Hironaka's Main Theorem I and its weak form #
Hironaka's Main Theorem I [Hir64, Main Theorem I, p. 132]: for X a reduced irreducible algebraic
k-scheme, k of characteristic zero, there is a closed subscheme D of X whose set of points
is exactly the singular locus and whose monoidal transformation (the blow-up of X in D) is
non-singular. It is derived from Kollár's functorial resolution [Kol07, Theorem 36]: the sequence
BR X of the resolution functor (Hironaka.Resolution.Algebraic.Kol07.Thm36.BR) is a finite
succession of blow-ups of X with smooth last stage (clause (1)) whose centres lie over Sing X (a
property of the construction, stronger than clause (2), an isomorphism over the smooth locus, proved
below). The enlargement of the centre (exists_center_support_eq_singularLocus,
Hironaka.Resolution.Algebraic.Hir64.SingleCenter; Hironaka's remark [Hir64, pp. 132–133] that a
finite succession of monoidal transformations is one monoidal transformation with a suitably chosen
centre) then produces ONE centre D with |D| = Sing X and D.blowUp non-singular.
The centres lie over the singular locus (stageMap_mem_singularLocus_of_mem_center_BR; the
argument of [Kol07, 4.2] and of the proof of [Kol07, Theorem 27]: a functorial resolution is an
isomorphism over the smooth points, read through the clauses). Let z be a point of the centre
Z_i of BR X and suppose x := Π_i z is a smooth point. Take a member U of the finite affine
cover of X through x (the cover of [Kol07, Proposition 37, proof]): U is integral and
affine, an algebraic k-scheme through U → X → Spec k, so a class member
(isReducedEquidimensional_of_isIntegral). Clause (4a)
(BR_pullback_smooth_of_class) for the open immersion U → X gives
BR U = ((BR X).pullback (U → X)).eraseEmpty, and on the integral affine U the resolution is
the embedding-free affine resolution BR_affine' k U with its empty blow-ups deleted
(BR_eq_BR_affine'_of_integral), that is, the restriction to U of a truncated run of BP on an
admissible embedding emb : U → A (exists_admissibleEmbedding). The point z transports along
the flat pullback (exists_restrictedCenterHasPointOver_pullback_iff), across the deletions of
empty blow-ups (exists_restrictedCenterHasPointOver_bot_eraseEmpty_iff) and along the closed
immersion (restrictedCenterHasPointOver_iff_of_isClosedImmersion) to a centre of the BP run
with a point over emb x, x a smooth point of U (Scheme.Hom.preimage_smoothLocus_eq) —
against not_restrictedCenterHasPointOver_BP_of_mem_smoothLocus
(Hironaka.Resolution.Algebraic.Kol07.Thm36.AffineTheorem36IsoSnc): no centre of the
principalization sequence of I_X has a point over a smooth point of X.
The single-blow-up form (resolution_eq_single_blowUp): Π : X_r → X is one blow-up
K.blowUp over X with |K| = ⋃ Π_i(|Z_i|) ⊆ Sing X
(AlgebraicGeometry.exists_blowUp_composite_iso: the composite of a sequence of blow-ups is a
single blow-up, the composition of blow-ups [TSPA, Tag 080B] iterated).
The theorem (exists_support_eq_singularLocus_isRegular_blowUp):
exists_center_support_eq_singularLocus at S := BR X, with clause (1)
(BR_smooth_last_of_class) and the centres over Sing X as its two hypotheses. The
theorem is AlgebraicGeometry.exists_support_eq_singularLocus_isRegular_blowUp, below.
The weak form (AlgebraicGeometry.exists_dense_isIso_restrict_isRegular_blowUp, at the
end of the file) is Hironaka's weakening of the first condition [Hir64, p. 132]: the blow-up is an
isomorphism over the complement of its centre (blowUp.isIso_π_restrict_compl_support), and the
complement of Sing X is a nonempty open of the irreducible X, hence dense, since the local ring
at the generic point is a field (isRegularAt_genericPoint_of_isIntegral).
