f_{d} (1, s) \in k [x, s]$ .

2.1F2F3step 1.1

The constant term in $x$ of $g$ is $f_{m} (1, s)$ , which is nonzero: the distinct degree- $m$ monomials $x^{m - j} y^{j}$ become the distinct monomials $s^{j}$ , so their nonzero coefficient vector cannot vanish. Consequently $g (0, s) = f_{m} (1, s) \neq 0$ , and the exact power of $x$ dividing $f (x, x s)$ is $m$ ; since $E = V (x)$ on this chart by [F3], the equation of the total transform on the chart is $x^{m} g$ with $g$ not divisible by $x$ .

3.1F3F4step 1.1step 2.1

By [F4] the total transform is $π^{*} C = C^{'} + m E$ , and in the chart its local equation is the product of a local equation of $C^{'}$ with the $m$ -th power $x^{m}$ of the exceptional equation; by step 2.1 the local equation of the total transform is $x^{m} g$ with $x ∤ g$ , so the strict transform is cut out by $g = 0$ in this chart, as claimed. The same computation with the roles of $x$ and $y$ interchanged, using the second chart with $x = y u$ , gives the symmetric description $f (y u, y) = y^{m} \tilde{g} (u, y)$ with $\tilde{g} (u, 0) = f_{m} (u, 1)$ and strict transform $\tilde{g} = 0$ ; the two chart equations glue to the strict transform by [F4] and Strict transform of a closed subscheme, since they are the saturations of the total transform by the exceptional equation on each chart.

3.2F1F2step 2.1

A closed point $q$ of $E$ in the first chart corresponds to an irreducible polynomial $p (s)$ , and its ambient maximal ideal is $(x, p (s))$ . Let $e$ be the exponent of $p$ in the nonzero polynomial $f_{m} (1, s)$ . Its image in $k [s]_{(p)}$ lies in $(p)^{e} ∖ (p)^{e + 1}$ . If $g$ belonged to $(x, p)^{e + 1}$ in the local chart ring, reduction modulo $x$ would put that polynomial in $(p)^{e + 1}$ , a contradiction. Thus the order of $g$ is at most $e \leq \deg f_{m} (1, s) \leq m$ . Points of $E$ outside the strict transform have unit equation and order zero. The second chart gives the identical bound, covering also the point at infinity. This proves the bound for every closed point, with arbitrary residue field; at the generic point of $E$ , $g$ is a unit as well.

4.1step 3.1step 3.2∎

Steps 3.1 and 3.2 prove the assertions: the strict transform equation in each chart is obtained from the total transform by dividing by the largest power of the exceptional equation, which is exactly $x^{m}$ in the first chart and $y^{m}$ in the second, with the leading form evaluated at $(1, s)$ (respectively $(u, 1)$ ) as the value along $E$ , and the strict transform has multiplicity at most $m$ at every point of $E$ .

Remarks

The result is the chart-level form of the standard fact that the strict transform of a plane curve of multiplicity $m$ at the origin meets the exceptional curve in the closed points determined by the irreducible homogeneous factors of the leading form, each with the corresponding multiplicity. Over a splitting field these factors are linear and describe the geometric tangent directions.
No reducedness or smoothness of $C$ away from the origin is used; only the finite multiplicity $m$ enters.

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Normalization of a reduced curve is finite

Statement

Let $k$ be a field and let $C$ be a reduced $k$ -scheme of finite type, of pure dimension one (for example a reduced projective plane curve or an open subscheme of one). Then there exists a finite morphism $ν ⁣ : C^{n u} \to C$ with the following properties:

$C^{n u}$ is regular of dimension one (equivalently normal: all local rings are discrete valuation rings or fields);
$ν$ is an isomorphism over the regular locus of $C$ and is birational on each irreducible component;
on an affine chart $Spec A$ of $C$ , $ν$ is the morphism corresponding to the integral closure of $A$ in its total ring of fractions;
$ν$ is unique up to a unique $C$ -isomorphism, and $ν_{*} O_{C^{n u}}$ is a coherent $O_{C}$ -module.

No separability or perfectness hypothesis on $k$ is needed.

Facts & Assumptions

Given: A field $k$ and a reduced finite-type $k$ -scheme $C$ of pure dimension one. The Axiom of Choice is inherited from the finiteness suppliers (The Axiom of Choice).

[F1]

A finite-type domain over a field has finite normalization: The integral closure of a finite-type domain over any field in its fraction field is a finite module. No perfectness or separability assumption is required.

[F2]

The integral closure of a domain in a field extension is integrally closed, A domain is integrally closed if and only if its prime localisations are, equivalently if and only if its maximal localisations are, and A module-finite algebra over a Noetherian ring is a Noetherian ring, and so is every ring between the two: An integral closure in a field is an integrally closed domain; its localizations are integrally closed; a module-finite algebra over a Noetherian ring is Noetherian.

[F3]

Finite normalization commutes with principal localization: If $B$ is the integral closure of a domain $A$ in its fraction field, then $B_{f}$ is the integral closure of $A_{f}$ in that field, and finiteness is preserved.

[F4]

Function field of an integral finite-type scheme: Every nonempty affine open of an integral finite-type scheme has fraction field equal to its generic stalk. This does not require separatedness.

[F5]

Every point of a Zariski-open set has a distinguished-open neighbourhood inside it, Gluing affine schemes along compatible open isomorphisms, and Morphisms of schemes are local on compatible open covers: Distinguished opens refine neighborhoods in affine schemes; schemes can be glued along open isomorphisms satisfying the cocycle condition, and morphisms agreeing on overlaps glue.

[F6]

Every algebra of finite type over a principal ideal domain is a Noetherian ring, A Noetherian ring has finitely many minimal prime ideals, Irreducible components of the spectrum correspond to minimal prime ideals, and The reduced quotient by the nilradical: Finite-type algebras over fields are Noetherian. A Noetherian scheme has finitely many irreducible components, and the minimal primes of a reduced Noetherian ring have zero intersection.

[F7]

The reduction of a scheme and Integral schemes: Each irreducible component with its reduced structure is integral.

[F8]

total ring of fractions: $Q (A)$ is the localization of $A$ at its nonzerodivisors; for a domain it is its fraction field.

[F9]

Integral closure in an extension ring and integrally closed domains: Integral closure consists of elements satisfying monic equations, and a domain is integrally closed if all such elements in its fraction field belong to it.

[F10]

Finite morphisms of schemes and Finite is affine and local on its target: A finite morphism is affine with module-finite coordinate algebras on affine target opens; module-finiteness on an affine open cover implies finiteness.

[F11]

Injective integral extensions preserve Krull dimension and Dimension can be computed on an open cover: An injective integral ring extension preserves dimension, and the dimension of a Noetherian space is the supremum of the dimensions of an open cover.

[F12]

normal noetherian ring, Height-one localizations of normal Noetherian domains are DVRs, and one dimensional regular local rings are dvrs, and Equivalent characterizations of a DVR: A normal Noetherian domain has DVR localizations at height-one primes; its zero-dimensional localizations are fields. A one-dimensional Noetherian local ring is regular exactly when it is a DVR, and a DVR is integrally closed.

[F13]

Quasi-coherent module on a scheme and Coherent module sheaves: An affine direct image of the structure sheaf is quasi-coherent; on a locally Noetherian scheme a finite-type quasi-coherent module is coherent.

Proof

1.1F1F2F4F6F7

Since $C$ is quasi-compact and its affine coordinate rings are Noetherian, it is Noetherian. Its finitely many reduced irreducible components $C_{i}$ are integral finite-type curves of dimension one. Choose a finite affine cover $U_{i j} = Spec A_{i j}$ of each component and identify all their fraction fields with its generic field $K_{i}$ . Let $B_{i j}$ be the integral closure of $A_{i j}$ in $K_{i}$ . These are finite Noetherian normal domains.

2.1F2F3F5F10F11step 1.1

The schemes $Spec B_{i j}$ glue over $C_{i}$ , even if $C_{i}$ is nonseparated. Here are the overlap identifications explicitly. For a point in $U_{i j} \cap U_{i l}$ choose distinguished neighborhoods $D (f) \subset U_{i j}$ and $D (g) \subset U_{i l}$ contained in that intersection. On $D (f)$ write $g = c / f^{r}$ , and on $D (g)$ write $f = d / g^{s}$ . Their common intersection is the distinguished open $D (f c)$ in $U_{i j}$ and $D (g d)$ in $U_{i l}$ ; both coordinate rings are the same subring of $K_{i}$ , since they are the sections of the same open subscheme. Their integral closures are therefore the same subring of $K_{i}$ , namely $(B_{i j})_{f c} = (B_{i l})_{g d}$ . These common distinguished opens cover the overlap. Their identities glue, and the triple-overlap identities satisfy the cocycle condition because all are identities inside $K_{i}$ . Thus scheme and morphism gluing produce an integral normal scheme $C_{i}^{n u}$ and a morphism $ν_{i} : C_{i}^{n u} \to C_{i}$ , with inverse image of $U_{i j}$ equal to $Spec B_{i j}$ . It is finite by the affine-cover criterion. Its generic field is $K_{i}$ , and its dimension is one by integral dimension preservation on the affine cover.

3.1F1F2F6F9F10step 2.1

Set $C^{n u} = ⨆_{i} C_{i}^{n u}$ and compose each $ν_{i}$ with the reduced closed immersion $C_{i} ↪ C$ to obtain $ν$ . This is finite: closed immersions are finite on affine charts by the quotient-ring description; composition is module-finite, and a finite disjoint union is module-finite. Empty affine opens have empty inverse image, with coordinate algebra zero; the following computation concerns nonempty affine opens. More explicitly, on any affine $U = Spec A \subset C$ , let $p_{1}, \dots, p_{m}$ be its minimal primes and $A_{i} = A / p_{i}$ . The corresponding component inverse images are affine with finite coordinate domains $D_{i}$ lying in $Frac (A_{i})$ , normal and birational over $A_{i}$ . Consequently $D_{i}$ is its integral closure $B_{i}$ : every element of $D_{i}$ is integral over $A_{i}$ , and every element of the fraction field integral over $A_{i}$ is also integral over $D_{i}$ , hence belongs to $D_{i}$ . Therefore $ν^{- 1} (U) = Spec B$ where $B = \prod_{i} B_{i}$ . Each factor has finitely many $A_{i}$ -module generators; placing these in their separate coordinates gives finitely many $A$ -module generators of $B$ . This uses no Chinese-remainder decomposition of $A$ .

4.1F6F8step 3.1

We identify $Q (A)$ correctly. Distinct minimal primes are incomparable. For each $i$ choose $a_{i j} \in p_{j} ∖ p_{i}$ for $j \neq i$ and put $g_{i} = \prod_{j \neq i} a_{i j}$ ; for one minimal prime put $g_{1} = 1$ . Its image is nonzero in $A_{i}$ and zero in every other $A_{j}$ . An element $s \in A$ is a nonzerodivisor exactly when its image is nonzero in every $A_{i}$ : the forward implication follows since otherwise $s g_{i} = 0$ with $g_{i} \neq 0$ , and the reverse follows from the injection $A ↪ \prod_{i} A_{i}$ . Hence localization gives an injection $Q (A) ↪ \prod_{i} Frac (A_{i})$ . For any tuple $u_{i} / v_{i}$ in this product, choose lifts $a_{i}, b_{i} \in A$ of $u_{i}, v_{i}$ , and set $a = \sum_{i} g_{i} a_{i}$ and $s = \sum_{i} g_{i} b_{i}$ . The image of $s$ in $A_{i}$ is the nonzero product $g_{i} v_{i}$ , so $s$ is a nonzerodivisor, and $a / s$ has the prescribed tuple of images. This proves surjectivity.

4.2F11F12step 2.1step 3.1

Each $C_{i}^{n u}$ is normal Noetherian of dimension one. All its local rings are therefore fields or DVRs, hence regular. Their disjoint union is regular of dimension one. The generic-field identifications give birationality on each component, meaning the restriction from the corresponding normalized component, not a claim that scheme-theoretic base change over $C_{i}$ removes the other branches.

4.3F13step 3.1

The affine description gives a finite module $B$ on each affine target chart. Thus $ν_{*} O_{C^{n u}}$ is quasi-coherent of finite type, hence coherent on the locally Noetherian scheme $C$ .

5.1F9step 3.1step 4.1

Under that identification $B = \prod_{i} B_{i}$ is the integral closure of $A$ in $Q (A)$ . Since $B$ is module-finite over $A$ , every $b \in B$ satisfies a monic equation over $A$ : multiplication by $b$ on a finite generating family and the determinant trick give a monic polynomial annihilating $B$ , hence annihilating $1$ . Conversely a tuple integral over $A$ has its $i$ th coordinate integral over $A_{i}$ , so that coordinate belongs to $B_{i}$ . This proves precisely the affine description in part (3).

6.1F3F6F9F12step 3.1step 5.1

At a regular point $x$ of $C$ , its local ring is a field (in dimension zero its maximal ideal has zero cotangent space and hence is zero by Nakayama) or a DVR, hence an integrally closed domain. Thus only one component passes through $x$ . Remove the other finitely many closed components to obtain a neighborhood with integral coordinate rings. For an affine neighborhood $Spec A$ therein, localization of its integral closure at the prime of $x$ equals $A_{x}$ . Indeed any element integral over $A_{x}$ has an equation with finitely many denominators outside that prime; clearing these denominators after multiplying the element by their product shows it belongs to a localization of the integral closure of $A$ . The reverse inclusion is immediate. As $B / A$ is a finite module, its zero stalk at $x$ implies it vanishes on a distinguished neighborhood of $x$ (annihilate each of finitely many generators with an element outside the prime). On that neighborhood $A = B$ , so $ν$ is an isomorphism. These neighborhoods cover the regular locus.

7.1F5F8step 2.1step 5.1step 4.3∎

Any other morphism with the stated properties has, by part (3), the same integral-closure algebra $B \subset Q (A)$ on each affine target chart. These canonical identifications commute with restriction: they are the same identifications inside the component function fields used in step 2.1. They therefore glue to a $C$ -isomorphism. It is unique: an $A$ -algebra automorphism of $B$ localizes to an automorphism of $Q (A)$ fixing $A$ , hence fixing every fraction $a / s$ . Here localization of $B$ at the nonzerodivisors of $A$ equals $Q (A)$ , since $A \subset B \subset Q (A)$ . Since $B$ embeds in $Q (A)$ , that automorphism is already the identity on $B$ . This proves part (4).

Remarks

Normalization separates the reduced irreducible components rather than gluing their normalizations along intersection points. The overlap construction above does not assume separatedness. The field-finiteness theorem [F1] applies to arbitrary fields, including imperfect fields, and regularity here means regularity of the local rings, not smoothness over $k$ .

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Normalization defect delta of a reduced curve

Definition

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field and let $C$ be a reduced curve of finite type over $k$ : a $k$ -scheme of finite type with $C = C_{r e d}$ (The reduction of a scheme) and pure dimension one, proper over $k$ in the applications. Let

$ν ⁣ : \tilde{C} ⟶ C$

be the finite normalization of Normalization of a reduced curve is finite, and let

$Q_{C} : = coker (O_{C} ⟶ ν_{*} O_{\tilde{C}})$

be the normalization defect sheaf, a coherent $O_{C}$ -module (Coherent module sheaves). The normalization defect of $C$ is

$δ_{k} (C) : = \dim_{k} H^{0} (C, Q_{C}),$

the $k$ -dimension of its space of global sections, a nonnegative integer.

