The monomial split under a smooth blow-up of order at least the mark #
[Kol07, 111, Step 1] observes that the birational transforms (Π_1)^{-1}_* N(I) and
(Π_1)^{-1}_*(I, m) differ only by an ideal sheaf of exceptional divisors of Π_1, hence only in
their monomial part, so that N((Π_1)^{-1}_*(I, m)) = (Π_1)^{-1}_* N(I). Here, for one smooth
blow-up
π : B_Z X → X of order ≥ m for the marked triple (X, I, m, E) (conditions (1′)–(2′) of
[Kol07, Definition 65], MarkedTriple.IsOrderGeBlowUp) with total transform E' = π^{-1}_{tot} E:
𝒪(mF) · π^{-1}_*(I, m) = π^* I = π^* M(I) · π^* N(I) (pow_dvd_comap_of_leOrdAlong,
pow_mul_markedTransform and the split), where 𝒪(−mF) and π^* M(I) are monomials of E',
being invertible ideal sheaves supported on E' (eq_monomialPart_of_isInvertible_of_support_le
on the marked triple induced after the blow-up), and the nonmonomial part ignores monomials
(nonmonomialPart_monomial_mul). The same argument with any control c such that
F^c ∣ π^* N(I) gives the birational transform (the exceptional order, whose power always divides)
and the marked transform π^{-1}_*(N(I), a) for ord_Z N(I) ≥ a.
blowUpTriple: the marked triple(B_Z X, π^{-1}_*(I, m), m, π^{-1}_{tot} E)induced after the blow-up (MarkedTriple.inducedon the one-step sequence).exists_comap_monomialPart_eq_monomial(π^* M(I)is a monomial ofE'),nonmonomialPart_controlledTransform_eq(the common core),nonmonomialPart_markedTransform(N(π^{-1}_*(I, m)) = N(π^* N(I))),nonmonomialPart_markedTransform_eq_weakTransform(with the birational transform ofN(I)),nonmonomialPart_markedTransform_of_le_ord(with the marked transformπ^{-1}_*(N(I), a)).
These identities are iterated along the rounds of Steps 1 and 2 of the proof of
[Kol07, Theorem 107]
(Hironaka/Resolution/Algebraic/MarkedOrderReduction/Step1NonmonomialPart.lean,
NonmonomialInvariant.lean). The general facts on exceptional orders and nonvanishing used here are
in Hironaka/Scheme/BlowUpSequence/MarkedTripleBlowUp.lean and
Hironaka/Resolution/Algebraic/Snc/DictionaryGenericPoints.lean.
The support of the monomial part lies in the support of the family: off E every member's
inner product is the unit ideal.
The marked triple (B_Z X, π^{-1}_*(I, m), m, π^{-1}_{tot} E) induced after the blow-up
(MarkedTriple.induced at stage 1 of the one-step sequence).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The identity (60.1) of [Kol07, Definition 60]: F^m · π^{-1}_*(I, m) = π^* I for a blow-up of
order ≥ m (pow_dvd_comap_of_leOrdAlong).
An invertible ideal sheaf on B_Z X supported on the exceptional divisor or over E is a
monomial of the total transform π^{-1}_{tot} E.
π^* M(I) is a monomial of π^{-1}_{tot} E ([Kol07, 111, Step 1]: the two transforms "differ
only by tensoring with an ideal sheaf of exceptional divisors"): it is invertible (the pull-back of
an invertible ideal sheaf along the blow-up) and supported on
π^{-1}(supp E) ⊆ supp(π^{-1}_{tot} E), a point off the exceptional divisor lying on the strict
transform of the member through its image.
The common core of the three identities below: for a control c with F^c ∣ π^* J and J
nonvanishing, the nonmonomial part of the controlled transform (π^* J : F^c) is that of π^* J;
they differ by the monomial F^c, which N ignores.
π^* I is nonvanishing: it is F^m · π^{-1}_*(I, m), both factors nonvanishing.
N(π^{-1}_*(I, m)) = N(π^* N(I)) ([Kol07, 111, Step 1], the identity
"N((Π_1)^{-1}_*(I, m)) = (Π_1)^{-1}_* N(I)" read modulo N on the right).
N(π^{-1}_*(I, m)) is the nonmonomial part of the birational transform π^{-1}_* N(I) (the
weak transform, the controlled transform at the exceptional order); [Kol07, 111, Step 1] read
modulo N.
For ord_Z N(I) ≥ a, N(π^{-1}_*(I, m)) = N(π^{-1}_*(N(I), a)): the marked transform of
N(I) at any admissible mark has the same nonmonomial part.