The universal property of the blow-up from the affine statements #
The exceptional ideal of the blow-up is invertible and the blow-up is universal among the
morphisms along which I becomes invertible [TSPA, Tags 0806 and 02OS]; Hau14, Definition 4.4
and Theorem 4.18. Both statements are proved by locality over the affine opens of X, from the
affine statements of the modules under Hironaka/Scheme/BlowUp/AffineBlowUp/:
blowUp.isInvertible_comap_π: the exceptional idealI·𝒪_Bis invertible. Onπ⁻¹(U) ≅ affineBlowUp (I.ideal U)it is the exceptional ideal of the affine blow-up (comap_exceptionalDivisor_ι, fromι_toBaseand the affine computation of inverse images throughfromSpec), invertible byaffineBlowUp.isInvertible_exceptionalIdeal; theπ⁻¹(U)coverB, and invertibility is local (IsInvertible.of_forall_comap_isOpenImmersion).blowUp.hom_ext: two liftsg, g'of an admissiblef : Y ⟶ Xare equal. Fory ∈ Ychoose an affineU ∋ f y; onf⁻¹(U)both lifts land inπ⁻¹(U) ≅ affineBlowUp (I.ideal U)over the admissiblef⁻¹(U) → U ≅ Spec Γ(X, U), so they agree by the affine uniqueness (affineBlowUp.hom_ext); conclude withScheme.hom_ext_of_forall.blowUp.exists_lift: an admissiblef : Y ⟶ Xlifts. For every affine openU ⊆ X,f⁻¹(U) → U ≅ Spec Γ(X, U)is admissible forI(U), so the affine universal property (affineBlowUp.lift) givesf⁻¹(U) → affineBlowUp (I.ideal U) → Boverf; the local lifts agree on the overlaps by uniqueness and glue over the cover{f⁻¹(U)}ofY(Scheme.Cover.glueMorphisms).
Together: the blow-up satisfies the predicate IsBlowUp of
Hironaka.Scheme.BlowUp.UniversalProperty (blowUp.isBlowUp). The reduction of the universal
property to the affine case, by uniqueness on overlaps, is the argument of
[Hau14, Theorem 4.18, proof (a)].
An ideal sheaf is invertible as soon as its inverse images along a family of open immersions
with jointly surjective images are — the local generator on an affine open of the source is
transported along Scheme.Hom.appIso.
On the component ι U, the exceptional ideal of the blow-up is the exceptional ideal of the
affine blow-up affineBlowUp (I.ideal U) (comap_fromSpec, ι_toBase).
Every point of the blow-up lies in the image of some component ι U.
The exceptional ideal of the blow-up is invertible — the exceptional divisor
I.exceptionalDivisor = I.comap (blowUpπ I) is an effective Cartier divisor [TSPA, Tag 02OS];
[Hau14, Definition 4.4]: the blow-up map is admissible for I.
The restriction of f over an affine open U ⊆ X, as a morphism f⁻¹(U) ⟶ Spec Γ(X, U).
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The ideal sheaf of I(U) pulled back along f⁻¹(U) → Spec Γ(X, U) is the inverse image
ideal sheaf I·𝒪_Y restricted to the open f⁻¹(U) (comap_fromSpec).
Admissibility restricts: if f is admissible for I, then f⁻¹(U) → Spec Γ(X, U) is
admissible for I(U).
The local lift f⁻¹(U) ⟶ blowUp I of an admissible f over an affine open U: the affine
lift into affineBlowUp (I.ideal U) followed by the component ι U.
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- One or more equations did not get rendered due to their size.
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A lift h of f maps f⁻¹(U) into π⁻¹(U), the image of ι U.
The factorization of a lift restricted to f⁻¹(U) through ι U lies over
f⁻¹(U) → Spec Γ(X, U).
Uniqueness of the lift [TSPA, Tag 0806]; [Hau14, Theorem 4.18, proof (a)]: two lifts of an
admissible f : Y ⟶ X to the blow-up are equal.
Existence of the lift [TSPA, Tag 0806]; [Hau14, Theorem 4.18 (a)]: an admissible
f : Y ⟶ X lifts to the blow-up.
The universal property of the blow-up [TSPA, Tag 0806]; [Hau14, Definition 4.4]: the
blow-up is final among the admissible X-schemes — an admissible f : Y ⟶ X has exactly one
lift.
The blow-up is a blow-up in the sense of the universal property [Hau14, Definition 4.4]: the
predicate IsBlowUp.