The following comments record why the definition is meaningful. Since $ν$ is finite, $ν_{*} O_{\tilde{C}}$ is a coherent $O_{C}$ -module and so is its quotient $Q_{C}$ ; this is the finite-pushforward case of Coherent higher direct images under proper morphisms, and also follows directly from the affine-local description of a finite morphism. The stalks of $Q_{C}$ vanish exactly at the points $p$ at which $O_{C, p} \to (ν_{*} O_{\tilde{C}})_{p}$ is an isomorphism, so $Q_{C}$ is supported on the non-normal locus of $C$ : a closed subset of the Noetherian one-dimensional space $C$ containing no generic point of $C$ , because the local ring of the reduced curve $C$ at a generic point is a field and hence normal. Such a closed set is a finite set of closed points; componentwise this is the finiteness of Proper closed subsets of a curve are finite. Consequently $H^{0} (C, Q_{C})$ is the direct sum of the finitely many stalk contributions $Q_{C, p}$ over the support of $Q_{C}$ , each of which is a finite-dimensional $k$ -vector space because $Q_{C, p}$ has finite length over the Noetherian local ring $O_{C, p}$ and the residue field $κ (p)$ is finite over $k$ . Thus $δ_{k} (C)$ is a well-defined nonnegative integer. When $C$ is proper over $k$ , the same finiteness is the statement of Euler characteristic of a coherent sheaf for the coherent sheaf $Q_{C}$ .

Remarks

The definition uses only the finite normalization $ν$ , its pushforward $ν_{*} O_{\tilde{C}}$ , and $H^{0}$ of the cokernel; no choice of a resolution of singularities or of a blowup sequence enters.
The defect can be read off pointwise as a sum of local contributions; the identity with the Euler characteristic difference $χ (O_{\tilde{C}}) - χ (O_{C})$ and the weighted sum of local lengths are proved later on this page, and are not part of the definition.

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The normalization defect is an Euler characteristic and a weighted sum of local lengths

Statement

Assume the Axiom of Choice. Let $k$ be a field, let $C$ be a reduced proper curve over $k$ with normalization $ν ⁣ : \tilde{C} \to C$ and defect sheaf $Q_{C}$ . Then: (1) $H^{q} (C, Q_{C}) = 0$ for $q \geq 1$ , so $δ_{k} (C) = χ_{k} (C, Q_{C}) = χ_{k} (O_{\tilde{C}}) - χ_{k} (O_{C})$ (using additivity of the Euler characteristic in the normalization sequence $0 \to O_{C} \to ν_{*} O_{\tilde{C}} \to Q_{C} \to 0$ and $χ (C, ν_{*} O) = χ (\tilde{C}, O)$ ); (2) $δ_{k} (C) = \sum$ over closed points $p$ of $[κ (p) : k]$ times the length of $Q_{C, p}$ over $O_{C, p}$ , a finite sum over the finite non-normal locus; (3) $δ_{k} (C) \geq 0$ , and $δ_{k} (C) = 0$ if and only if $C$ is regular, in which case the irreducible components of $C$ are disjoint and normal.

Facts & Assumptions

Given: A field $k$ , a reduced proper curve $C$ over $k$ (reduced in the sense of (The reduction of a scheme), pure dimension one), its finite normalization $ν ⁣ : \tilde{C} \to C$ of (Normalization of a reduced curve is finite), the defect sheaf $Q_{C} = coker (O_{C} \to ν_{*} O_{\tilde{C}})$ and the defect $δ_{k} (C) = \dim_{k} H^{0} (C, Q_{C})$ of (Normalization defect delta of a reduced curve).

[F1]

Normalization defect delta of a reduced curve: $Q_{C} = coker (O_{C} \to ν_{*} O_{\tilde{C}})$ is a coherent $O_{C}$ -module supported on the finite non-normal locus, $δ_{k} (C) = \dim_{k} H^{0} (C, Q_{C})$ , and $H^{0} (C, Q_{C})$ is the direct sum of the finitely many stalk contributions $Q_{C, p}$ over the support.

[F2]

Normalization of a reduced curve is finite: $ν ⁣ : \tilde{C} \to C$ is finite, $\tilde{C}$ is regular of dimension one, on an affine chart $ν$ corresponds to the inclusion of $A$ into its integral closure in the total ring of fractions, $ν$ is unique up to a unique $C$ -isomorphism, and $ν_{*} O_{\tilde{C}}$ is coherent.

[F3]

Euler characteristic is additive in short exact sequences: For a short exact sequence $0 \to F^{'} \to F \to F^{''} \to 0$ of coherent sheaves on a scheme proper over $k$ , $χ (X, F) = χ (X, F^{'}) + χ (X, F^{''})$ .

[F4]

Euler characteristic of a coherent sheaf: For $X$ proper over $k$ and $F$ coherent, $χ (X, F) = \sum_{q \geq 0} (- 1)^{q} \dim_{k} H^{q} (X, F)$ is a finite alternating sum of finite $k$ -dimensions.

[F5]

Affine pushforward is compatible with sheaf cohomology: For an affine morphism $f ⁣ : X \to S$ and a quasi-coherent $O_{X}$ -module $F$ , the natural maps $H^{q} (S, f_{*} F) \to H^{q} (X, F)$ are isomorphisms for all $q \geq 0$ .

[F6]

Composition series and length of a module: A composition series of a module is a finite chain with simple successive factors; the length $ℓ_{R} (M)$ is the number of factors, is independent of the series, and the zero module has length $0$ .

[F7]

Module length is additive in short exact sequences: For a short exact sequence $0 \to N \to M \to Q \to 0$ , $M$ has finite length if and only if $N$ and $Q$ do, and then $ℓ_{R} (M) = ℓ_{R} (N) + ℓ_{R} (Q)$ .

[F8]

A skyscraper sheaf of abelian groups at a point, Flasque sheaf and Flasque abelian sheaves are Γ-acyclic: A skyscraper sheaf $i_{x, *} A$ on a topological space $X$ is flasque, because its restriction maps are either identities or zero maps; hence, under the Axiom of Choice inherited from that acyclicity theorem, $H^{q} (X, i_{x, *} A) = 0$ for every $q > 0$ .

[F9]

one dimensional regular local rings are dvrs: A nonzero Noetherian local ring of dimension one is regular if and only if it is a discrete valuation ring; fields are excluded from the term DVR.

[F10]

Valuation rings are integrally closed: Every valuation ring is an integrally closed domain; in particular every discrete valuation ring is an integrally closed domain.

[F11]

normal noetherian ring: A commutative Noetherian ring is normal if every prime localization is an integrally closed domain.

Proof

1.1F1F2F3

The normalization sequence $0 \to O_{C} \to ν_{*} O_{\tilde{C}} \to Q_{C} \to 0$ is short exact: $Q_{C}$ is the cokernel by definition [F1], and $O_{C} \to ν_{*} O_{\tilde{C}}$ is injective because on an affine chart it is the inclusion of $A$ into its integral closure in the total ring of fractions [F2]. All three terms are coherent ( $O_{C}$ , $ν_{*} O_{\tilde{C}}$ by [F2], and $Q_{C}$ by [F1]), so the sequence satisfies the hypotheses of [F3].

1.2F2F4F5

One has $χ (C, ν_{*} O_{\tilde{C}}) = χ (\tilde{C}, O_{\tilde{C}})$ : the finite morphism $ν$ is affine by [F2] and its inverse image of an affine open is affine, so [F5] identifies $H^{q} (C, ν_{*} O_{\tilde{C}})$ with $H^{q} (\tilde{C}, O_{\tilde{C}})$ for every $q \geq 0$ , and the two alternating sums of [F4] agree.

1.3F1F8

Let $F$ be the finite closed support of $Q_{C}$ . The germ maps define a sheaf map $Q_{C} \to ⨁_{p \in F} i_{p, *} Q_{C, p}$ , where each summand is the skyscraper of the underlying abelian group. It is an isomorphism on stalks: at $p \in F$ its $p$ -component is the identity and the other summands have zero stalk since their points are closed; outside $F$ both stalks are zero. Hence it is an isomorphism of abelian sheaves. A finite sum of skyscrapers is flasque, because every restriction is a direct sum of identities and maps onto zero. Flasque acyclicity proves $H^{q} (C, Q_{C}) = 0$ for $q \geq 1$ and also proves directly the degree-zero stalk sum used below.

2.1F3F4step 1.2step 1.3

Consequently $δ_{k} (C) = \dim_{k} H^{0} (C, Q_{C}) = χ (C, Q_{C})$ by [F4], and applying [F3] to the sequence of step 1.1 gives $χ (C, ν_{*} O_{\tilde{C}}) = χ (C, O_{C}) + χ (C, Q_{C})$ , so the two identities combined with step 1.2 yield $δ_{k} (C) = χ (\tilde{C}, O_{\tilde{C}}) - χ (C, O_{C})$ ; this proves (1).

3.1F1F6F7step 2.1

For the length formula of (2): by [F1], $H^{0} (C, Q_{C})$ is the direct sum of the stalks $Q_{C, p}$ over the finite non-normal locus, and each $Q_{C, p}$ has finite length over the Noetherian local ring $O_{C, p}$ ; fixing a composition series [F6] with successive quotients $κ (p)$ and using additivity of $k$ -dimension in short exact sequences together with [F7], one gets $\dim_{k} Q_{C, p} = ℓ_{O_{C, p}} (Q_{C, p}) \cdot [κ (p) : k]$ . Summing gives $δ_{k} (C) = \sum_{p} [κ (p) : k] ℓ_{O_{C, p}} (Q_{C, p})$ over the closed points of the finite non-normal locus.

4.1F2F9F10F11step 3.1∎

For (3): the sum in step 3.1 has nonnegative terms, so $δ_{k} (C) \geq 0$ ; if $δ_{k} (C) = 0$ then every local length vanishes, hence $Q_{C} = 0$ and the injective map $O_{C} \to ν_{*} O_{\tilde{C}}$ of step 1.1 is an isomorphism, so the affine morphism $ν$ is an isomorphism and $C ≅ \tilde{C}$ is regular of dimension one by [F2]. Conversely, if $C$ is regular, then each one-dimensional local ring $O_{C, p}$ is a discrete valuation ring by [F9], hence an integrally closed domain by [F10], and the zero-dimensional stalks are fields, so $C$ is normal in the sense of [F11]; the identity $C \to C$ is then a finite morphism from a normal curve that is an isomorphism over the regular locus, so the uniqueness clause of [F2] makes the normalization isomorphic to the identity over $C$ , whence $Q_{C} = 0$ and $δ_{k} (C) = 0$ . Finally, in the regular case every local ring is a discrete valuation ring or a field, hence a domain, so each point of $C$ lies in a unique irreducible component and distinct components are disjoint; a component, having everywhere the local ring of $C$ , is itself regular and hence normal.

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Normalization is unchanged under finite birational maps of reduced curves

Statement

Assume the Axiom of Choice. Let $f ⁣ : X \to Y$ be a finite birational morphism of reduced curves of finite type over a field $k$ (for instance the restriction to a curve of a proper quasi-finite birational map; such a map is finite in the applications by A proper quasi-finite morphism is finite). Then $f$ induces an isomorphism of normalizations $\tilde{X} \to \tilde{Y}$ over $Y$ ; equivalently, the normalizations of $X$ and $Y$ are canonically identified with the same finite birational model of $Y$ .

Facts & Assumptions

Given: A field $k$ , reduced curves $X, Y$ of finite type over $k$ (pure dimension one, reduced), and a finite birational morphism $f ⁣ : X \to Y$ , where birational means that $f$ bijects the generic points of irreducible components and induces an isomorphism on their local rings (the function fields); this extends the integral-scheme convention of [F7]. Let $\tilde{X} \to X$ and $\tilde{Y} \to Y$ be the finite normalizations of (Normalization of a reduced curve is finite).

[F1]

Normalization of a reduced curve is finite: For a reduced $k$ -scheme $C$ of finite type and pure dimension one there is a finite morphism $ν ⁣ : \tilde{C} \to C$ with $\tilde{C}$ regular of dimension one, $ν$ an isomorphism over the regular locus, $ν$ corresponding on an affine chart $Spec A$ to the integral closure of $A$ in its total ring of fractions, and $ν$ unique up to a unique $C$ -isomorphism.

[F2]

Finite morphisms of schemes: A morphism $f ⁣ : X \to S$ is finite if for every affine open $U = Spec A \subseteq S$ its inverse image is affine, $f^{- 1} (U) = Spec B$ , and $B$ is module-finite over $A$ .

[F3]

Finite morphisms are integral and universally closed: For a finite morphism, every ring map $A \to B$ induced on an affine chart is integral.

[F4]

Birational morphisms restrict to isomorphisms between principal affine opens: For integral $k$ -schemes of finite type and a birational morphism $g ⁣ : X \to Y$ that is locally of finite type, there are nonempty affine opens $U = Spec A \subseteq X$ , $V = Spec B \subseteq Y$ with $g (U) \subseteq V$ and an element $σ \in B ∖ {0}$ such that the localised ring map $B_{σ} \to A_{σ}$ is an isomorphism and $g$ restricts to an isomorphism $g^{- 1} (D (σ)) \cap U \to D (σ)$ .

[F5]

Integral closure in an extension ring and integrally closed domains: For a domain $A$ with fraction field $K$ , the integral closure of $A$ in $K$ is the set of elements of $K$ integral over $A$ ; $A$ is integrally closed when it contains every such element.

[F6]

Integral closure is unchanged across an integral intermediate domain: For domains $A \subseteq B \subseteq L$ with $B$ integral over $A$ , an element $z \in L$ is integral over $A$ if and only if it is integral over $B$ .

[F7]

Birational morphisms of integral finite-type schemes: For integral $k$ -schemes of finite type, a morphism is birational when it maps the generic point to the generic point and induces an isomorphism of the function-field stalks.

Proof

1.1givenF2F7

By the stated birationality convention, each reduced component $X_{i}$ corresponds to exactly one reduced component $Y_{j}$ , with the same generic field. The restriction $f_{i} : X_{i} \to Y_{j}$ exists: the ideal of $Y_{j}$ pulls back to zero on the generic point of the reduced integral scheme $X_{i}$ , hence to zero everywhere on $X_{i}$ . It is finite, since on affine charts its coordinate algebra is a quotient of the finite $A$ -algebra for $f$ , and the $A$ -action factors through the quotient defining $Y_{j}$ . Thus $f_{i}$ is finite and birational in the integral sense of [F7].

2.1F1F5step 1.1algebra

The affine normalization construction of [F1] separates the reduced components, so $\tilde{X} = ∐_{i} {\tilde{X}}_{i}$ and $\tilde{Y} = ∐_{j} {\tilde{Y}}_{j}$ . Indeed for a reduced Noetherian affine curve with minimal primes $p_{i}$ , its total ring of fractions is $\prod_{i} Frac (A / p_{i})$ , as established in the construction of [F1]. Its integral closure is $\prod_{i} \overline{A / p_{i}}$ : projection of a monic equation proves one inclusion; conversely, lift a monic equation for each coordinate to $A [T]$ and multiply the finitely many lifted polynomials, obtaining a monic equation annihilating the tuple. These identifications commute with restrictions and give the claimed decompositions. It therefore suffices to compare the normalizations for each $f_{i}$ .

