The exceptional divisor of a blowup #
Let I be an ideal of a commutative ring R and let π : Proj R[It] ⟶ Spec R be the blowup of
Spec R along V(I). The exceptional divisor is the inverse image π⁻¹(V(I)), the closed
subscheme of Proj R[It] cut out by the ideal sheaf I·𝒪_{Proj R[It]}. This file proves that it
is an effective Cartier divisor. On the affine chart Spec R[I/a] of an element a ∈ I, it is
the principal divisor of the nonzerodivisor a, because I·R[I/a] = a·R[I/a]
(Ideal.map_algebraMap_affineBlowup, Ideal.isRegular_algebraMap_affineBlowup), and these charts
cover the blowup.
Together with the universal property of TauCeti/AlgebraicGeometry/Blowup/UniversalProperty.lean,
this characterizes the blowup: π is the terminal morphism to Spec R along which the centre
pulls back to an effective Cartier divisor. In particular the lift of π itself is the identity,
and π is an isomorphism exactly when V(I) is already an effective Cartier divisor on Spec R,
for instance when I is generated by a single nonzerodivisor: blowing up an effective Cartier
divisor changes nothing.
The same holds for the blowup Bl_I X ⟶ X of an arbitrary scheme X along an arbitrary ideal
sheaf I (AlgebraicGeometry.Scheme.IdealSheafData.blowup). Over an affine open U of X, the
exceptional divisor restricted to the chart Proj Γ(X, U)[I(U)t] is the exceptional divisor of the
affine blowup of Spec Γ(X, U) along V(I(U)), and these charts cover Bl_I X. Since being an
effective Cartier divisor is local, the exceptional divisor of Bl_I X is an effective Cartier
divisor. With the universal property of the blowup along an ideal sheaf, Bl_I X ⟶ X is thus the
terminal morphism to X along which V(I) becomes an effective Cartier divisor, and it is an
isomorphism exactly when V(I) is already one.
Main results #
The results on affine blowups are in the namespace TauCeti.AlgebraicGeometry.
comap_affineBlowupι_eq_ofIdealTop: on the chartSpec R[I/a], the exceptional divisor is cut out by the single equationa.isEffectiveCartier_comap_toSpecZero_reesAlgebra: the exceptional divisor of the blowup is an effective Cartier divisor.projReesAlgebraLift_eq_id: the lift of the blowup morphism through itself is the identity.isIso_toSpecZero_reesAlgebra_iff: the blowup morphism is an isomorphism if and only if the centre is an effective Cartier divisor.isIso_toSpecZero_reesAlgebra_span_singleton: blowing up the ideal generated by a nonzerodivisor gives an isomorphism.
The results on the blowup along an ideal sheaf are in the namespace
AlgebraicGeometry.Scheme.IdealSheafData.
comap_blowupChart_comap_blowupToBase: on the chart over an affine openU, the exceptional divisor ofBl_I Xis the exceptional divisor of the blowup ofSpec Γ(X, U)alongV(I(U)).isEffectiveCartier_comap_blowupToBase: the exceptional divisor ofBl_I Xis an effective Cartier divisor.blowupLift_eq_id: the lift ofBl_I X ⟶ Xthrough itself is the identity.isIso_blowupToBase_iff:Bl_I X ⟶ Xis an isomorphism if and only ifV(I)is an effective Cartier divisor onX.
References #
- The Stacks Project, Section Blowing up, the
lemmas that the exceptional divisor is an effective Cartier divisor, that it is cut out by
aon the affine chart ofa, and that blowing up an effective Cartier divisor is an isomorphism. - The Stacks Project, Lemma 0806, the universal property of the blowing up of a scheme along a quasi-coherent sheaf of ideals.
- R. Hartshorne, Algebraic Geometry, GTM 52, Springer, 1977, Proposition II.7.13(b).
The exceptional divisor on an affine chart. For a ∈ I, the inverse image of the centre
V(I) on the chart Spec R[I/a] ⟶ Proj R[It] is the zero scheme of the single function a.
The exceptional divisor is an effective Cartier divisor. The inverse image of the centre
V(I) under the blowup Proj R[It] ⟶ Spec R is an effective Cartier divisor on Proj R[It].
The blowup is its own lift. The lift of the blowup morphism Proj R[It] ⟶ Spec R through
itself, given by the universal property, is the identity.
Blowing up an effective Cartier divisor changes nothing. The blowup
Proj R[It] ⟶ Spec R is an isomorphism if and only if the centre V(I) is an effective Cartier
divisor on Spec R.
Blowing up a nonzerodivisor changes nothing. If a is a nonzerodivisor of R, the blowup
of Spec R along V(a) is an isomorphism.
The blowup of a scheme along an ideal sheaf #
The exceptional divisor on an affine chart. Over an affine open U of X, the inverse
image of the centre V(I) under the blowup Bl_I X ⟶ X, restricted to the chart
Proj Γ(X, U)[I(U)t] over U, is the exceptional divisor of the blowup of Spec Γ(X, U) along
V(I(U)).
The exceptional divisor is an effective Cartier divisor. The inverse image of the centre
V(I) under the blowup Bl_I X ⟶ X along an arbitrary ideal sheaf I is an effective Cartier
divisor on Bl_I X.
The blowup is its own lift. The lift of the blowup morphism Bl_I X ⟶ X through itself,
given by the universal property, is the identity.
Blowing up an effective Cartier divisor changes nothing. The blowup Bl_I X ⟶ X is an
isomorphism if and only if the centre V(I) is an effective Cartier divisor on X.