The clauses of Włodarczyk's Theorem 1.0.2 for the functor EDFunctor #
The clauses of [Wlo05, Theorem 1.0.2] for the embedded desingularization functor
Hironaka.Resolution.EDFunctor (Hironaka.Resolution.Algebraic.Wlo05.EmbeddedFunctor), in the form
used by AlgebraicGeometry.exists_functorial_embeddedDesingularization, stated and proved at the
end of this file: for a smooth, quasi-compact, separated X of finite type over
k, equidimensional (hX), and a REDUCED closed subscheme Y. Nothing is proved twice: the proofs
live here and the main theorem applies them.
Why the statements hold. The functor runs BED (Hironaka.Resolution.Algebraic.Wlo05.Embedded)
on the CORE of Y — the reduced ideal of V(Y) ∩ X₁, X₁ the union of the irreducible components
of X not inside V(Y) (coreIdeal, componentsOutside) — which is nonzero everywhere, as the
triples of BED require (EDFunctor_seq_of_pos). On the smooth X the components are pairwise
disjoint, so X = X₀ ⊔ X₁ is a clopen split with Y|_{X₀} = 0 and core|_{X₁} = Y|_{X₁}
(Hironaka.Resolution.Algebraic.Wlo05.EmbeddedFunctorSplit). Hence:
- the clauses about the run alone — smooth centres, and clause (a) — are those of
BEDat the core triple (BED_center_smooth,BED_isSnc_totalTransformSeq,BED_hasSncWith_center,Hironaka.Resolution.Algebraic.Wlo05.EmbeddedOrderGe); - clause (b): a point of a centre lies over
X₁(no centre lies overX₀: the run has order≥ 1for the core, whose cosupport missesX₀), where the strict transforms and the smooth points ofYand of its core agree; thenBED_center_disjoint_regat the core (Hironaka.Resolution.Algebraic.Wlo05.EmbeddedRegWindow); - clause (c) and the fine form of clause (e): the end data of the CP loop at the core
(
BED_strictTransform_last_smooth_of_embeddedEnd,BED_hasSncWith_strictTransform_last_of_embeddedEnd,BED_comap_composite_eq_strictTransform_mul_sncDivisor_of_embeddedEndatembeddedEnd_BED_of_invCE,Hironaka.Resolution.Algebraic.Wlo05.EmbeddedEndBridgeandHironaka.Resolution.Algebraic.Wlo05.EmbeddedCPBridge), transported toY: overX₁the last strict transforms agree, overX₀that ofYis zero on a smooth stage (smooth_strictTransformSeq_last_of_coreIdeal,hasSncWith_strictTransformSeq_last_of_coreIdeal,comap_composite_eq_strictTransform_mul_of_coreIdeal); - clause (d), in its two forms: the core commutes with a smooth
h(coreIdeal_comap, the exchange of components for flath), the empty boundary pulls back to itself, and the hypothesis that the image ofhmeets every component transfers to the core (hmeets_coreIdeal); thenbed_pullback_of_surjectiveandbed_pullback_eraseEmptyat the two core triples (Hironaka.Resolution.Algebraic.Wlo05.EmbeddedPullback,Hironaka.Resolution.Algebraic.Wlo05.EmbeddedEraseEmpty; the commutation with smooth morphisms of [Kol07, 34.1]).
The triples are those of [Kol07, Notation 64]. The reduction to the core is not in the literature.
Smooth centres for the functor: BED_center_smooth at the core triple
(EDFunctor_seq_of_pos).
Clause (a) of [Wlo05, Theorem 1.0.2], first part, for the functor:
BED_isSnc_totalTransformSeq at the core triple.
Clause (a) of [Wlo05, Theorem 1.0.2], second part, for the functor: BED_hasSncWith_center at
the core triple.
Clause (b) of [Wlo05, Theorem 1.0.2] for the functor: a point of a centre lies over X₁,
where the strict transforms and the smooth points of Y and of its core agree
(stageMap_notMem_image_smoothLocus_of_coreIdeal); then BED_center_disjoint_reg at the core
triple.
Clause (c) of [Wlo05, Theorem 1.0.2], first part, for the functor: the clause for BED at the
core triple (BED_strictTransform_last_smooth_of_embeddedEnd at embeddedEnd_BED_of_invCE),
transported to Y (smooth_strictTransformSeq_last_of_coreIdeal: over X₁ the last strict
transforms agree, over X₀ that of Y is zero on the smooth last stage).
Clause (c) of [Wlo05, Theorem 1.0.2], second part, for the functor: the clause for BED at
the core triple (BED_hasSncWith_strictTransform_last_of_embeddedEnd at
embeddedEnd_BED_of_invCE), transported to Y
(hasSncWith_strictTransformSeq_last_of_coreIdeal).
Clause (e) of [Wlo05, Theorem 1.0.2] in the fine form for the functor: the clause for BED at
the core triple (BED_comap_composite_eq_strictTransform_mul_sncDivisor_of_embeddedEnd at
embeddedEnd_BED_of_invCE), with the same snc divisor J; the factorisation transfers stalkwise
(comap_composite_eq_strictTransform_mul_of_coreIdeal).