3.1F1F2F3F4F5step 2.1

Fix $i, j$ and an affine open $U = Spec A \subseteq Y_{j}$ with $f_{i}^{- 1} (U) = Spec B$ ; then $A$ and $B$ are domains of dimension one, finite type over $k$ , the map $A \to B$ is injective, module-finite by [F2] and integral by [F3], and the birationality of $f_{i}$ gives $Frac (A) = Frac (B)$ as subfields of the common function field $K (Y_{j}) = K (X_{i})$ . Indeed [F4] applied to $f_{i}$ supplies a nonempty affine open of $Y_{j}$ on which the localised map is an isomorphism, and localising a domain at a nonzero element does not change its fraction field. By [F1] the normalization ${\tilde{Y}}_{j}$ over $U$ is the spectrum of the integral closure $\overline{A}$ of $A$ in $Frac (A)$ and ${\tilde{X}}_{i}$ over $f_{i}^{- 1} (U)$ is the spectrum of the integral closure $\overline{B}$ of $B$ in $Frac (B)$ ([F5]).

4.1F6step 3.1

In the situation of step 3.1 one has $\overline{A} = \overline{B}$ as subrings of the common field $Frac (A) = Frac (B)$ : since $A \subseteq B \subseteq Frac (B)$ and $B$ is integral over $A$ , [F6] says that an element is integral over $A$ exactly when it is integral over $B$ . Hence the affine normalizations agree canonically over $U$ , and the identification is the identity on the common function field.

5.1F1step 4.1∎

The identifications of step 4.1 are canonical on affine charts (both sides are the same integral closure inside the same function field), so they agree on overlaps and glue to an isomorphism ${\tilde{X}}_{i} \to {\tilde{Y}}_{j}$ over $Y_{j}$ ; assembling over the components by step 2.1 gives the isomorphism $\tilde{X} \to \tilde{Y}$ over $Y$ , and the uniqueness clause of [F1] makes it the canonical identification of the two normalizations with the same finite birational model of $Y$ .

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Euler characteristic and normalization defect under a point blowup

Statement

Assume the Axiom of Choice. Let $k$ be a field, let $S$ be a regular surface proper over $k$ with an ample invertible sheaf, let $C \subseteq S$ be a reduced curve (an effective Cartier divisor), let $p$ be a closed point of $S$ with residue degree $r = [κ (p) : k]$ and let $m = {ord}_{m_{p}} (f) \geq 1$ be the order of a local equation $f$ of $C$ in $O_{S, p}$ , namely $f \in m_{p}^{m} ∖ m_{p}^{m + 1}$ . This is the intrinsic multiplicity, agreeing with Multiplicity of a hypersurface equation at a rational point in its affine rational-point setting. Let $π ⁣ : S^{'} \to S$ be the blowup of $p$ with exceptional curve $E$ and let $C^{'}$ be the strict transform of $C$ . Then $π^{*} C = C^{'} + m E$ as effective Cartier divisors on $S^{'}$ , and writing $χ_{k} (O_{X}) : = χ_{k} (X, O_{X})$ for the Euler characteristic of the structure sheaf (Euler characteristic of a coherent sheaf), one has $χ_{k} (O_{C^{'}}) = χ_{k} (O_{C}) + r (\binom{m}{2}) .$ Consequently, for the normalization defect $δ_{k}$ of Normalization defect delta of a reduced curve, $δ_{k} (C^{'}) = δ_{k} (C) - r (\binom{m}{2}) .$

Facts & Assumptions

Given: A field $k$ , a regular surface $S$ proper over $k$ with an ample invertible sheaf, a reduced effective Cartier divisor $C \subseteq S$ , a closed point $p$ with residue field $K = κ (p)$ of degree $r$ over $k$ , the multiplicity $m = {mult}_{p} (C) \geq 1$ , the blowup $π ⁣ : S^{'} \to S$ of $p$ with exceptional curve $E$ , and the strict transform $C^{'}$ . The Axiom of Choice is assumed as in the statement, inherited from the Proj and cohomology constructions (The Axiom of Choice).

[F1]

Total transform equals strict transform plus multiplicity times the exceptional divisor: For a reduced curve $C$ on a regular surface, a point blowup gives $π^{*} C = C^{'} + m E$ with $C^{'}$ the strict transform, and $C^{'}$ meets $E$ in the $0$ -cycle of degree $m$ over $K$ cut out by the degree- $m$ leading form. Both $C$ and $C^{'}$ are effective Cartier divisors (Cartier divisor).

[F2]

Point blowups of regular surfaces stay regular, with rational exceptional fibre over the residue field: $S^{'}$ is regular of pure dimension two, $E$ is an effective Cartier divisor isomorphic to $P_{K}^{1}$ , and $O_{E} (E) ≅ O_{P_{K}^{1}} (- 1)$ .

[F3]

Blowups of finite type ideals are locally H-projective, and proper and Strict transform of a closed subscheme: The blowup is proper over $S$ , hence proper over $k$ ; $C$ and $C^{'}$ are closed subschemes of $S$ and $S^{'}$ respectively, hence proper over $k$ ; $C^{'}$ is reduced because $C$ is reduced; and properness makes all these schemes of finite type over $k$ (Proper morphisms). Their affine coordinate rings are Noetherian because $k$ is Noetherian (Every algebra of finite type over a Noetherian ring is a Noetherian ring); thus their structure sheaves, and all finite locally free sheaves on them, are coherent (Coherent module sheaves).

[F4]

Pushforward and vanishing for point blowups on a surface: $π_{*} O_{S^{'}} = O_{S}$ and $R^{q} π_{*} O_{S^{'}} = 0$ for all $q > 0$ .

[F5]

Projection formula for invertible twists and Cohomology comparison when higher direct images vanish: For an invertible sheaf $L$ on $S$ the projection formula identifies $R^{q} π_{*} (π^{*} L) ≅ R^{q} π_{*} O_{S^{'}} \otimes L$ , so by [F4] the sheaf $π^{*} L$ has vanishing higher direct images and $π_{*} (π^{*} L) = L$ ; the vanishing-direct-image comparison gives $H^{q} (S^{'}, π^{*} L) ≅ H^{q} (S, L)$ for all $q$ , whence $χ_{k} (S^{'}, π^{*} L) = χ_{k} (S, L)$ since $S, S^{'}$ are proper over $k$ and the sheaves are coherent.

[F6]

Twisting the exact sequence of an effective Cartier divisor and Effective Cartier divisors give a short exact sequence: For an effective Cartier divisor $D$ on a scheme $X$ with closed immersion $i ⁣ : D \to X$ and an invertible sheaf $L$ there is a short exact sequence $0 ⟶ L (- D) ⟶ L ⟶ i_{*} (L ∣_{D}) ⟶ 0,$ where $L (- D) = L \otimes O_{X} (- D)$ is again invertible; for $L = O_{X}$ this is the standard sequence $0 \to O_{X} (- D) \to O_{X} \to i_{*} O_{D} \to 0$ .

[F7]

Euler characteristic is additive in short exact sequences: On a scheme proper over $k$ , the Euler characteristic of coherent sheaves is additive in short exact sequences.

[F8]

Closed immersion preserves cohomology and coherent pushforward: For a closed immersion $i ⁣ : Z \to X$ and a quasi-coherent $O_{Z}$ -module $F$ there are isomorphisms $H^{q} (Z, F) ≅ H^{q} (X, i_{*} F)$ for all $q \geq 0$ , and $i_{*} F$ is coherent when $X$ is locally Noetherian and $F$ is coherent.

[F9]

Euler characteristic of line bundles on a projective line over a finite field extension: For an invertible sheaf $M$ of degree $d$ over $K$ on $E ≅ P_{K}^{1}$ one has $χ_{k} (E, M) = r (1 + d)$ ; in particular a line bundle of degree $- j$ has $k$ -Euler characteristic $r (1 - j)$ .

[F10]

Normalization defect delta of a reduced curve, The normalization defect is an Euler characteristic and a weighted sum of local lengths, Normalization is unchanged under finite birational maps of reduced curves, A proper quasi-finite morphism is finite and Morphisms from a proper scheme to a separated one are proper: For a reduced proper curve $X$ over $k$ with normalization $\tilde{X}$ one has $δ_{k} (X) = χ_{k} (O_{\tilde{X}}) - χ_{k} (O_{X})$ ; a finite birational morphism of reduced curves induces an isomorphism of normalizations; and a morphism from a proper $k$ -scheme to a separated $k$ -scheme is proper, while a proper quasi-finite morphism is finite.

[F11]

The blowup is an isomorphism off the center: $π$ restricts to an isomorphism over $S ∖ {p}$ .

Proof

1.1F1F2F3

The curve $C$ is an effective Cartier divisor on the regular surface $S$ , so [F1] gives the divisor identity $π^{*} C = C^{'} + m E$ and shows that $C^{'}$ is again an effective Cartier divisor, meeting $E$ in a finite $0$ -cycle of degree $m$ over $K$ . By [F2] the exceptional curve is $E ≅ P_{K}^{1}$ with $O_{E} (E) ≅ O_{P_{K}^{1}} (- 1)$ , and by [F3] the schemes $S, S^{'}, C, C^{'}$ are proper and locally Noetherian of finite type over $k$ with coherent structure sheaves; $r = [K : k]$ is finite because $S$ is of finite type over $k$ .

2.1F6F7F8step 1.1

For $j = 0, 1, \dots, m$ put $L_{j} : = O_{S^{'}} (- π^{*} C + j E) : = O_{S^{'}} (- π^{*} C) \otimes O_{S^{'}} (E)^{\otimes j}$ , an invertible sheaf with $L_{m} = O_{S^{'}} (- C^{'})$ by the divisor identity of step 1.1. For $j \geq 1$ apply the twisted sequence of [F6] on $S^{'}$ to the effective Cartier divisor $E$ and the invertible sheaf $L_{j}$ : since $L_{j} (- E) = L_{j - 1}$ and writing $i ⁣ : E \to S^{'}$ for the closed immersion and $Q_{j} : = L_{j} ∣_{E}$ , one gets the short exact sequence $0 \to L_{j - 1} \to L_{j} \to i_{*} Q_{j} \to 0$ , whose three terms are coherent because $S^{'}$ is locally Noetherian. Additivity [F7] gives $χ_{k} (S^{'}, L_{j}) - χ_{k} (S^{'}, L_{j - 1}) = χ_{k} (S^{'}, i_{*} Q_{j})$ , and [F8] identifies the last term with $χ_{k} (E, Q_{j})$ .

2.2F6F7F8step 1.1

Apply the untwisted sequence of [F6] to the effective Cartier divisor $C$ on $S$ and to the effective Cartier divisor $C^{'}$ on $S^{'}$ , whose structure sheaves are coherent by step 1.1: $0 \to O_{S} (- C) \to O_{S} \to i_{*} O_{C} \to 0$ and $0 \to O_{S^{'}} (- C^{'}) \to O_{S^{'}} \to i_{*}^{'} O_{C^{'}} \to 0$ . Additivity [F7] and the identification $H^{q} (C, O_{C}) ≅ H^{q} (S, i_{*} O_{C})$ , respectively $H^{q} (C^{'}, O_{C^{'}}) ≅ H^{q} (S^{'}, i_{*}^{'} O_{C^{'}})$ , from [F8], give $χ_{k} (S, O_{S}) = χ_{k} (S, O_{S} (- C)) + χ_{k} (O_{C})$ and $χ_{k} (S^{'}, O_{S^{'}}) = χ_{k} (S^{'}, O_{S^{'}} (- C^{'})) + χ_{k} (O_{C^{'}})$ .

3.1F2F9algebrastep 2.1

The restriction $Q_{j} = L_{j} ∣_{E}$ is computed as follows. First, $O_{S^{'}} (j E) ∣_{E} = O_{E} (E)^{\otimes j}$ is isomorphic to $O_{P_{K}^{1}} (- j)$ by [F2]. Second, $(π^{*} O_{S} (- C)) ∣_{E} ≅ O_{E}$ : the morphism $π ∣_{E} ⁣ : E \to S$ factors as $E \to Spec K \to S$ , and the restriction of the invertible sheaf $O_{S} (- C)$ to the residue point is a free rank-one $K$ -module, whose pullback along $E \to Spec K$ is free of rank one. Hence $Q_{j} ≅ O_{P_{K}^{1}} (- j)$ , a line bundle of degree $- j$ over $K$ , and [F9] gives $χ_{k} (E, Q_{j}) = r (1 - j)$ .

4.1step 2.1step 3.1algebra

Summing the identities of step 2.1 over $j = 1, \dots, m$ and substituting step 3.1 gives $χ_{k} (S^{'}, L_{m}) - χ_{k} (S^{'}, L_{0}) = \sum_{j = 1}^{m} r (1 - j) = - r (\binom{m}{2})$ , that is, $χ_{k} (S^{'}, O_{S^{'}} (- C^{'})) = χ_{k} (S^{'}, π^{*} O_{S} (- C)) - r (\binom{m}{2})$ because $L_{0} = π^{*} O_{S} (- C)$ .

5.1F4F5step 4.1

By [F5] applied to the invertible sheaf $O_{S} (- C)$ the Euler characteristics agree: $χ_{k} (S^{'}, π^{*} O_{S} (- C)) = χ_{k} (S, O_{S} (- C))$ , and likewise $χ_{k} (S^{'}, O_{S^{'}}) = χ_{k} (S^{'}, π^{*} O_{S}) = χ_{k} (S, O_{S})$ . Combining with step 4.1, $χ_{k} (S^{'}, O_{S^{'}} (- C^{'})) = χ_{k} (S, O_{S} (- C)) - r (\binom{m}{2})$ .

6.1step 1.1step 2.2step 5.1

Subtracting the two identities of step 2.2 and substituting step 5.1 yields $χ_{k} (O_{C^{'}}) - χ_{k} (O_{C}) = [χ_{k} (S^{'}, O_{S^{'}}) - χ_{k} (S^{'}, O_{S^{'}} (- C^{'}))] - [χ_{k} (S, O_{S}) - χ_{k} (S, O_{S} (- C))] = χ_{k} (S^{'}, O_{S^{'}}) - χ_{k} (S, O_{S}) + r (\binom{m}{2}) = r (\binom{m}{2})$ , since $χ_{k} (S^{'}, O_{S^{'}}) = χ_{k} (S, O_{S})$ . This proves $χ_{k} (O_{C^{'}}) = χ_{k} (O_{C}) + r (\binom{m}{2})$ and, together with the divisor identity $π^{*} C = C^{'} + m E$ of step 1.1, the first assertions.

7.1F10F11step 1.1step 6.1

The morphism $C^{'} \to C$ induced by $π$ is proper: $C^{'}$ is a closed subscheme of the proper $k$ -scheme $S^{'}$ , hence proper over $k$ , and $C$ is separated over $k$ as a closed subscheme of the separated scheme $S$ ; by [F10] a morphism from a proper $k$ -scheme to a separated one is proper. It is quasi-finite: by [F11] it is an isomorphism over $C ∖ {p}$ , and over $p$ its fibre is the finite $0$ -cycle $C^{'} \cap E \subseteq E$ of step 1.1. It is birational: it is an isomorphism over the dense open $C ∖ {p}$ , $p$ being a closed point of the reduced curve $C$ . Hence $C^{'} \to C$ is finite by [F10], and [F10] identifies the normalizations of $C$ and $C^{'}$ over $C$ .

