The single-centre form of Main Theorem I: the enlargement of the centre #
Hironaka's Main Theorem I [Hir64, Main Theorem I, p. 132] resolves a reduced irreducible algebraic
k-scheme X by ONE monoidal transformation whose centre D has the singular locus of X as its
set of points. In the strong form [Hir64, p. 132] the resolving morphism is obtained by a finite
succession of monoidal transformations with non-singular centres containing no simple point, and
Hironaka remarks that any finite succession is a single monoidal transformation with a suitably
chosen centre [Hir64, pp. 132–133]. This file makes that remark precise in the form needed: given a
finite succession S of monoidal transformations of X with smooth last stage X_r (hypothesis
hsmooth) whose centres lie over Sing X (hypothesis hcenters), it produces one closed subscheme
D ⊆ X with |D| = Sing X and D.blowUp non-singular (exists_center_support_eq_singularLocus).
This is the conclusion of Main Theorem I. The main theorem, with the succession provided by
Kollár's resolution functor, is in Hironaka.Resolution.Algebraic.Hir64.MainTheoremI.
The argument #
Let 𝒥 := X.singIdeal be the ideal of Sing X with its reduced structure. X_r is integral
(Hironaka.Resolution.Algebraic.Hir64.SingleCenterTools), and T := (X_r, 𝒥·𝒪_{X_r}, ∅) is a
triple (enlargementTriple). Let P := BP T be its principalization [Kol07, Theorem 35] and
S' := S.concat P. The composite S'.last → X is one blow-up K.blowUp with |K| ⊆ Sing X
(AlgebraicGeometry.exists_blowUp_composite_iso; the centres of P lie over Π⁻¹(Sing X) by
stageMap_mem_support_of_mem_center_BP and forall_center_concat_mem). Set D := K * 𝒥. Then
|D| = |K| ∪ Sing X = Sing X, and D.blowUp ≅ K.blowUp, because 𝒥·𝒪_{K.blowUp} corresponds
to (𝒥·𝒪_{X_r})·𝒪_{P.last}, which is invertible by clause (2) of [Kol07, Theorem 35]
(isInvertible_comap_composite_BP), so the second blow-up is an isomorphism
(exists_iso_blowUp_mul_of_isInvertible_comap). Finally D.blowUp ≅ S'.last is smooth over k
(smooth_composite_BP), hence regular (isRegular_of_smooth, IsRegular.of_isIso).
Conventions #
The helpers are stated for a structure morphism f : X ⟶ Spec k; the two that need neither k
nor f are stated without them. The theorem itself is stated in the X.Over form of the main
theorems.
Every centre lies over Sing X, which misses the (regular) generic point of the integral scheme
X, so the last stage X_r has a point over the generic point
(exists_stage_point_over_of_forall_center_mem).
The last stage X_r is integral: no centre is a whole stage
(support_center_ne_top_of_forall_mem_singularLocus) and blow-ups of an integral locally
Noetherian scheme in proper centres stay integral (isIntegral_last_of_forall_support_ne_top).
The pullback 𝒥·𝒪_{X_r} of the ideal of Sing X has nonzero stalks everywhere (condition (2)
of [Kol07, Notation 64] for the triple T): its support Π⁻¹(Sing X) is a proper closed subset of
the irreducible X_r (it misses a point over the generic point), hence nowhere dense
(isNonzeroEverywhere_of_isNowhereDense_support).
The triple (X_r, 𝒥·𝒪_{X_r}, ∅) to which the principalization theorem is applied in the
enlargement of the centre (not Hironaka's step: his single centre [Hir64, p. 133] is defined by
J(m)J_0 with f_0⁻¹(J(m)) = f_0⁻¹(J_0)^m J_1): X_r → Spec k is smooth (hypothesis
hsmooth), proper over X so quasi-compact and separated, and equidimensional since X_r is
irreducible (exists_smoothOfRelativeDimension_of_preconnectedSpace); the ideal is nonzero
everywhere (isNonzeroEverywhere_comap_singIdeal); the boundary is empty, a simple normal
crossing divisor on a smooth scheme (isSnc_empty_of_smooth).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The enlargement of the centre [Hir64, pp. 132–133]: for X a reduced irreducible algebraic
k-scheme and a finite succession S of monoidal transformations with smooth last stage
(hsmooth) whose centres lie over Sing X (hcenters), there is a closed subscheme D with
|D| = Sing X and D.blowUp non-singular.