Clause (d) of [Wlo05, Theorem 1.0.2], for a smooth surjection: the core commutes with the
smooth h (coreIdeal_comap), so the two values are BED at the two core triples, one the
pull-back data of the other; then bed_pullback_of_surjective.
Clause (d) of [Wlo05, Theorem 1.0.2], for a smooth morphism whose image meets every irreducible
component of Y: as for a surjection, with the image hypothesis transferred to the core
(hmeets_coreIdeal); then bed_pullback_eraseEmpty.
The side condition of the deletion form of clause (d) of [Wlo05, Theorem 1.0.2] in the form
EDFunctor.pullback_eraseEmpty takes it: if the image of h meets every irreducible component of
the closed subscheme Y, it meets the closure of every point of V(Y) maximal for
specialization. Such a point η = ι y is the generic point of the component closure {y} of Y
(a generic point g of an irreducible Z ⊇ closure {y} specializes to y, so ι g = η by
maximality and g = y), and ι carries closure {y} into closure {η}.
Włodarczyk's embedded desingularization with smooth centres [Wlo05, Theorem 1.0.2]. There
is a family ED of blow-up sequence functors on embedded pairs
(BlowUpSequenceFunctor (EmbeddedPair k)), one for each field k of characteristic zero, such
that
- every value
S = ED(X, Y), with compositeσ : X_r → X, exceptional divisorsE_i(the total transforms of the empty divisor family) and strict transformsY_iofY, is an embedded desingularization of the pair (EmbeddedPair.IsDesingularizedBy): every centreC_iis smooth overk; (a) everyE_i, the last included, is a simple normal crossing family, andC_ihas simple normal crossings withE_i; (b) no point ofC_i ∩ Y_ilies over a point ofReg(Y), the image inXof the smooth locus ofY → Spec k(the points whereY, notY_i, is smooth); (c)Ỹ = Y_ris smooth overkand has simple normal crossings withE_r; (e)σ^*(I_Y) = I_Ỹ · Jfor the ideal sheafJof a simple normal crossing divisor supported inE_r; - (d) for a morphism of embedded pairs
h : (X', Y') ⟶ (X, Y)whose underlying morphism is smooth,ED(X', Y')is the pull-backh^* Sifhis surjective, andh^* Swith the empty blow-ups deleted if the image ofhmeets every irreducible component ofY(CommutesWithSmoothMorphismsMeetingComponents).
An embedded pair (EmbeddedPair) is a reduced closed subscheme Y of a smooth, equidimensional
algebraic k-scheme X (smooth of one relative dimension over k); a morphism
(X', Y') ⟶ (X, Y) is a k-morphism h : X' → X along which the ideal sheaf of Y pulls back to
that of Y'. If Y = X (the ideal sheaf ⊥), (b) forces every centre to be empty. If Y is
empty or smooth, the clauses on S alone hold for the empty succession.
Relation to the source.
- Translation.
S.totalTransformSeq (DivisorFamily.empty _) iis Włodarczyk's exceptional divisor $E_i$,S.strictTransformSeq P.Y ihis strict transform $Y_i$, and the image inXof the smooth locus ofY → Spec kis his $\operatorname{Reg}(Y)$. - Interpretation. Włodarczyk's "subvariety $Y$ of a smooth variety $X$" is read as a reduced
closed subscheme
Yof a smooth, equidimensional algebraick-schemeX(EmbeddedPair). His Bravo–Villamayor strengthening, the source of clause (e), is stated for "a reduced closed subscheme" [Wlo05, Theorem 4.7.1]. - Gap. In clause (e), $\sigma^*(I_Y) = I_{\tilde Y} I_{\tilde E}$, Włodarczyk's divisor
$\tilde E$ is "a natural combination of the irreducible components of the divisor $E_r$"; here
Jis only asked to be the ideal sheaf of a simple normal crossing divisor supported in $E_r$. - Gap. In clause (d) the deletion form, in which
ED(X', Y')is the pull-back ofED(X, Y)with the empty blow-ups deleted, is asserted only when the image ofhmeets every component ofY, whereas Włodarczyk's (d), read through his Definition 2.1.5 (extensions by isomorphisms) and Proposition 2.4.2, asserts it for every smoothh. The construction handles the components ofYin rounds whose boundary data depend on all of them, and over an open subset missing a component the runs fall out of step: forYthe disjoint union of a smooth surface and the three coordinate axes through a point off it, in a smooth threefold, the pull-back ofED(X, Y)to the complement of the surface has three non-empty centres that the run on that complement lacks. - Gap. The parts of Włodarczyk's (d) on embeddings of ambient varieties and on equivariance
under group actions have no counterpart; in particular, although the family
EDhas one functor for each field, no clause relates the functors of different fields, as Kollár's change of fields [Kol07, 34.2] (CommutesWithChangeOfFields) does. The functor built here is not Włodarczyk's single invariant-driven run, and functoriality for closed embeddings is, in Kollár's words, "quite delicate" [Kol07, 34 and Claim 71.2]: he proves it only for an empty boundary, whereas the later rounds here run with the exceptional divisors of the earlier ones as boundary.
Source statement: [Wlo05, Theorem 1.0.2]