8.1F10step 6.1step 7.1∎

By step 1.1 both $C$ and $C^{'}$ are reduced proper curves over $k$ , so [F10] computes their defects on the common normalization $\tilde{C}$ : $δ_{k} (C) = χ_{k} (O_{\tilde{C}}) - χ_{k} (O_{C})$ and $δ_{k} (C^{'}) = χ_{k} (O_{\tilde{C}}) - χ_{k} (O_{C^{'}})$ . Subtracting and using step 6.1, $δ_{k} (C^{'}) - δ_{k} (C) = χ_{k} (O_{C}) - χ_{k} (O_{C^{'}}) = - r (\binom{m}{2})$ . Thus $π^{*} C = C^{'} + m E$ , $χ_{k} (O_{C^{'}}) = χ_{k} (O_{C}) + r (\binom{m}{2})$ and $δ_{k} (C^{'}) = δ_{k} (C) - r (\binom{m}{2})$ .

Remarks

The factor $r = [κ (p) : k]$ records the residue degree of the blown-up point: the successive quotients of the filtration are line bundles of degree $- j$ on a projective line over $κ (p)$ , and their $k$ -Euler characteristic is measured through $r$ .
Summing the identity over the singular points of a reduced curve on a regular surface gives the strictly decreasing invariant that drives the resolution algorithm; for $m = 1$ the correction vanishes, matching the fact that blowing up a regular point of a reduced curve does not change $χ_{k} (O_{C})$ .

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

Contact order of two regular components at a point

Definition

Assume the Axiom of Choice (The Axiom of Choice), inherited from the regular-local prerequisites. Let $S$ be a regular Noetherian scheme of pure dimension two over a field $k$ (embedding dimension and regular local ring, Left and right Noetherian rings, Chain dimension and the empty-space convention). Let $Y, Z \subseteq S$ be reduced closed subschemes of pure dimension one, with no common irreducible component, and suppose every branch of either curve through the chosen closed point $p$ is regular at $p$ . Write $A = O_{S, p}$ and $I_{Y}, I_{Z} \subseteq A$ for their germ ideals. If $p$ misses either curve, set $n_{p} (Y, Z) = 0$ . Otherwise the contact order is $n_{p} (Y, Z) = {length}_{A / I_{Y}} (A / (I_{Y} + I_{Z})) .$ Length means composition-series length (Composition series and length of a module).

Here the required local dimension follows from the geometry, rather than from the global dimension alone. A closed point of a pure one-dimensional Noetherian curve has local dimension one: a zero-dimensional local ring would make it the generic point of a zero-dimensional component, since the point is closed. At a closed point on $Y$ , the ambient regular local ring cannot have dimension zero. If it had dimension one, it would be a DVR (one dimensional regular local rings are dvrs), and a branch prime with one-dimensional quotient would be zero. The closed curve would then contain the generic point, hence the whole two-dimensional ambient component, contradicting its pure dimension one. Thus $\dim A = 2$ at every actual contact point. Generic local rings of $S$ need not have dimension two.

The displayed length is finite. The minimal primes of $A / I_{Y}$ are the branch primes of $Y$ through $p$ . No common component means that $I_{Z}$ is contained in none of these primes: an inclusion would make a one-dimensional branch of $Y$ a component of $Z$ . Thus the quotient has no generic point of a curve branch in its support, and its support is only the closed point. A finite module over a Noetherian local ring with this support has finite length. This uses noncontainment in every branch prime, not the weaker assertion that the image ideal is merely nonzero.

Local equations. A regular branch prime $P \subset A$ is principal. Indeed, its regular quotient $A / P$ has cotangent dimension one, so choose $f \in P$ with nonzero class in $m / m^{2}$ . Then $A / (f)$ is regular of dimension one (regular local quotient by parameter is regular) and hence a DVR. The prime $P / (f)$ must be zero since its quotient still has dimension one. Therefore $P = (f)$ , with $f$ a prime element of the regular local domain $A$ (regular local rings are domains and cohen macaulay). The reduced curve ideal is the intersection of its finitely many distinct branch primes, so it is their product: if an element divisible by a product of some distinct prime elements is also divisible by a new prime element, primality forces divisibility of its remaining factor by that new element. Induction gives the intersection/product equality. Consequently $I_{Y} = (y)$ and $I_{Z} = (z)$ , with $y, z$ products of the respective branch equations, and $n_{p} (Y, Z) = {length}_{A / (y)} (A / (y, z)) .$ These nonzero equations are regular sections; they define effective Cartier data. Changing an equation by a unit does not change the quotient or its length (Effective cartier divisor, Cartier divisor, Cartier divisor local equation equivalence).

Symmetry and transversality. The length equals the length of $A / (I_{Y} + I_{Z})$ as an $A$ -module, and similarly as an $A / I_{Z}$ -module, since all composition factors are the same residue field. Thus contact is symmetric. It equals one precisely when $(y, z) = m$ : a nonzero local quotient has length one precisely when it is the residue field. In that case the classes of $y, z$ form a basis of the two-dimensional cotangent space. Their regular parameter quotients are one-dimensional regular local rings, and their tangent lines are distinct. Conversely, if both curve germs are regular and their tangent lines are distinct, their equations have independent cotangent classes and generate $m$ by Nakayama (Assuming the Axiom of Choice, generators modulo an ideal in the Jacobson radical lift to generators). Hence their contact is one. This is exactly transversal meeting at $p$ . If either curve has at least two branches through $p$ , its product equation lies in $m^{2}$ ; its cotangent class cannot be part of a parameter basis, so the positive contact length is at least two, even if individual pairs of branches have distinct tangents.

The total contact order is $n (Y, Z) = \sum_{p \in Y \cap Z} n_{p} (Y, Z) .$ The intersection is a zero-dimensional closed subscheme of a Noetherian scheme because there is no common component, so it has finitely many closed points. The sum is therefore a finite nonnegative integer. This definition includes all regular finite-type surface cases and uses no perfectness or rationality assumption on the residue fields.

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A point blowup lowers pairwise contact order by one and separates transverse branches

Statement

Assume the Axiom of Choice, inherited from the blowup construction (The Axiom of Choice). Let $S$ be a regular surface over a field $k$ (Contact order of two regular components at a point), let $p$ be a closed point, and let $Y, Z \subseteq S$ be distinct curves that are regular at $p$ and pass through $p$ , with contact order $n = n_{p} (Y, Z) \geq 1$ (Contact order of two regular components at a point). Let $π ⁣ : S^{'} \to S$ be the blowup of $p$ , with exceptional curve $E$ , and let $Y^{'}, Z^{'}$ be the strict transforms of $Y, Z$ . Then:

if $n = 1$ , then $Y^{'}$ and $Z^{'}$ meet $E$ at distinct points, so they are disjoint in a neighbourhood of $E$ ;
if $n > 1$ , then $Y^{'}$ and $Z^{'}$ meet at the point of $E$ corresponding to their common tangent direction, the contact order of $Y^{'}$ and $Z^{'}$ there is $n - 1$ , and every intersection of a strict transform with $E$ has order one.

Facts & Assumptions

Given: A regular surface $S$ over $k$ , a closed point $p$ with local ring $A = O_{S, p}$ , a regular system of parameters $x, y \in A$ , local equations $y$ of $Y$ and $z$ of $Z$ at $p$ , the blowup $π ⁣ : S^{'} \to S$ of $p$ with exceptional curve $E$ and strict transforms $Y^{'}, Z^{'}$ , and the contact order $n = n_{p} (Y, Z)$ of Contact order of two regular components at a point.

[F1]

Contact order of two regular components at a point: $n = {length}_{A / (y)} ((A / (y)) / (z))$ and $O_{Y, p} = A / (y)$ is a one-dimensional reduced Noetherian local ring; every component of $Y$ and of $Z$ through $p$ is regular at $p$ .

[F2]

Affine blowup standard charts and overlaps, Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains, Flat base change for blowups, and failure without flatness, The exceptional divisor is the projectivized normal cone and associated graded ring of a regular local ring: Localizing the base at $p$ gives the charts $A [(x, y) / x]$ and $A [(x, y) / y]$ , with inverse ratio overlap. The exceptional curve is $Proj {gr}_{m} A$ , hence $P_{κ (p)}^{1}$ after choosing parameters, since $\dim A = 2$ at the contact point. The quotient presentations follow from the regular-sequence torsion calculation below.

[F3]

Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: The chart ring is the affine blowup algebra $A [I / x]$ with $I A [I / x] = x A [I / x]$ and $x$ a nonzerodivisor, so on the first chart the inverse image ideal of the centre is $(x)$ .

[F4]

Total transform equals strict transform plus multiplicity times the exceptional divisor: For a reduced curve $C \subseteq S$ through $p$ whose local equation has multiplicity $m$ at $p$ , one has $π^{*} C = C^{'} + m E$ and $C^{'}$ meets $E$ in the $0$ -cycle of degree $m$ cut out by the degree- $m$ leading form of a local equation of $C$ at $p$ .

[F5]

Contact order of two regular components at a point and regular local quotient by parameter is regular: In a two-dimensional regular local ring, a regular curve germ has a prime equation with nonzero cotangent class, as shown in the definition's local-equation argument. Conversely, quotienting by an equation with nonzero cotangent class gives a regular one-dimensional local ring. Thus regularity of such a curve germ is equivalent to its equation having order one.

[F6]

Effective cartier divisor and Cartier divisor: The exceptional curve $E$ is an effective Cartier divisor on $S^{'}$ , so the total transform $π^{*} C$ and the expression $C^{'} + m E$ of [F4] are well defined as divisors.

[F7]

regular local quotient by parameter is regular, one dimensional regular local rings are dvrs and regular local rings are domains and cohen macaulay: A quotient of a regular local ring by an element with nonzero cotangent class is regular of dimension one less; in dimension one it is a discrete valuation domain.

[F8]

dimension at most embedding dimension: The dimension of a nonzero Noetherian local ring is at most its embedding dimension.

Proof

1.1F1F5F7

The regular curve germ $Y$ has prime ideal $P$ with $A / P$ regular of dimension one. Its cotangent space has dimension one, so the kernel of $m / m^{2} \to m / (P + m^{2})$ is nonzero. Choose $y \in P$ with nonzero cotangent class and extend it to a parameter system $x, y$ . The ring $A / (y)$ is a one-dimensional regular local domain, hence a DVR. The prime $P / (y)$ must be zero, since its quotient has dimension one, whereas the only nonzero prime in a DVR is maximal and has zero-dimensional quotient. Thus $P = (y)$ . The same argument gives a principal equation $z$ of $Z$ . In the DVR $A / (y)$ , $\overset{ˉ}{x}$ is a uniformizer and the contact order is $n = {ord}_{\overset{ˉ}{x}} \overset{ˉ}{z}$ .

2.1F1F5step 1.1

Write $ℓ_{Y}$ and $ℓ_{Z}$ for the leading forms of $y$ and $z$ in the symmetric algebra of $m / m^{2}$ , so $ℓ_{Y} = y$ and $ℓ_{Z} = α x + β y$ with $(α, β) \neq (0, 0)$ by the regularity of $Z$ at $p$ in [F5]. Since $\overset{ˉ}{z} = z (x, 0) = α x + O (x^{2})$ in the DVR $A / (y)$ with uniformizer $x$ , the order is $n = 1$ exactly when $α \neq 0$ : the tangent directions of $Y$ and $Z$ at $p$ , cut out by $ℓ_{Y}$ and $ℓ_{Z}$ , agree exactly when $ℓ_{Z}$ is a nonzero multiple of $y$ , that is exactly when $α = 0$ and $β \neq 0$ ; hence $n = 1$ if and only if the tangent directions differ, and in that case $β$ may be zero or not, while for $n > 1$ the two curves have the common tangent direction cut out by $y$ .

3.1F2F3F4step 2.1F6

The first chart is $A [T] / (x T - y)$ : if $x g = (x T - y) h$ , reduction modulo $x$ and regularity of $y$ modulo $x$ give $h = x h_{1}$ , then cancellation gives $g = (x T - y) h_{1}$ . Thus no $x$ -power torsion remains in the incidence quotient. Work in the first chart $Spec A [T] / (x T - y)$ , $T = y / x$ , so $y = x T$ and $E = V (x)$ ; by [F2] and [F3] this chart contains the point of $E$ corresponding to the tangent direction cut out by $y$ , namely $T = 0$ , and the other chart covers the remaining points, so the two charts together see all of $E$ . By [F4] applied to $Y$ and $Z$ , whose local equations have multiplicity one at $p$ , one has $π^{*} Y = Y^{'} + E$ and $π^{*} Z = Z^{'} + E$ as identities of effective Cartier divisors, well defined by [F6]; and $Y^{'}$ meets $E$ in the reduced point cut out by $ℓ_{Y}$ , $Z^{'}$ in the reduced point cut out by $ℓ_{Z}$ ; explicitly in this chart the total transform of $Z$ is $V (z)$ with $z = x h$ , $h = z / x \in A [y / x]$ , and $Z^{'} = V (h)$ .

4.1F4step 2.1step 3.1

If $n = 1$ , then $α \neq 0$ by step 2.1, so $h (0, 0) = α \neq 0$ : the strict transform $Z^{'}$ does not pass through the point $Y^{'} \cap E = {T = 0}$ , and its intersection with $E$ is cut out by $ℓ_{Z}$ at the point of $E$ corresponding to the tangent direction of $Z$ , which differs from that of $Y$ by step 2.1. On the open complement of $Z^{'} \cap Y^{'}$ (a closed subset missing $E$ ) the curves $Y^{'}$ and $Z^{'}$ are disjoint: they meet $E$ at distinct points and are therefore disjoint near $E$ .

4.2F2F4F5F7F8step 1.1step 2.1step 3.1

If $n > 1$ , then $α = 0$ and $β \neq 0$ , so $ℓ_{Z} = β y$ : both $Y^{'}$ and $Z^{'}$ meet $E$ at the single point $q$ corresponding to the common tangent direction cut out by $y$ , namely $T = 0$ ; and $h (x, 0) = \overset{ˉ}{z} / x = x^{n - 1} u (x)$ for a unit $u$ of the DVR $A / (y)$ , because ${ord}_{x} \overset{ˉ}{z} = n$ by step 1.1. The ambient local ring at $q$ is regular: the domain chart $B = A [T] / (x T - y)$ has $B / x B = κ (p) [T]$ , and $O_{S^{'}, q} = B_{(x, T)}$ has maximal ideal generated by $x, T$ . The strict prime chain $(0) ⊊ (x) ⊊ (x, T)$ gives dimension at least two, while [F8] bounds it above by its embedding dimension at most two. Hence it is regular, with $x, T$ a cotangent basis. Since $h m o d x = β T$ is a nonzero linear form, $Z^{'}$ is regular at $q$ by [F5], and $Y^{'} = V (T)$ is regular there; in the DVR $O_{Y^{'}, q} ≅ (A / (y))$ with uniformizer $x$ , the ideal of $Z^{'}$ is generated by $h (x, 0) = x^{n - 1} u (x)$ , so the contact order of $Y^{'}$ and $Z^{'}$ at $q$ is $n - 1$ . Finally each of $Y^{'}, Z^{'}$ meets $E$ in a $0$ -cycle of degree one by [F4], so every intersection of a strict transform with $E$ has order one.