The local step: no centre of BR X has a point over a smooth point x = f w lying in an
affine open U of X (the argument of [Kol07, 4.2] and of the proof of [Kol07, Theorem 27]).
Through BR U = ((BR X).pullback f).eraseEmpty (clause (4a) for the open immersion f),
BR U = (BR_affine' k U).eraseEmpty on the integral affine U, the transports of the centre
point, and not_restrictedCenterHasPointOver_BP_of_mem_smoothLocus on an admissible embedding of
U.
No centre of BR X has a point over a smooth point of X — clause (2) of [Kol07, Theorem 36]
read as "no centre over the smooth locus": the local step on the member of the finite affine cover
through the point.
Every point of every centre of BR X lies over the singular locus of X, the complement of
the smooth locus (Scheme.singularLocus_eq_compl_smoothLocus).
The single-blow-up form: the resolution Π : X_r → X of the resolution functor is one blow-up
of X in a centre whose support lies in Sing X (AlgebraicGeometry.exists_blowUp_composite_iso:
the support of the centre is the union of the images of the centres of the sequence).
Hironaka's Main Theorem I [Hir64, Main Theorem I, p. 132]. Let X be a reduced and
irreducible (IsIntegral) algebraic k-scheme (AlgScheme k), k of characteristic zero. Then
there is a closed subscheme D of X, given by its ideal sheaf, whose set of points is exactly the
singular locus of X (the points whose local ring is not regular), such that the monoidal
transformation D.blowUp of X with centre D is non-singular. If X is non-singular, D is the
empty subscheme (D = ⊤) and D.blowUpπ is an isomorphism.
Relation to the source.
- Translation.
(D.support : Set X.left) = X.left.singularLocusis Hironaka's (i), "the set of points of $D$ is exactly the singular locus of $X$";IsRegular D.blowUpis his (ii), the non-singularity of the result of the monoidal transformation of $X$ with centre $D$. - Interpretation. Hironaka's "say reduced and irreducible" is read as a hypothesis
(
IsIntegral). His Introduction describes Main Theorem I as "the resolution of singularities of an arbitary [sic] reduced algebraic B-scheme X" [Hir64, p. 112], irreducible or not; reducibleXis not covered here. - Out of scope. Hironaka's strong form [Hir64, pp. 132–133], that
D.blowUpπis the composite of a finite succession of monoidal transformations whose centres $D_i$ are non-singular (a), contain no simple point of $X_i$ (b), and along which $X_i$ is normally flat (c), and Main Theorem I*: both rest on the Hilbert–Samuel function and normal flatness, whereas the library follows the route through the order of an ideal and hypersurfaces of maximal contact, as in Włodarczyk's and Kollár's proofs. The statement here is Main Theorem I as printed.
Source statement: [Hir64, Main Theorem I, p. 132]
Hironaka's Main Theorem I, weak form [Hir64, p. 132, the paragraph after Main Theorem I].
Let X be a reduced and irreducible (IsIntegral) algebraic k-scheme (AlgScheme k), k of
characteristic zero. Then there is a closed subscheme D of X whose open complement U is dense
in X, such that the monoidal transformation f : D.blowUp → X with centre D restricts to an
isomorphism f⁻¹(U) → U and D.blowUp is non-singular.
Density of U excludes D = X.
Relation to the source.
- Translation.
D.support.complis the open complement $U = X \setminus D$, andIsIso (D.blowUpπ ∣_ U)says that the restriction $f^{-1}(U) \to U$ of the projectionD.blowUpπover $U$ is an isomorphism. - Interpretation. As in Main Theorem I, Hironaka's "say reduced and irreducible" is read as a
hypothesis (
IsIntegral), although his Introduction [Hir64, p. 112] describes Main Theorem I for every reduced $X$.
Source statement: [Hir64, the paragraph after Main Theorem I, p. 132]