The quotient I_F^{-m} π⁻¹(I) exists; the birational transform of a marked ideal #
The exceptional ideal I_F = π^*I_Y (hY.idealSheaf.pullback π h.contMDiff) is the ideal sheaf of
the
exceptional divisor F = π⁻¹(Y): in a blow-up chart of index i its stalk is generated by the
scaling coordinate u_i, and off F it is the unit ideal (isIdealSheafOf_exceptionalIdealSheaf,
stalkIdeal_exceptionalIdealSheaf_eq_span_coord, stalkIdeal_exceptionalIdealSheaf_of_notMem:
the results of Hironaka.Manifold.BlowUp.Divisor on the pullback of the ideal sheaf of the
centre, applied to I_Y).
If the order of I along the centre Y is at least m, the total transform π⁻¹(I) lies in
I_F^m (totalTransform_le_pow_exceptional; stalkwise, I_a ⊆ I_{Y,a}^m maps to
(π^*I)_{a'} ⊆ (π^*I_Y)^m_{a'}, and off the exceptional divisor I_F is the unit ideal), and the
quotient, specified stalkwise by the colon ideals (π⁻¹(I)_{a'} : I_{F,a'}^m)
(IdealSheaf.IsDivExceptional), exists as a locally finitely generated ideal sheaf
(exists_isDivExceptional), uniquely (IsDivExceptional.unique). This is the division of the
total transform by a power of the ideal of the exceptional divisor in Hironaka's weak transform
[Hir64, Ch. 0, §5, p. 142], in Bierstone–Milman's [BM97, Remark 1.8] and in Kollár's birational
transform of a marked ideal [Kol07, Definition 60, (60.1)]. The local generators are Kollár's
u_i^{-m}(f ∘ π) [Kol07, Definition 60, (60.3)]: on a blow-up chart Φ of index i, f_j ∘ π
(the pulled-back local generators f_j of I) is divisible by u_i^m as a germ at every point
of the exceptional divisor {u_i = 0}, so the iterated canonical quotient of
Hironaka.Manifold.BlowUp.Transform.Division gives analytic sections g_j on the chart with
u_i^m g_j = f_j ∘ π; at a point of the exceptional divisor the colon (span(u_i^m g_j) : u_i^m)
is span(g_j) because the stalk is a domain (it is isomorphic to the ring of convergent power
series) and u_i ≠ 0, and off it u_i is a unit, so the colon by the unit ideal is
span(u_i^m g_j) = span(g_j). The exponent is allowed to vary with the point as long as it is
locally constant on the exceptional divisor, so that the same lemma serves the birational
transform of a marked ideal (constant m, here) and Hironaka's weak transform (the generic order
along each component of the centre, Hironaka.Manifold.BlowUp.Transform.Weak). The birational
transform MarkedIdealSheaf.birationalTransform has the colon stalks whenever m ≤ ν_Y(I)
(isDivExceptional_birationalTransform) and keeps the marking (birationalTransform_m); in a
blow-up chart the quotient's stalk is spanned by any germs g_j with u_i^m g_j = f_j ∘ π,
whatever the generators (IsDivExceptional.stalkIdeal_eq_span).
ν_{D,a}(J) ≥ p iff J_a ⊆ D_a^p: the order along D is the largest p with J_a ⊆ D_a^p
(the set of such p is downward closed) [BM97, §3].
A coordinate germ of a chart is nonzero: its own coordinate derivative is 1.
The stalk of the structure sheaf is a domain: it is isomorphic to the ring of convergent power series.
The exceptional ideal #
I_F = π^*I_Y is the ideal sheaf of the exceptional divisor F = π⁻¹(Y).
In a blow-up chart of index i the stalk of I_F is generated by the scaling coordinate
u_i.
Off the exceptional divisor the stalk of I_F is the unit ideal.
The quotient I_F^{-m} π⁻¹(I) #
Stalkwise: if ν_Y(I) ≥ m at π a' then π⁻¹(I)_{a'} ⊆ I_{F,a'}^m.
If ν_Y(I) ≥ m along Y then π⁻¹(I) ⊆ I_F^m [Hir64, Ch. 0, §5, p. 142].
The scaling coordinate of a blow-up chart is a unit off the exceptional divisor.
The colon stalks (π⁻¹(I)_{a'} : I_{F,a'}^{m a'}) have local generators, for any exponent m
that is locally constant on the exceptional divisor and at most the order of I along the
centre: the divided generators u_i^{-m}(f_j ∘ π) on a blow-up chart
[Kol07, Definition 60, (60.3)], the pulled-back generators off the exceptional divisor.
The quotient I_F^{-m} π⁻¹(I) exists as a locally finitely generated ideal sheaf when
ν_Y(I) ≥ m along Y ([Hir64, Ch. 0, §5, p. 142]; [BM97, Remark 1.8]).
The quotient is determined by its stalks.
The birational transform of (I, m), m ≤ ν_Y(I), has the colon stalks
[Kol07, Definition 60, (60.1)].
The birational transform keeps the marking.
In a blow-up chart of index i, the quotient I_F^{-m} π⁻¹(I) is generated by germs g_j
with u_i^m g_j = f_j ∘ π, for any local generators f_j of I, whatever the generators
[Kol07, Definition 60, (60.3)].