5.1step 4.1step 4.2∎

Steps 4.1 and 4.2 prove the two assertions: for transverse branches ( $n = 1$ ) the strict transforms meet $E$ at distinct points and are disjoint near $E$ , while for $n > 1$ they meet at the common tangent direction with contact order $n - 1$ , every intersection with $E$ having order one.

Remarks

The computation uses only the first chart because the point of $E$ cut out by $y$ is $T = 0$ there; when the common tangent direction is the other coordinate direction the same argument runs in the second chart with $x$ and $y$ interchanged.
The statement is the local input for resolving plane curve singularities by repeated point blowups, where it shows that the contact order of two branches drops by exactly one at each step at which they still share a tangent direction.

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Blowing up a multiple point separates pairwise transverse components

Statement

Assume the Axiom of Choice. Let $S$ be a regular surface over a field $k$ and let $p$ be a closed point through which pass $s \geq 2$ distinct regular curves $Y_{1}, \dots, Y_{s}$ , pairwise meeting transversally at $p$ (contact orders one) and pairwise disjoint away from $p$ . Let $π ⁣ : S^{'} \to S$ be the blowup of $p$ with exceptional curve $E$ . Then the strict transforms $Y_{i}^{'}$ meet $E$ at $s$ distinct points, no three support curves meet at a point of $S^{'}$ (in particular at most two components pass through any point of $E$ ), and the only new intersections are the transverse intersections $Y_{i}^{'} \cap E$ at distinct points. If $s = 2$ the two strict transforms become disjoint.

Facts & Assumptions

Given: A regular surface $S$ over $k$ (a Noetherian scheme of dimension two regular at every point, in the sense of Contact order of two regular components at a point), a closed point $p \in S$ , distinct regular curves $Y_{1}, \dots, Y_{s}$ through $p$ with $s \geq 2$ , pairwise of contact order one at $p$ and pairwise disjoint away from $p$ , and the blowup $π ⁣ : S^{'} \to S$ of $p$ with exceptional curve $E$ .

[A1]

Choice. The Axiom of Choice is assumed, as in the statement; the cited suppliers used below are stated under it (The Axiom of Choice).

[F1]

Contact order of two regular components at a point: For distinct reduced curves $Y, Z$ through a closed point $p$ of a regular surface, the total contact order is the sum of the local lengths $n_{p} (Y, Z)$ and the definition records that $n_{p} (Y, Z) = 1$ if and only if $Y$ and $Z$ meet transversally at $p$ , meaning $p \in Y \cap Z$ , each of $Y$ and $Z$ is regular at $p$ , and their tangent lines are distinct one-dimensional subspaces of the two-dimensional $k (p)$ -vector space $m_{p} / m_{p}^{2}$ ; in that case the local contact order is computed in the local-equation form $n_{p} (Y, Z) = {length}_{O_{Y, p}} (O_{Y, p} / z O_{Y, p})$ for a local equation $z$ of $Z$ .

[F2]

A point blowup lowers pairwise contact order by one and separates transverse branches: Let $Y, Z$ be distinct regular curves through a closed point $p$ of a regular surface with contact order $n \geq 1$ , and let $Y^{'}, Z^{'}$ be their strict transforms under the blowup of $p$ with exceptional curve $E$ . If $n = 1$ , then $Y^{'}$ and $Z^{'}$ meet $E$ at distinct points and are disjoint near $E$ ; if $n > 1$ , the strict transforms meet at the point of $E$ corresponding to their common tangent direction with contact order $n - 1$ ; every intersection of a strict transform with $E$ has order one.

[F3]

Affine blowup standard charts and overlaps, Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains, Flat base change for blowups, and failure without flatness, The exceptional divisor is the projectivized normal cone and associated graded ring of a regular local ring: Localizing the base at $p$ gives the charts $A [(x, y) / x]$ and $A [(x, y) / y]$ , with inverse ratio overlap. The exceptional curve is $Proj {gr}_{m} A$ , hence $P_{κ (p)}^{1}$ after choosing parameters, since $\dim A = 2$ at the contact point. The quotient presentations follow from the regular-sequence torsion calculation below.

[F4]

The blowup is an isomorphism off the center: The restriction of the blowup to the complement of the center is an isomorphism: $π ⁣ : π^{- 1} (S ∖ {p}) \to S ∖ {p}$ is an isomorphism of schemes.

[F6]

Strict transform of a closed subscheme: The strict transform of a closed subscheme is the scheme-theoretic closure of its inverse image minus the exceptional divisor; on a chart where the ideal of $E$ is invertible it is cut out by the saturation of the inverse-image ideal by the ideal of $E$ .

[F7]

dimension at most embedding dimension, regular local rings are domains and cohen macaulay, one dimensional regular local rings are dvrs and regular local quotient by parameter is regular: Regular local rings are domains, local dimension is at most embedding dimension, a regular parameter quotient is regular of dimension one less, and a regular hypersurface equation has multiplicity one.

[F8]

Total transform equals strict transform plus multiplicity times the exceptional divisor: For a reduced curve of multiplicity one at the blown-up point, the strict transform is given by dividing its equation by the exceptional equation; its intersection with $E$ is the divisor of its nonzero linear leading form.

Proof

1.1A1F1given

By [F1], since each pair $Y_{i}, Y_{j}$ with $i \neq j$ has contact order $n_{p} (Y_{i}, Y_{j}) = 1$ , every $Y_{i}$ is regular at $p$ and the tangent lines $T_{i} \subseteq m_{p} / m_{p}^{2}$ are pairwise distinct one-dimensional $k (p)$ -subspaces; moreover $Y_{i} \cap Y_{j} \subseteq {p}$ by the hypothesis that the curves are pairwise disjoint away from $p$ .

2.1F1F3F6F7F8step 1.1

Fix $i$ . Its regular prime quotient $A / P$ has cotangent dimension one, so choose $u \in P$ with nonzero cotangent class. The regular one-dimensional quotient $A / (u)$ is a DVR; its prime $P / (u)$ is zero because $(A / (u)) / (P / (u))$ still has dimension one. Thus $P = (u)$ and $u$ is a principal equation of the curve germ. Regularity makes its initial form a nonzero linear form $a X + b Y$ . Choose regular parameters $x, y$ so that $b \neq 0$ . The $x$ -chart is $B = A [T] / (x T - y)$ : reducing $x g = (x T - y) h$ modulo $x$ forces $h = x h_{1}$ , and cancellation proves the incidence quotient has no $x$ -power torsion. In this ring its strict transform is cut by $w = u / x$ , with $w m o d x = a + b T$ . Thus $Y_{i}^{'} \cap E$ is the single reduced point $T = - a / b$ . The other chart $A [U] / (y U - x)$ has equation $w^{'} = u / y$ with $w^{'} m o d y = a U + b$ ; it gives the same point if $a \neq 0$ , and none if $a = 0$ . This proves there are no other intersections with $E$ . At that point the ambient local ring has maximal ideal $(x, T + a / b)$ and prime chain $(0) ⊊ (x) ⊊ (x, T + a / b)$ ; it has dimension and embedding dimension two, so is regular with this cotangent basis, and $w$ has a nonzero coefficient on $T + a / b$ . Hence $Y_{i}^{'}$ is regular there and its tangent line differs from $E$ 's. Equivalently its quotient by $x$ is the residue field, giving contact length one.

3.1F2step 2.1

Applying [F2] with $n = 1$ to each pair $Y_{i}, Y_{j}$ , $i \neq j$ , the strict transforms meet $E$ at distinct points; with the uniqueness of step 2.1 this says $e_{i} \neq e_{j}$ whenever $i \neq j$ , so $Y_{1}^{'}, \dots, Y_{s}^{'}$ meet $E$ at the $s$ distinct points $e_{1}, \dots, e_{s}$ .

4.1F2F4step 1.1step 3.1

For $i \neq j$ the strict transforms $Y_{i}^{'}$ and $Y_{j}^{'}$ are disjoint: near $E$ this is [F2] with $n = 1$ , and outside $E$ the blowup restricts to an isomorphism of $S^{'} ∖ E$ with $S ∖ {p}$ by [F4], so a common point of $Y_{i}^{'}$ and $Y_{j}^{'}$ outside $E$ would map to a common point of $Y_{i}$ and $Y_{j}$ different from $p$ , which does not exist; hence $Y_{i}^{'} \cap Y_{j}^{'} = \emptyset$ , and in particular the two strict transforms are disjoint when $s = 2$ .

5.1F3F4step 2.1step 4.1

Consequently no three of the support curves $E, Y_{1}^{'}, \dots, Y_{s}^{'}$ meet at a point of $S^{'}$ : each $Y_{i}^{'}$ meets $E$ only in $e_{i}$ , the points $e_{i}$ are distinct, and the $Y_{i}^{'}$ are pairwise disjoint, so a point of $E$ lies on at most one strict transform and a point outside $E$ lies on at most one curve; every $e_{i}$ is a transverse intersection of $E$ with $Y_{i}^{'}$ by step 2.1. The only intersections not present before the blowup are these points $e_{i}$ : the original curves met one another only at $p$ , and each such intersection has been separated, while outside $E$ the blowup creates no new intersections because it is an isomorphism there [F4].

6.1step 3.1step 4.1step 5.1∎

Therefore $Y_{1}^{'}, \dots, Y_{s}^{'}$ meet $E$ at the $s$ distinct points $e_{1}, \dots, e_{s}$ , no three support curves meet at a point of $S^{'}$ , the only new intersections are the transverse intersections $Y_{i}^{'} \cap E = {e_{i}}$ at distinct points, and $Y_{1}^{'}$ and $Y_{2}^{'}$ are disjoint when $s = 2$ ; this proves every clause of the statement.

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Resolution of reduced plane curves by point blowups and the delta recurrence

Statement

Assume the Axiom of Choice. Let $C_{0} \subseteq P_{k}^{2}$ be a reduced projective plane curve over a field $k$ (equivalently, a reduced hypersurface; reducible $C_{0}$ is allowed). Let $S$ be the regular projective surface obtained from $P_{k}^{2}$ by finitely many point blowups at closed points, and let $C \subseteq S$ be the reduced strict transform of $C_{0}$ . Then there is a finite sequence of blowups of closed points of the current regular projective surface after which the following hold: (a) every irreducible component of the resulting strict transform $C^{*}$ is regular; (b) the total support of $C^{*}$ together with the exceptional curves is a regular embedded normal-crossing support: all its components are regular, every intersection of two distinct components is transverse (pairwise contact order at most one at each intersection point), and at most two components pass through any point of the regular ambient surface; (c) at a blowup centered at a closed point $p$ of multiplicity $m$ and residue degree $r = [κ (p) : k]$ , the strict transform $C^{'}$ satisfies $δ_{k} (C^{'}) = δ_{k} (C) - r m (m - 1) / 2$ , where $δ_{k}$ is the normalization defect of the reduced curve (Normalization defect delta of a reduced curve); this identity is the exact statement used for termination, and the multiplicity $m$ is the $m_{p}$ -adic order of a local reduced equation, with $m = 0$ when the center misses the current strict transform. This is regular embedded normal-crossing support over the residual residue fields; it does NOT assert that the components are smooth over an imperfect $k$ , does not produce a relative SNC divisor with components smooth over $k$ , and makes no claim about resolution of singularities in dimension greater than two.

Facts & Assumptions

Given: The Axiom of Choice, a reduced projective plane curve $C_{0} \subseteq P_{k}^{2}$ over $k$ , a regular projective surface $S$ obtained from $P_{k}^{2}$ by finitely many point blowups, and its current reduced strict transform $C \subseteq S$ .

[A1]

Choice. The Axiom of Choice is assumed, as in the statement; the cited suppliers used below are stated under it.

[F1]

Point blowups of regular surfaces stay regular, with rational exceptional fibre over the residue field: Point blowups of a regular surface at closed points are regular of pure dimension two; the exceptional curve is an effective Cartier divisor.

[F2]

Blowups of finite type ideals are locally H-projective, and proper: Every finite-type ideal blowup is proper over its base. For an integral projective base, an ample twist makes the point ideal globally generated (Eventual generation of coherent projective twists); invertible rescaling preserves its relative Proj (Invariance of the blowup under invertible (fractional) rescaling of the ideal), and a finite degree-one generating family embeds it in relative projective space (Relative Proj of a graded quasi-coherent algebra, Closed subschemes of projective space and saturated ideals). Closed immersions remain closed after base change (Closed immersions are affine quotients and survive base change).

[F3]

Total transform equals strict transform plus multiplicity times the exceptional divisor: At a point of multiplicity $m$ , the total transform of a reduced curve is $C^{'} + m E$ with $C^{'}$ the strict transform, and $C^{'}$ is obtained by dividing a local equation by the $m$ -th power of an exceptional equation.

[F4]

Euler characteristic and normalization defect under a point blowup: For a reduced curve $C$ on a regular proper surface with an ample invertible sheaf, a point blowup at a closed point of multiplicity $m$ and residue degree $r$ gives $δ_{k} (C^{'}) = δ_{k} (C) - r (\binom{m}{2})$ .

[F5]

Normalization defect delta of a reduced curve: $δ_{k} (C) = \dim_{k} H^{0} (C, Q_{C})$ is a nonnegative integer, finite for curves of finite type over $k$ .

[F6]

Normalization is unchanged under finite birational maps of reduced curves: A finite birational morphism of reduced curves induces an isomorphism of their normalizations; the strict transform $C^{'} \to C$ is proper and quasi-finite, hence finite by A proper quasi-finite morphism is finite.

[F7]

A point blowup lowers pairwise contact order by one and separates transverse branches: Blowing up a point of contact order $n \geq 1$ between two regular curves: for $n = 1$ the strict transforms meet $E$ at distinct points and are disjoint near $E$ ; for $n > 1$ they meet at the point of $E$ of their common tangent direction with contact order $n - 1$ , and every strict transform meets $E$ with order one.

[F8]

Blowing up a multiple point separates pairwise transverse components: If $s \geq 2$ pairwise transversal regular curves pass through the blown-up point, their strict transforms meet $E$ at $s$ distinct points, no three support curves meet at a point, and the only new intersections are transverse intersections with $E$ .

[F9]

Contact order of two regular components at a point: Contact order is the length of the quotient of the local ring of one curve by the ideal of the other; it is one exactly for a transversal crossing of two regular branches, zero for disjoint germs, and at least two for a positive nontransversal contact.

[F10]

Normalization defect delta of a reduced curve and The normalization defect is an Euler characteristic and a weighted sum of local lengths: The non-normal locus of a reduced finite-type curve is finite. Its complement is exactly the regular locus, by the normalization's regularity and its being an isomorphism there. For two distinct integral curve components their proper closed intersection is finite by Proper closed subsets of a curve are finite.

[F11]

regular local quotient by parameter is regular: In a regular local ring an equation with nonzero cotangent class has regular quotient of dimension one less. Conversely, for a hypersurface in a two-dimensional regular local ring, a regular one-dimensional quotient has cotangent dimension one, so its equation has nonzero cotangent class.

