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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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Sources

The Stacks Project, Divisors, Sections 31.33-31.36 (Blowing up; Strict transform; Admissible blowups; Blowing up and flatness) (standard reference, not scraped)
Ravi Vakil, Foundations of Algebraic Geometry, June 27, 2011 draft (author-hosted 'Early (out-of-date) version of The Rising Sea') (standard reference, not scraped)