Tools for the loop of BED along a smooth surjection #
Clause (d) of [Wlo05, Theorem 1.0.2] ([Wlo05, 4.1]; the first clause of [Kol07, 34.1]): the
embedded desingularization sequence commutes with smooth morphisms. The loop of
Hironaka.Resolution.Algebraic.Wlo05.Embedded runs BMO_1 and stops at the first stage where a
centre contains the strict transform of one of the remaining components Yᵢ of Y. Along a smooth
surjection h : X' → X carrying the pullback data, BMO_1 itself commutes ([Kol07, Theorem 69]),
and this module supplies the bookkeeping of the STOP RULE and of the COMPONENTS:
- the irreducible components of
h⁻¹(Y)are the components of theh⁻¹(Yᵢ)(mem_genericPoints_preimage_iff); the relationIsPullbackComponents h C C'records "C'= the components of the preimages of the members ofC", the loop's invariant alongh; - the strict transform of a finite union is the union of the strict transforms
(
coe_support_strictTransformSeq_biUnion), and strict transforms are monotone in the ideal (strictTransformSeq_mono); - the stop rules coincide stagewise (
stopRule_aux): downstairs "Z_n ⊇ X̄_n(Yᵢ)" gives upstairs "Z'_n ⊇ X̄'_n(Y'_{ij})" for every componentY'_{ij}ofh⁻¹(Yᵢ)by monotonicity (h⁻¹ X̄_n(Yᵢ) = X̄'_n(h⁻¹ Yᵢ) ⊆ X̄'_n(Y'_{ij})); conversely the generic point ofX̄'_n(Y'_{ij})(the proof of [Kol07, Corollary 22]: the strict transform of an integral scheme has a generic point over the original one before its first containing centre,exists_isGenericPoint_strictTransformSeq) lies over the generic point ofX̄_n(Yᵢ), so a centreZ'_n = h_n⁻¹(Z_n)containing the former forcesZ_nto contain the latter — both directions need that no earlier centre absorbed the component, which is exactly the loop's situation at its first absorbing stage (proved by strong induction on the stage); - the truncation
S.take nhas the same stop rule asSbelown(centerContains_take_iff,Hironaka.Resolution.Algebraic.Kol07.Thm36.CentersMiss), and the last-stage lift of a truncation carries the strict transforms and stage maps (strictTransformSeq_pullback_last,pullbackLastHom_stageMap).
The pullback and pushforward of blow-up sequences are those of [Kol07, Definition 30]. Used by
Hironaka.Resolution.Algebraic.Wlo05.EmbeddedPullback (the functoriality clause) and by the CP
modules.
Generic points of finite unions and of preimages #
A closed set is the union of the closures of its generic points (every point is a specialization of one of them).
The generic point of a member of a finite family of closed sets that is maximal among the members is a generic point of the union.
A generic point of a finite union of irreducible closed sets is the generic point of one of them.
Under a continuous open map from a Noetherian sober space, the generic points of the preimage
of Z are the generic points of the preimages of the components of Z ([Wlo05, 4.1], read on the
components of Y).
Reduced ideals of closed sets #
A reduced ideal sheaf is the reduced ideal of its support.
The reduced ideal of the closure of a point cuts out an integral closed subscheme.
A reduced strict transform with a generic point is the reduced ideal of the closure of that point.
The components relation along h #
C' is the family of the irreducible components of the preimages h⁻¹(V(c)), c ∈ C — the
reduced ideals of the closures of the generic points of the h⁻¹(V(c)): the loop's invariant along
a smooth h ([Wlo05, 4.1]).
Equations
- One or more equations did not get rendered due to their size.
Instances For
A component of h⁻¹(V(I_{closure {η}})) along a smooth h lies over η.
The inverse image of the reduced ideal of a component is contained in the reduced ideal of
each component of its preimage (V(c') ⊆ h⁻¹(V(c))).
The support of h⁻¹(V(c)) is the union of the components' supports (indexed by their
generic points).
The base case: the components of V(h^* I) #
The members of componentIdeals T are reduced ideals of closures of points.
Along a smooth h carrying the pullback data the components of V(h^* I_Y) are the
components of the preimages of the components of V(I_Y) ([Wlo05, 4.1]).
The stop rules coincide #
Downstairs to upstairs (monotonicity): if V(c') ⊆ h⁻¹(V(c)), a centre containing the strict
transform of V(c) pulls back to a centre containing the strict transform of V(c').
Upstairs to downstairs (the proof of [Kol07, Corollary 22]): before any absorption of V(c) or
of the component V(c') of its preimage, the generic point of the strict transform of V(c') lies
over the generic point of the strict transform of V(c); a centre h_n⁻¹(Z_n) containing the
former forces Z_n to contain the latter.
The stop rules coincide (the proof of [Wlo05, Theorem 4.7.1] along h) — for a family C
of reduced ideals of irreducible closed sets and C' the components of their preimages, if no
member of C is absorbed before stage n, then no member of C' is absorbed before stage n,
and at stage n a component V(c') of h⁻¹(V(c)) is absorbed by the pulled-back run exactly when
V(c) is absorbed by the run. Strong induction on n.
Transport to the truncation and to the last-stage lift #
No absorption of V(c) before stage n gives the history hypothesis of the generic point
lemma on the truncation S.take n.
The last-stage lift lies over h ([Kol07, Definition 30, 30.1]).
The proof of [Kol07, Corollary 22] on both sides of h: for a component V(I_{closure {η'}})
of h⁻¹(closure {η}), with no absorption of either along the truncated run Q, the generic points
ηQ and ηQ' of the two strict transforms at the last stage exist, the last-stage lift carries
ηQ' to ηQ, and ηQ' lies over η'.