Proof

1.1F1F2F3given

The ambient surface $S$ is regular of pure dimension two and projective over $k$ by hypothesis and [F1], [F2]; each point blowup is regular and proper by [F1, F2]. It is also projective over $k$ : the current surface is integral, since it is obtained from the integral plane by point blowups (Blowing up a nonzero ideal on an integral scheme is birational); choose an embedding of it into $P_{k}^{n}$ and twist its coherent point ideal by a power $L$ of the hyperplane bundle. Finitely many global generators of $I \otimes L$ surject $O [z_{0}, \dots, z_{N}]$ onto $⨁ I^{d} \otimes L^{d}$ , giving a closed embedding of the blowup into $P_{S}^{N}$ . This is closed in $P_{k}^{N} \times P_{k}^{n}$ ; the Segre map embeds the product as a closed subscheme of projective space. Indeed on each open $z_{i j} \neq 0$ the rank-one minor equations solve $z_{a b} / z_{i j} = (z_{a j} / z_{i j}) (z_{i b} / z_{i j})$ , exactly the product affine chart, and these chart identifications glue. Thus iteration preserves all the required hypotheses. Since $C$ is realized as the reduced strict transform on $S$ , [F3] applies at every center: the strict transform is obtained by dividing a local reduced equation by the appropriate power of an exceptional equation, and its scheme-theoretic support is the curve we blow up further.

2.1F3F4F5F10F11step 1.1

Termination of the regularization stage. Let $p$ be a closed point of the current surface at which the strict transform $\tilde{C}$ is not regular, and let $m$ be the order of its local reduced equation. The quotient criterion of [F11] shows that order one is equivalent to regularity of this curve germ; hence $m \geq 2$ ; the residue degree $r = [κ (p) : k]$ is at least one. Blowing up $p$ yields, by [F4] applied to the reduced curve $\tilde{C}$ on the regular proper surface with the ample invertible sheaf of [F2], $δ_{k} ({\tilde{C}}^{'}) = δ_{k} (\tilde{C}) - r (\binom{m}{2})$ , a strict decrease because $m \geq 2$ ; by [F5] the defect is a nonnegative integer, so only finitely many such blowups at singular points are possible along any branch of the construction. Each blowup at a singular point is legitimate (step 1.1) and keeps every component reduced by [F3]; regularizing the finitely many singular points of the current curve, and iterating the strictly decreasing invariant, terminates after finitely many blowups with a strict transform whose components are all regular, which is clause (a).

3.1F7F9F10step 2.1

Crossing stage reduction. Every existing exceptional component remains regular under subsequent point blowups: at a point on a regular curve, choose parameters with its equation $y$ ; its strict transform is $T = 0$ in $A [T] / (x T - y)$ , with quotient $A / (y)$ , and meets the new exceptional curve transversally. Thus after stage 2, consider the entire reduced support consisting of the regularized curve and all strict transforms of old exceptional curves. These components are regular and finite in number. The number of intersection points of this entire support is finite by [F10]. If there are no pairwise intersections, the crossing stage is finished and there are no multiple points to treat. Otherwise let $N \geq 1$ be the maximum contact order among intersecting pairs of distinct components, computed as in [F9]. While $N > 1$ , let $R_{N}$ be the finite set of points at which some pair has contact order exactly $N$ , and blow up every point of $R_{N}$ : by F7 a pair of contact order $N > 1$ at such a point is replaced by a pair of contact order $N - 1$ at the point of $E$ of their common tangent direction, and every strict transform meets $E$ transversally (order one); pairs of smaller contact order and the newly created intersections with $E$ have order at most $N - 1$ or one. After each finite round, stop if the contact set is empty; otherwise its positive maximum strictly decreases. Since this maximum is a positive integer and each round is finite by [F10], after finitely many rounds either there are no intersections or their maximum is one. In both cases every remaining pairwise intersection is transverse.

4.1F3F8F9step 3.1

Multiple points. Once every pairwise contact has order at most one, blow up each point through which $s \geq 3$ regular components pass. Locally at each such point, [F8] applies after shrinking away from all other pairwise intersections, and its strict transforms meet the new exceptional curve in $s$ distinct points with no triple intersection over this center. Elsewhere the old support is unchanged, and the only new intersections are the transverse intersections with $E$ ; hence the number of points where at least three components meet strictly decreases, no new such point is created, and the process terminates after finitely many blowups. The result is a finite sequence (steps 2.1-4.1) after which all components are regular, all pairwise intersections are transverse, and at most two components pass through any point of the ambient regular surface; together with the effective Cartier property of the components and of $E$ from [F1] and [F3], this is the regular embedded normal-crossing support of clause (b).

5.1F3F4F6step 4.1∎

Clause (c) is the invariant used in steps 2.1-4.1, stated separately: at a center $p$ of multiplicity $m$ and residue degree $r$ , when $p$ lies on the curve, the strict transform satisfies $δ_{k} (C^{'}) = δ_{k} (C) - r m (m - 1) / 2$ by [F4]. When $p$ misses it, the blowup restricts to the identity on the curve by The blowup is an isomorphism off the center, so its defect is unchanged and the same formula holds with $m = 0$ . For a center on the curve, the total-transform identity $π^{*} C = C^{'} + m E$ is [F3]. The identity is meaningful because the strict transform ${\tilde{C}}^{'} \to \tilde{C}$ is proper and quasi-finite, hence finite, so [F6] identifies the normalizations of the two curves and the defect is computed on the same normal model; the multiplicity is the $m_{p}$ -adic order of a local reduced equation by [F3]. The sequence constructed in steps 2.1-4.1 is finite and consists of point blowups of regular projective surfaces, and no smoothness of the components over an imperfect field and no statement in dimension greater than two is asserted.

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

Invariance of the blowup under invertible (fractional) rescaling of the ideal

Definition

Assume the Axiom of Choice. Let $X$ be integral, let $I \subseteq O_{X}$ be a quasi-coherent ideal of finite type, and let $J \subseteq K_{X}$ be an invertible fractional ideal. Its product $F = J I$ is a subsheaf of $K_{X}$ ; it need not be contained in $O_{X}$ . Define its fractional Rees algebra and blowup by $R (F) = ⨁_{n \geq 0} F^{n}, F^{0} = O_{X}, {Bl}_{F} X = {Proj}_{X} R (F) .$ Here $F^{n} ≅ J^{\otimes n} \otimes I^{n}$ , with multiplication induced inside $K_{X}$ , so the algebra is quasi-coherent and generated in degree one. When $F \subseteq O_{X}$ , this is the ordinary ideal blowup of Blowup of a scheme along an ideal sheaf.

On a nonempty affine open $U = Spec A$ trivializing $J$ , choose $g \in Frac (A)^{\times}$ with $J ∣_{U} = g A$ . Multiplication by $g^{n}$ is an $A$ -module isomorphism $I^{n} \to (g I)^{n}$ for every $n$ , with inverse division by $g^{n}$ . These maps respect multiplication and give a graded algebra isomorphism $R (I) \to R (g I)$ . The inverse of its contravariantly induced Proj map defines $ρ_{I, J} ∣_{U} : {Bl}_{I} U ⟶ {Bl}_{J I} U .$ No assertion that $g \in A$ is needed.

Replacing $g$ by $u g$ , $u \in A^{\times}$ , changes the degree- $n$ map by $u^{n}$ . This automorphism induces the identity on Proj: on any homogeneous localization, numerator and denominator of a degree-zero fraction acquire the same power of $u$ , which cancels. Thus the local maps agree on overlaps and glue to a canonical isomorphism of $X$ -schemes $ρ_{I, J} : {Bl}_{I} X \tilde{\to} {Bl}_{J I} X .$ Division by $g^{n}$ gives its inverse, including when the original ideal is zero and both blowups are empty. This construction uses relative Proj and is compatible with restriction to opens.

Remarks

An invertible sheaf $L$ on an integral scheme can be realized as an invertible fractional ideal by choosing a nonzero basis of its generic fiber: local sections inject into that fiber, since locally $L$ is free and the coordinate rings are domains. Consequently the algebra $⨁_{n \geq 0} I^{n} \otimes L^{\otimes n}$ has the same relative Proj as $R (I)$ . In particular one may use an ample twist that makes $I \otimes L$ globally generated, without treating that sheaf as an ordinary ideal. For $J = (1 / x)$ and $I = O$ on an affine domain containing a nonunit $x$ , the product is fractional, illustrating why the distinction is necessary.

The ordinary effective Cartier rescaling also works without integrality of $X$ . For any scheme $X$ , a quasi-coherent ideal $I$ and an effective Cartier divisor with ideal $J$ (Effective cartier divisor), use here the Rees Proj ${Proj}_{X} (⨁_{n \geq 0} I^{n})$ even if $I$ is not of finite type: ideal powers commute with affine localization, so this is a quasi-coherent graded algebra to which Relative Proj of a graded quasi-coherent algebra applies. Write $J = g O$ locally, where $g$ is a nonzerodivisor. Multiplication by $g^{n}$ is an isomorphism $I^{n} \to g^{n} I^{n}$ in every degree, with inverse on its image, so it gives a graded Rees algebra isomorphism and a local blowup isomorphism. On overlaps $g$ changes by a unit; the same degree-zero cancellation proves that these isomorphisms glue canonically to ${Bl}_{I} X ≅ {Bl}_{J I} X$ . Thus multiplying the center ideal by an effective Cartier ideal leaves the blowup scheme canonically unchanged on arbitrary $X$ , including $I = 0$ . This assertion concerns the blowup object; the center and the open complement used to define a strict transform may change.

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Blowing up I and I^d agree

Statement

Assume the Axiom of Choice, inherited from the relative Proj construction (The Axiom of Choice). Let $X$ be a scheme, let $I$ be a quasi-coherent ideal sheaf of finite type on $X$ (Quasi-coherent ideal sheaves) and let $d \geq 1$ . Then there is a canonical isomorphism of $X$ -schemes ${Bl}_{I^{d}} X ⟶ {Bl}_{I} X;$ more precisely $R (I^{d})$ is the Veronese regrading $R (I)^{(d)}$ of the Rees algebra sheaf, the degree- $n$ piece of $R (I^{d})$ being $I^{d n}$ , and the canonical identification $Proj R (I) = Proj R (I)^{(d)}$ of Proj is invariant under Veronese regrading glues over $X$ .

Facts & Assumptions

Given: A scheme $X$ , a quasi-coherent ideal sheaf $I$ of finite type, its powers $I^{n}$ , the Rees algebra sheaves $R (I) = ⨁_{n \geq 0} I^{n}$ and $R (I^{d}) = ⨁_{n \geq 0} I^{d n}$ (Rees algebra sheaf of a finite type ideal), and the blowups ${Bl}_{I} X = {Proj}_{X} R (I)$ and ${Bl}_{I^{d}} X = {Proj}_{X} R (I^{d})$ of Blowup of a scheme along an ideal sheaf.

[F1]

Rees algebra sheaf of a finite type ideal: The Rees algebra sheaf of a quasi-coherent ideal sheaf is the graded $O_{X}$ -algebra $⨁_{n \geq 0} I^{n}$ with degree- $n$ piece $I^{n}$ , and $R (I)^{(d)} : = ⨁_{n \geq 0} R (I)_{d n} = ⨁_{n \geq 0} I^{d n}$ is its Veronese regrading; the graded pieces are quasi-coherent.

[F2]

Proj is invariant under Veronese regrading: For a commutative nonnegatively graded ring $S$ and $d \geq 1$ there is a canonical isomorphism $Proj S ≅ Proj S^{(d)}$ mapping the chart $D_{+} (f)$ of $Proj S$ , for homogeneous $f \in S_{+}$ of positive degree, to the chart $D_{+} (f^{d})$ of $Proj S^{(d)}$ with the same coordinate ring $S_{(f)} = S_{(f^{d})}^{(d)}$ ; it is the identity for $d = 1$ and sends the empty Proj to the empty Proj.

[F3]

Affine blowup standard charts and overlaps: For $I = (f_{0}, \dots, f_{r})$ the standard opens $D_{+} (f_{i} t) = Spec A [I / f_{i}]$ cover ${Bl}_{I} Spec A$ with the stated overlap identifications, and the presentation is independent of the chosen generating family.

[F4]

Relative Proj of a graded quasi-coherent algebra: The relative Proj ${Proj}_{X} A$ of a quasi-coherent graded $O_{X}$ -algebra is constructed by gluing the spectra of the degree-zero localisations over affine opens of $X$ , compatibly with restriction to smaller affine opens.

[F5]

Blowups restrict to open subschemes of the base: For an open subscheme $U ↪ X$ there is a canonical isomorphism ${Bl}_{I ∣_{U}} U \to {Bl}_{I} X \times_{X} U$ , and the blowup is determined up to canonical isomorphism by its restrictions to an open cover.

Proof

1.1F1

The graded $O_{X}$ -algebras $R (I^{d})$ and $R (I)^{(d)}$ are canonically isomorphic: their degree- $n$ pieces are $(I^{d})^{n} = I^{d n}$ in both cases, the multiplications are the multiplication of $O_{X}$ , and the identifications are compatible with restriction to open subschemes.

1.2F2

On an affine open $U = Spec A \subseteq X$ with $I ∣_{U} = \tilde{I}$ and $R (I) = ⨁_{n \geq 0} I^{n} t^{n}$ , [F2] applied to the graded ring $S = R (I)$ gives a canonical isomorphism $Proj R (I) ≅ Proj R (I)^{(d)}$ that maps a chart $D_{+} (f)$ , for homogeneous $f \in R (I)_{+}$ of positive degree, to $D_{+} (f^{d})$ with the same coordinate ring $R (I)_{(f)} = R (I)_{(f^{d})}^{(d)}$ .

2.1F1F3step 1.1step 1.2

The isomorphism of step 1.2 identifies the standard charts of the two blowups: for $a \in I$ the chart $D_{+} (a t)$ of $Proj R (I)$ corresponds to $D_{+} ((a t)^{d}) = D_{+} (a^{d} t^{d})$ in $Proj R (I)^{(d)}$ , and under step 1.1 this is the standard chart $D_{+} (a^{d} t^{'})$ of $Proj R (I^{d})$ , with coordinate ring $A [I / a] = (R (I))_{(a t)} = R (I^{d})_{(a^{d} t^{'})} = A [I^{d} / a^{d}]$ via the identification $a t \leftrightarrow a^{d} t^{'}$ ; the charts $D_{+} (a t)$ cover ${Bl}_{I} U$ for any generating family by [F3], and their images cover ${Bl}_{I^{d}} U$ .

3.1F2F4step 2.1

The chartwise identifications are canonical: on the overlap of two charts they are the identity of the common localisation of $R (I)$ , and on restriction to a smaller affine open $V \subseteq U$ the identification for $V$ is the restriction of the identification for $U$ , because both sides are computed by the same graded localisations and the Veronese isomorphism of [F2] is natural in the graded ring; hence the identifications are compatible with the gluing data of the standard charts.

4.1F4F5step 1.1step 3.1∎

The identifications of step 3.1 glue over an affine cover of $X$ to an isomorphism of $X$ -schemes ${Proj}_{X} R (I) \to {Proj}_{X} R (I^{d})$ by [F4], and over arbitrary open subschemes the restriction compatibility of [F5] gives the same isomorphism; since $R (I^{d}) ≅ R (I)^{(d)}$ by step 1.1, this proves the stated canonical isomorphism ${Bl}_{I^{d}} X \to {Bl}_{I} X$ , which is the identity when $d = 1$ .

