The dictionary, part II: total transforms and boundaries, and simple normal crossings as only #
normal crossings
The second part of the translation between Kollár's divisor families and Hironaka's boundaries
(definitions in Hironaka/Scheme/Snc/Dictionary.lean):
- Total transforms. The members of Kollár's total transform
π⁻¹_tot(F) = (π_*^{-1} F^j)_j + F_exc[Kol07, Definition 25] cover exactlyπ⁻¹(V(F)) ∪ π⁻¹(V(D)): the support of a strict transform is the closure of the part off the exceptional divisor (coe_support_saturate, on a locally Noetherian scheme), so it lies inπ⁻¹(V(F^j)), and the part ofπ⁻¹(V(F^j))over the center is inside the exceptional divisorπ⁻¹(V(D)), the last member. Hence the reduced union of the total transform is Hironaka'sred(f⁻¹(E) ∪ f⁻¹(D))(reducedTransform) of the reduced union, and along a sequence Hironaka's boundariesE_iof [Hir64, Main Theorem II (iii)] are the reduced unions of Kollár'sE_i; for a reducedE, of the total transforms of the component family ofE. - Kollár's simple normal crossings give Hironaka's only normal crossings, for any family. At a
point
xofZwith Kollár's coordinatesz[Kol07, Definition 24], the members throughxare(z_{c(i)} = 0)withcinjective; the reduced union of the family is the radical of the product of the members (vanishingIdeal_support), its stalk atxhas the minimal primes of(∏ z_{c(i)})(stalks commute with products,stalkIdeal_mul; radicals do not change minimal primes,Ideal.radical_minimalPrimes, and the stalk of a radical is sandwiched between the stalk and its radical), and these are the(z_{c(i)})(IsLocalRing.minimalPrimes_span_prod, eachz_{c(i)}prime since𝒪_{X,x}/(z_{c(i)})is a regular local ring). This is [Hir64, Definition 2] atx; withD = X(⊥) it is "only normal crossings". - Clause (iii) of Main Theorem II. Condition (3) of [Kol07, Definition 66] for the total
transform of the component family of
Eat stageigives Hironaka's clause (iii) for the boundaryE_i: the stage is smooth overk(IsSmooth.stageMap_smoothOfRelativeDimension) and the boundary is the reduced union.
The converse direction, for the family of irreducible components, is in
Hironaka/Resolution/Algebraic/Snc/DictionaryComponents.lean.
The support of a family, as a set, is the finite union of the supports of its members.
The support of the strict transform of J under the blow-up with center D is the closure of
π⁻¹(V(J)) ∖ π⁻¹(V(D)), on a locally Noetherian scheme (coe_support_saturate); this is the
strict transform of [Hau14, Definition 6.2].
The members of the total transform [Kol07, Definition 25] of a family under the blow-up with
center D cover exactly π⁻¹(V(F)) ∪ π⁻¹(V(D)): each strict transform lies in π⁻¹(V(F^j)), and
the part of π⁻¹(V(F^j)) over the center lies in the exceptional divisor, the last member.
The reduced union of the total transform of a family [Kol07, Definition 25] is Hironaka's
reduced total transform red(f⁻¹(E) ∪ f⁻¹(D)) [Hir64, Main Theorem II (iii)] of its reduced union
(reducedTransform).
boundarySeq_eq_unionIdeal_totalTransformSeq, the induction, with the locally Noetherian
hypothesis as an explicit argument so that it is carried to the blow-ups.
Hironaka's boundaries E_i [Hir64, Main Theorem II (iii)], started at the reduced union of a
family, are the reduced unions of Kollár's total transforms [Kol07, Definition 66 (1)] at every
stage of a sequence.
For a reduced E, the boundaries E_i of [Hir64, Main Theorem II] are the reduced unions of
Kollár's total transforms of the component family of E.
Kollár's simple normal crossings give Hironaka's only normal crossings (any family) #
The support of a finite product of ideal sheaves is the union of the supports (support_mul
iterated).
The reduced union of a family is the radical of the product of its members: both are the
vanishing ideal sheaf of ⋃ᵢ V(E^i) (vanishingIdeal_support).
The stalk of an ideal sheaf lies in the stalk of its radical.
The stalk of the radical lies in the radical of the stalk (Ideal.map_radical_le on an affine
open computing the stalk).
The stalk of the radical of an ideal sheaf has the same minimal primes as the stalk of the ideal sheaf (both stalks have the same radical).
A coordinate of a regular system of parameters that generates the stalk of an ideal sheaf is a
prime element: the quotient by it is a regular local ring, hence a domain, and the coordinate is
nonzero (it is not in 𝔪²).
From Kollár's simple normal crossings [Kol07, Definition 24 (4)] to Hironaka's only normal
crossings [Hir64, Definition 2], for an arbitrary family: if Z has simple normal crossings with
F then the reduced union of F has only normal crossings with Z. At x ∈ Z Kollár's
coordinates make the members through x the hyperplanes (z_{c(i)} = 0), whose union has the
minimal primes (z_{c(i)}) (prod_stalkIdeal_eq_span with minimalPrimes_span_prod).
The reduced union of a simple normal crossing family [Kol07, Definition 24 (1)–(3)] has only
normal crossings Hir64, Definition 2 with D = X.
From condition (3) of [Kol07, Definition 66] to clause (iii) of [Hir64, Main Theorem II]:
along a smooth blow-up sequence, if the center Z_i has simple normal crossings with Kollár's
total transform E_i of the component family of E, then Hironaka's boundary E_i has only
normal crossings with Z_i; the one-directional dictionary at the stage X_i (smooth over k)
with the boundary identified as the reduced union of the total transform.
Along a smooth blow-up sequence of order d starting with (X, J, E.componentFamily), every
boundary E_i has only normal crossings with the center D_i: clause (iii) of
[Hir64, Main Theorem II] in the form used by its proof.