Remarks

The identification matches the relative twists $O_{{Bl}_{I^{d}} X} (1)$ with $O_{{Bl}_{I} X} (d)$ under the Veronese isomorphism, as recorded in Proj is invariant under Veronese regrading.
Together with Invariance of the blowup under invertible (fractional) rescaling of the ideal this shows that the blowup depends on the ideal sheaf only up to the equivalence generated by invertible rescaling and positive powers.

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Integrality and reducedness of blowups from the Rees charts

Statement

Assume the Axiom of Choice, inherited from the relative Proj construction (The Axiom of Choice). Let $X$ be an integral scheme (Integral schemes) and let $I$ be a nonzero quasi-coherent ideal sheaf of finite type on $X$ (Quasi-coherent ideal sheaves). Then the blowup ${Bl}_{I} X$ of Blowup of a scheme along an ideal sheaf is integral: the affine blowup algebras $A [I / a]$ are domains because $A$ is a domain and $a$ is nonzero, and they glue along localisations. More generally, if $X$ is reduced then ${Bl}_{I} X$ is reduced, because the affine blowup algebras of a reduced ring are reduced.

Facts & Assumptions

Given: An integral (respectively reduced) scheme $X$ , a nonzero quasi-coherent ideal sheaf $I$ of finite type, and for an affine open $U = Spec A \subseteq X$ with $I = Γ (U, I)$ and $a \in I$ the affine blowup algebra $A [I / a] = (R (I))_{(a)} \subseteq A_{a}$ , the degree-zero part of the localisation of $R (I) = ⨁_{n \geq 0} I^{n} t^{n}$ (Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains, Rees algebra sheaf of a finite type ideal).

[F1]

Integral schemes: $X$ is integral exactly when it is nonempty and every nonempty affine open of $X$ is the spectrum of a domain; equivalently $X$ is reduced and its underlying space is irreducible.

[F2]

Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: For $a \in I$ the affine blowup algebra $A [I / a]$ has $I A [I / a] = a A [I / a]$ with $a$ a nonzerodivisor and $(A [I / a])_{a} = A_{a}$ . Its construction as the degree-zero part of the localisation of $R (I) \subseteq A [t]$ at the degree-one element $a t$ embeds $A [I / a]$ into $A_{a}$ as the subring generated by $A$ and the fractions $i / a$ , $i \in I$ ; if $A$ is a domain and $a \neq 0$ then $A [I / a]$ is a domain, and if $A$ is reduced then $A [I / a]$ is reduced.

[F3]

Affine blowup standard charts and overlaps: For $I = (f_{0}, \dots, f_{r})$ the standard opens $D_{+} (f_{i} t) = Spec A [I / f_{i}]$ cover ${Bl}_{I} Spec A$ and the presentation is independent of the chosen generating family.

[F4]

A principal localization identifies its spectrum with a distinguished open: For $a \in A$ the principal open $D (a)$ is $Spec A_{a}$ and the localisation morphism is an open immersion $D (a) ↪ Spec A$ with image the complement of $V (a)$ .

[F5]

Blowups restrict to open subschemes of the base: For an open subscheme $U ↪ X$ there is a canonical isomorphism ${Bl}_{I ∣_{U}} U \to {Bl}_{I} X \times_{X} U$ ; the blowup is covered by the restrictions over an affine cover of $X$ .

[F6]

The reduction of a scheme: On $Spec A$ , the reduction is $Spec (A / \sqrt{(0)})$ . Consequently reducedness is affine-local: a nilpotent section on a reduced affine chart is zero, and these charts cover all stalks. Also a subring of a reduced ring is reduced, since its nilpotent elements are nilpotent in the larger ring and hence zero.

Proof

1.1F2F3F4F5F6

Over an affine base $U = Spec A$ , discard generators $f_{i} = 0$ , whose charts are empty. If $A$ is a domain, every remaining chart $B_{i} = A [I / f_{i}] \subset A_{f_{i}}$ is a domain; if $A$ is reduced, every chart is reduced (including empty charts). Hence the blowup is reduced in either case, by affine-local reducedness. Moreover $D (f_{i})$ in chart $i$ is $Spec A_{f_{i}}$ by the chart localization identity.

2.1F1F2F3F4step 1.1

Suppose $X$ is integral and $I \neq 0$ . Then $X ∖ Z$ is a nonempty open: a nonzero local section of the ideal in a domain remains nonzero at the generic point. Put $W = π^{- 1} (X ∖ Z)$ . On chart $i$ , the inverse-image ideal is $f_{i} B_{i}$ , so $W \cap Spec B_{i} = D (f_{i})$ . The identifications $D (f_{i}) = Spec A_{f_{i}}$ are the structural morphism and agree on intersections: after both denominators are inverted, the ratio transition maps fix $A$ and the ordinary fractions. Thus they glue to $W ≅ X ∖ Z$ . This is a nonempty irreducible open.

3.1F1F3F6step 1.1step 2.1∎

Every nonempty chart is a domain chart with a nonzero denominator, so its nonempty principal open $D (f_{i}) \subset W$ is dense. The closure of $W$ therefore contains every chart and is the whole blowup. A closure of an irreducible set is irreducible. Together with reducedness and nonemptiness, this proves integrality. If the ideal is zero, all charts are empty and the reducedness assertion still holds.

Remarks

The argument does not need $X$ to be Noetherian or $I$ to be principal anywhere; it only uses that the affine blowup algebra sits inside the localisation $A_{a}$ .
If $I = 0$ the blowup is empty and hence reduced, while irreducibility and nonemptiness fail; this is why the integral statement assumes $I \neq 0$ .

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Strict transforms of closed subschemes are blowups of the subscheme

Statement

Assume the Axiom of Choice. Let $I$ be a quasi-coherent ideal sheaf of finite type on a scheme $X$ (Quasi-coherent ideal sheaves), let $π ⁣ : {Bl}_{I} X \to X$ be the blowup of Blowup of a scheme along an ideal sheaf with center $Z = V (I)$ , and let $W ↪ X$ be a closed subscheme (Closed immersions of schemes). Write $J = I O_{W}$ for the inverse image ideal of $I$ in $W$ ; it is a quasi-coherent ideal sheaf of finite type, and it cuts out the scheme-theoretic intersection $W \cap Z$ . Then the strict transform $W^{'}$ of $W$ (Strict transform of a closed subscheme), the scheme-theoretic closure of $π^{- 1} (W ∖ Z)$ in $W \times_{X} {Bl}_{I} X$ , is canonically isomorphic over $W$ to the blowup ${Bl}_{J} W$ ; equivalently, $W^{'}$ is the blowup of $W$ along the closed subscheme $W \cap Z$ . On the standard affine charts $Spec A \subseteq X$ with $I = (f_{0}, \dots, f_{r})$ and $W = Spec (A / K)$ , the chart $Spec A [I / f_{i}]$ of the blowup meets $W^{'}$ in $Spec (A [I / f_{i}] / (K A [I / f_{i}] : f_{i}^{\infty}))$ : $W^{'}$ is cut out by the saturation of the pullback ideal of $W$ by the exceptional equation. Moreover the strict transform of a finite scheme-theoretic union is the union of the strict transforms; in particular a finite union of components is transformed componentwise.

Facts & Assumptions

Given: A scheme $X$ , a quasi-coherent ideal sheaf $I$ of finite type with zero scheme $Z$ , the blowup $π ⁣ : {Bl}_{I} X \to X$ , a closed subscheme $i ⁣ : W ↪ X$ with inverse image ideal $J = I O_{W}$ , and the Axiom of Choice, inherited from the relative Proj constructions (The Axiom of Choice).

[F1]

Strict transform of a closed subscheme: With $U = (W \times_{X} {Bl}_{I} X) ∖ E$ , the strict transform $W^{'}$ is the scheme-theoretic closure of $U$ in $W \times_{X} {Bl}_{I} X$ ; when the closure is computed by Schematic closure and agreement on a dense open, it is the smallest closed subscheme through which $U ↪ W \times_{X} {Bl}_{I} X$ factors. On a standard affine chart $Spec A [I / a]$ on which the exceptional subscheme is cut by $a$ , and on which $W$ is cut by $K \subseteq A$ , the strict transform is cut out by the saturation $(K A [I / a] : a^{\infty})$ , and these local descriptions glue.

[F2]

Blowup of a scheme along an ideal sheaf, Quasi-coherent ideal sheaves and Quasi-coherent module on a scheme: For a quasi-coherent ideal sheaf $J$ of finite type the blowup ${Bl}_{J} W = {Proj}_{W} R (J)$ exists, and $J = I O_{W}$ is quasi-coherent of finite type because $I$ is and $i^{- 1}$ and quotients of quasi-coherent modules preserve these properties; its zero scheme is $W \times_{X} Z \subseteq W$ .

[F3]

The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier and Universal property of the blowup: On the blowup of $W$ along $J$ the pullback $J O_{{Bl}_{J} W}$ is invertible, so for the composite ${Bl}_{J} W \to W \to X$ the inverse image of $Z$ is an effective Cartier divisor; consequently there is a unique $X$ -morphism $ψ ⁣ : {Bl}_{J} W \to {Bl}_{I} X$ . Equivalently ${Bl}_{I} X$ is final among $X$ -schemes in which the inverse image of $Z$ is an effective Cartier divisor.

[F4]

Affine blowup standard charts and overlaps and Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: For $I = (f_{0}, \dots, f_{r})$ the standard opens $Spec A [I / f_{i}]$ cover ${Bl}_{I} Spec A$ and $(A [I / f_{i}])_{f_{i}} = A_{f_{i}}$ ; the image of $f_{i}$ is a nonzerodivisor in $A [I / f_{i}]$ , so $(L : f_{i}^{\infty}) = {b : f_{i}^{n} b \in L for some n}$ is the preimage of the extension of an ideal $L$ to $A [I / f_{i}] [1 / f_{i}]$ .

[F5]

Closed immersions are local on the target: A morphism is a closed immersion if and only if its restrictions over the members of an open cover of the target are closed immersions.

[F6]

Uniqueness of the blowup: Two blowups of the same scheme along the same ideal sheaf are isomorphic by a unique isomorphism compatible with the structural morphisms.

Proof

1.1F2F3

By [F2] the inverse image ideal $J = I O_{W}$ is quasi-coherent of finite type with zero scheme $W \times_{X} Z$ , so the blowup ${Bl}_{J} W$ and its structural morphism $σ ⁣ : {Bl}_{J} W \to W$ are defined; composing with $W \to X$ exhibits it as an $X$ -scheme. By [F3] the inverse image of $Z$ on ${Bl}_{J} W$ is cut by the invertible ideal $J O_{{Bl}_{J} W}$ , hence is an effective Cartier divisor, so the universal property supplies a unique $X$ -morphism $ψ ⁣ : {Bl}_{J} W \to {Bl}_{I} X$ . Together with $σ$ it defines a morphism $u = (σ, ψ) ⁣ : {Bl}_{J} W \to W \times_{X} {Bl}_{I} X$ over $X$ .

2.1F4step 1.1algebra

Chart computation. Let $Spec A \subseteq X$ be affine with $I = (f_{0}, \dots, f_{r})$ and $W = Spec (A / K)$ . Write $B_{i} = A [I / f_{i}]$ and $C_{i} = (A / K) [J / {\overset{ˉ}{f}}_{i}]$ , so that $Spec C_{i}$ is the chart of ${Bl}_{J} W$ over $Spec (A / K)$ . The morphism $u$ on this chart is the $A$ -algebra homomorphism $φ_{i} ⁣ : B_{i} \to C_{i}$ with $φ_{i} (a) = \overset{ˉ}{a}$ for $a \in A$ and $φ_{i} (f_{j} / f_{i}) = {\overset{ˉ}{f}}_{j} / {\overset{ˉ}{f}}_{i}$ . It is surjective because $C_{i}$ is generated over $A / K$ by the elements ${\overset{ˉ}{f}}_{j} / {\overset{ˉ}{f}}_{i}$ . Its kernel is the saturation $(K B_{i} : f_{i}^{\infty})$ : indeed $b \in \ker φ_{i}$ if and only if the image of $b$ in $C_{i} [1 / {\overset{ˉ}{f}}_{i}]$ vanishes, and by [F4] applied to $A / K$ and $J$ this localized ring is $(B_{i} / K B_{i}) [1 / f_{i}]$ , the localization of $B_{i} / K B_{i}$ at $f_{i}$ , so $b$ lies in the kernel precisely when $f_{i}^{n} b \in K B_{i}$ for some $n$ , which is the saturation.

3.1F1F5step 2.1

By step 2.1 the restriction of $u$ to the chart $Spec C_{i}$ is the closed immersion $Spec C_{i} ↪ Spec B_{i}$ cut out by the saturation $(K B_{i} : f_{i}^{\infty})$ , whose image is exactly the piece of $W^{'}$ over $Spec A$ described in [F1]; this is the full preimage of chart $i$ : on a source chart $j$ , membership in target chart $i$ means the pulled-back ratio ${\overset{ˉ}{f}}_{i} / {\overset{ˉ}{f}}_{j}$ is a unit, precisely the overlap with source chart $i$ . Equivalently, if the image lies in target chart $i$ , the pullback of its center ideal is generated by ${\overset{ˉ}{f}}_{i}$ ; comparison with a regular generator ${\overset{ˉ}{f}}_{j}$ on a source chart forces their ratio to be a unit, as in the universal-property proof. Thus $Spec C_{i}$ is the full preimage. Therefore these chart descriptions agree on overlaps and cover the target, so by [F5] the morphism $u$ is a closed immersion and its image is exactly $W^{'}$ . Hence $u$ identifies $W^{'}$ with ${Bl}_{J} W$ over $W$ , and in particular $W^{'}$ is the blowup of $W$ along $J = I O_{W}$ , equivalently along $W \cap Z$ .

4.1F3F6step 3.1

Canonicity. The morphism $ψ$ is the unique $X$ -morphism from ${Bl}_{J} W$ to ${Bl}_{I} X$ provided by the universal property in [F3], and $σ$ is the structural morphism of the blowup, so $u$ is determined by the data of the two blowups; conversely the inverse $W^{'} \to {Bl}_{J} W$ is obtained by gluing the inverse chart isomorphisms $B_{i} / (K B_{i} : f_{i}^{\infty}) ≅ C_{i}$ from steps 2.1–3.1, determined by the same data, and any two isomorphisms with these properties agree by [F6]. Thus the identification of $W^{'}$ with ${Bl}_{J} W$ is canonical.

5.1F1step 2.1∎

Union statement. Suppose $W = W_{1} \cup W_{2}$ is the scheme-theoretic union of two closed subschemes, so on an affine chart their ideals satisfy $K = K_{1} \cap K_{2}$ . The saturation of $K$ with respect to $f_{i}$ is the preimage of the ideal $K B_{i} [1 / f_{i}]$ under $B_{i} \to B_{i} [1 / f_{i}]$ by [F4], and $B_{i} [1 / f_{i}] = A_{f_{i}}$ . Flat localization gives $(K_{1} \cap K_{2}) A_{f_{i}} = K_{1} A_{f_{i}} \cap K_{2} A_{f_{i}}$ ; taking preimages under $B_{i} \to A_{f_{i}}$ commutes with finite intersections; hence $(K B_{i} : f_{i}^{\infty}) = (K_{1} B_{i} : f_{i}^{\infty}) \cap (K_{2} B_{i} : f_{i}^{\infty})$ on every chart. By step 2.1 the right-hand side cuts out the union of the chart pieces of $W_{1}^{'}$ and $W_{2}^{'}$ , and these chartwise identifications glue, so $W^{'} = W_{1}^{'} \cup W_{2}^{'}$ as closed subschemes of $W \times_{X} {Bl}_{I} X$ . Induction gives the result for every finite scheme-theoretic union. In particular a finite union of components is transformed componentwise, and $W^{'}$ is the union of the strict transforms of its parts; this union may be empty or have a single component.

Remarks

The hypothesis that $I$ has finite type is used only to know that the blowups and the inverse image ideal are defined as in Blowup of a scheme along an ideal sheaf; the identification itself is chartwise.
The saturation in the chart description is exactly what removes the components of the pullback of $W$ that lie inside the exceptional divisor, which is why a subscheme contained in the center has empty strict transform: for $W \subseteq Z$ the ideal $J$ is zero on the charts, $W^{'} = \emptyset$ , and ${Bl}_{J} W$ is the relative Proj of a graded algebra concentrated in degree zero, which is empty. This convention is forced by the closure definition, and the theorem covers it.

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Blowing up the base ideal resolves a rational map to projective space

Statement

Assume the Axiom of Choice. Let $X$ be an integral finite-type $k$ -scheme, let $L$ be invertible, and let $s_{0}, \dots, s_{n}$ be meromorphic sections of $L$ , not all zero. Their ratios define $φ : X ⇢ P_{k}^{n}$ . Put $F = \sum_{i} O_{X} s_{i} \subseteq K_{X} (L)$ and define the finite-type quasi-coherent fractional ideal $I = F \otimes L^{- 1} \subseteq K_{X}$ . Its fractional blowup is $B = {Proj}_{X} (⨁_{d \geq 0} I^{d}), I^{0} = O_{X} .$ Then $B$ is integral, its projection $π : B \to X$ resolves $φ$ , and $(π, ψ) : B ↪ X \times_{k} P_{k}^{n}$ is the schematic closure of its graph. The generating line bundle is $M = π^{*} L \otimes (I O_{B})$ . If $f : Y \to X$ is a $k$ -morphism from an integral scheme, $D \subseteq X$ is the domain of a representative of $φ$ , $f^{- 1} (D) \neq \emptyset$ , and $θ : Y \to P_{k}^{n}$ extends that representative composed with $f$ , then $f$ factors uniquely through $B$ . The nonempty inverse-image condition ensures that this induced rational map is defined.

Facts & Assumptions

Given: The Axiom of Choice, the integral finite-type $k$ -scheme $X$ , $L$ , the meromorphic tuple $(s_{i})$ , its fractional ideal $I$ , and $B$ as above.

[F1]

Rational maps of integral finite-type schemes and Rational section line bundle: Meromorphic sections are elements of the one-dimensional generic fiber of $L$ (zero is allowed here); a nonzero tuple gives projective ratios on a nonempty open. Representatives agree on nonempty opens and their target is separated.

[F2]

Invariance of the blowup under invertible (fractional) rescaling of the ideal and Relative Proj of a graded quasi-coherent algebra: A fractional ideal of form $a^{- 1} J$ on an affine domain, with $a \neq 0$ and $J$ an ordinary ideal, has Rees Proj canonically isomorphic to the blowup of $J$ , by degreewise rescaling. Relative Proj glues quasi-coherent graded algebras.

[F3]

The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier: The extended ordinary center ideal on its blowup is invertible.

[F4]

Maps to projective space equal generating line-bundle data: An invertible sheaf with $n + 1$ generating global sections gives a morphism to projective $n$ -space; its coordinates on chart $j$ are the section ratios.

[F5]

Affine blowup standard charts and overlaps and Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains: For $J = (p_{0}, \dots, p_{n})$ in a domain $A$ , chart $j$ is $A [J / p_{j}] \subseteq A_{p_{j}}$ , generated by the ratios $p_{i} / p_{j}$ ; zero generators give empty charts.

[F6]

Blowups of finite type ideals are locally H-projective, and proper: The surjection $A [T_{0}, \dots, T_{n}] \to R (J)$ gives a closed immersion of the blowup into $Spec A \times P_{k}^{n}$ with the displayed chart ratios.

[F7]

Integrality and reducedness of blowups from the Rees charts: A blowup of a nonzero finite-type ideal on an integral scheme is integral. Its unchanged nonempty open is dense in every nonempty domain chart.

Proof

1.1F1F2F7

Trivialize $L$ by $e$ on an affine $U = Spec A$ . Write $s_{i} = h_{i} e$ and clear denominators by $a \in A ∖ {0}$ , obtaining $p_{i} = a h_{i} \in A$ . Then $I ∣_{U} = a^{- 1} J$ with $J = (p_{i})$ , so it is quasi-coherent of finite type. Its powers are also quasi-coherent. Degreewise multiplication by $a^{d}$ identifies its Rees algebra with $R (J)$ . Changing $a$ or the frame rescales these maps in each degree, inducing the same degree-zero ratio maps on Proj. Thus the relative Proj is defined and locally the ordinary blowup of $J$ ; $J \neq 0$ , so $B$ is integral.

2.1F3F4F5step 1.1

The extended fractional ideal $I O_{B} = a^{- 1} (J O_{B})$ is invertible by [F3]. The sections $π^{*} s_{i}$ lie in $M = π^{*} L \otimes (I O_{B})$ and generate it. If $b$ generates $J O_{B}$ on a local chart, the frame of $M$ is $π^{*} e \otimes (b / a)$ , and the coefficients of these sections are $p_{i} / b$ , which are regular and generate the unit ideal. Therefore [F4] gives $ψ : B \to P_{k}^{n}$ , with coordinates $p_{i} / p_{j} = h_{i} / h_{j}$ on chart $j$ , resolving $φ$ .

3.1F1F5F6F7step 2.1

Locally on $X$ , [F6] makes $(π, ψ)$ a closed immersion with chart rings $A [J / p_{j}]$ . These immersions agree on overlaps because their coordinate ratios agree, hence glue. On the nonempty open where a nonzero $p_{j}$ is invertible, it is the graph of $φ$ ; this open is dense in the corresponding domain chart by [F5]. Consequently $B$ is the schematic closure of that graph: a function on a domain chart vanishing on this dense principal open is zero. For any representative $φ_{D}$ the graph over $D$ is closed in $D \times P_{k}^{n}$ by separatedness, and its nonempty dense part is the same graph just considered. The closure restricted to $D$ is therefore exactly that graph, with its reduced scheme structure.

4.1F1step 3.1∎

Let $f, θ, D$ satisfy the stated hypotheses. The nonempty open $f^{- 1} (D)$ is dense in the integral scheme $Y$ , and $(f, θ)$ lands in $B$ there by step 3.1. The pullback of the ideal of the closed immersion $B ↪ X \times P_{k}^{n}$ vanishes on this dense open. It vanishes everywhere: on each nonempty affine open of an integral scheme, a regular function zero on a dense open is zero in its domain coordinate ring. Thus $(f, θ)$ factors through $B$ . Any other $X$ -lift has the same projective component on $f^{- 1} (D)$ , since $B$ over $D$ is the graph. Two maps from a reduced integral scheme to separated projective space agreeing on a dense open agree everywhere, by the same ideal-vanishing argument applied to the diagonal. Hence the two lifts agree, as $B$ is a closed subscheme of the product.

Remarks

The base ideal is fractional when the sections have poles. Clearing denominators supplies ordinary ideals locally; their rescalings need not define a single ordinary ideal globally. Multiplying the tuple by a nonzero rational scalar preserves its ratios and its blowup. A morphism whose image is entirely outside every representative's domain has no induced rational map to extend; no factoring claim is made for it.

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Blowing up replaces the center by its projectivized normal directions

Remark

A blowup is not the deletion of the center. Let $I$ be a quasi-coherent ideal sheaf of finite type on a scheme $X$ with zero scheme $Z = V (I)$ , and let $π ⁣ : {Bl}_{I} X \to X$ be the blowup of Blowup of a scheme along an ideal sheaf, with exceptional subscheme $E = π^{- 1} (Z)$ (Exceptional subscheme of a blowup). Then:

$π$ is proper over $X$ (Blowups of finite type ideals are locally H-projective, and proper); no properness of the base over a field or global H-projective embedding is required;
$π$ is an isomorphism over the open complement $X ∖ Z$ (The blowup is an isomorphism off the center). It can also be an isomorphism over the center: an effective Cartier center has identity blowup, as is seen on its single principal regular chart;
the pullback ideal $I O_{Bl}$ is invertible (The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier), so after the blowup the center is an effective Cartier divisor and further blowups along it do nothing;
the center is replaced, not deleted: $E$ is identified with the projectivized normal cone ${Proj}_{Z} ({gr}_{I} O_{X})$ (The exceptional divisor is the projectivized normal cone), so the points of $E$ lying over $z \in Z$ record the normal directions to $Z$ at $z$ . For a closed point center on a regular finite-type surface of pure dimension two over a field this is a projective line over the residue field, $P_{κ (p)}^{1}$ , and the blowup stays a regular surface (Point blowups of regular surfaces stay regular, with rational exceptional fibre over the residue field).

Surjectivity is part of the same picture but carries a hypothesis. If the center is empty ( $I = O_{X}$ ) then $π$ is an isomorphism; if $I$ vanishes identically on an open set, then the blowup has empty fibres over that set, so no unconditional surjectivity holds. In the integral finite-type situations used on this page, a nonzero center ideal has a nonempty dense complement. The proper image of the blowup is closed and contains that complement, hence is all of $X$ . Thus every fiber is nonempty, including the projectivized normal-cone fibers over the center.

Finally, strict transforms record how subvarieties approach the center: a closed subscheme $W \subseteq X$ has strict transform (Strict transform of a closed subscheme) cut out on the standard affine charts by the saturation of the pullback of its ideal by a local equation of the center, so parts supported entirely in the exceptional locus are removed. A subscheme contained in the center has empty strict transform; a single blowup need not separate all remaining branches.

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No inference to general resolution of singularities

Remark

The results of this page prove resolution for reduced projective plane curves by point blowups (Resolution of reduced plane curves by point blowups and the delta recurrence) and the regularity of point blowups of regular surfaces; the blowups used are proper and, in the point case, have the explicit projective chart descriptions of Blowups of finite type ideals are locally H-projective, and proper. These statements do not imply resolution of singularities for arbitrary varieties, nor for surfaces over imperfect fields with smoothness assertions about the resulting components, nor for schemes of dimension at least three. The termination argument for the curve case uses two features special to curves on surfaces: the one-dimensional normalization defect $δ_{k}$ , which decreases by $r m (m - 1) / 2$ at each singular point blowup, and the pairwise contact order of two regular branches at a point of a regular surface, which decreases by one under an appropriate point blowup. In higher dimension there is no such defect count, and the embedded normal-crossing support produced for curves does not control the singularities of a general ambient scheme. The page therefore claims only the plane-curve resolution and the regularity statements stated and proved in its items, and the normal-crossing conclusion is asserted over the residual residue fields, not as smoothness over an imperfect base field.

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Cohomology comparison when higher direct images vanish

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $f : X \to Y$ be a morphism of schemes and $F$ an $O_{X}$ -module. If $R^{q} f_{*} F = 0$ for all $q > 0$ , then the natural cohomology comparison is an isomorphism $H^{n} (Y, f_{*} F) \tilde{\to} H^{n} (X, F) (n \geq 0) .$ The comparison is natural in $F$ , agrees in degree zero with the identity $Γ (Y, f_{*} F) = Γ (X, F)$ , and applies also to $f$ restricted over any open subset of $Y$ where the same vanishing holds. No quasi-coherence, separatedness or properness of $f$ is required.

Facts & Assumptions

Given: The Axiom of Choice, a morphism of schemes $f : X \to Y$ and an $O_{X}$ -module $F$ with vanishing higher direct images.

[F1]

Godement terms are flasque and compute cohomology: Under Choice the Godement resolution of an abelian sheaf is functorial, exact, has flasque terms, and computes its sheaf cohomology.

[F2]

Enough injective sheaves of modules: The category of modules on a ringed space is abelian with supplied functorial injective resolutions under Choice; forgetting the module structure preserves kernels, cokernels and exactness.

[F3]

Flasque sheaf and Flasque abelian sheaves are Γ-acyclic: A flasque sheaf has surjective restrictions and has zero positive cohomology on every open subset.

[F4]

Direct image of a sheaf along a continuous map and Local-section formula for derived direct image: Direct image sections on $V$ are sections on $f^{- 1} V$ , and higher direct images of a module are sheafifications of $V \mapsto H^{q} (f^{- 1} V, F)$ , with no restriction on the morphism.

[F5]

The acyclic-resolution theorem for right derived functors, Higher direct image of a sheaf and Sheaf cohomology as right derived global sections: An exact resolution by objects acyclic for a left exact functor computes its right derived functors, with canonical comparison, provided its syzygies lie in the domain of the supplied resolution datum.

[F6]

The Axiom of Choice and AC implies DC implies countable choice: Choice supplies Dependent Choice, as needed for acyclic-resolution comparisons.

Proof

1.1F1F2F6given

Apply the Godement construction to the underlying abelian sheaf of $F$ , retaining its module structure. For a module $G$ , the first term on an open $U$ is $\prod_{x \in U} G_{x}$ , with $a \in O_{X} (U)$ acting through its germ on each factor. The germ map is module-linear; take its module cokernel and repeat. Since these cokernels have the same underlying abelian sheaves, this yields an exact functorial module resolution $F \to G^{∙}$ whose terms are flasque and whose section complex computes $H^{n} (X, F)$ . All modules and syzygies lie in the supplied datum's domain, and Choice supplies the required Dependent Choice.

1.2F3F4

Every flasque module $G$ is $f_{*}$ -acyclic: on every open $V \subset Y$ , flasque acyclicity gives $H^{q} (f^{- 1} V, G) = 0$ for $q > 0$ , and the local-section formula gives $R^{q} f_{*} G = 0$ . Moreover $f_{*} G$ is flasque, since its restrictions are the restrictions of $G$ along inverse-image opens. These facts hold for arbitrary $f$ .

2.1F2F4F5step 1.1step 1.2

Apply the acyclic-resolution theorem to $f_{*}$ and $G^{∙}$ . It identifies the cohomology sheaves of $f_{*} G^{∙}$ with $R^{q} f_{*} F$ . The hypothesis makes this an exact coaugmented resolution of $f_{*} F$ , and its terms are flasque by step 1.2. Forget the $O_{Y}$ -module structure, preserving exactness by [F2]. All terms and syzygies are then in $A b (Y)$ , the full domain of the supplied cohomology datum. Apply the acyclic-resolution theorem to $Γ (Y, -)$ on that category; this resolution computes $H^{n} (Y, f_{*} F)$ . Its global section complex equals $Γ (X, G^{∙})$ term by term, so step 1.1 gives the claimed comparison isomorphism.

3.1F1F4F5step 2.1∎

Functoriality of Godement and of the acyclic-resolution comparisons makes these identifications natural and independent of presentations. In degree zero they are exactly the equality of direct-image global sections. Restricting over an open of $Y$ repeats the same proof. Empty schemes and the zero module give zero complexes, and for the identity morphism the comparison is the identity.

5 · Examples, counterexamples and false statements

None yet.