Kollár's Lemma 62 for one blowing-up #
For Z ⊆ H ⊂ X and π : B_Z X → X with π_H : B_Z H → H the restricted blow-down,
(π_H)_*^{-1}(I|_H, m) = π_*^{-1}(I, m)|_{B_Z H} when m ≤ ord_Z I [Kol07, Lemma 62]. The proof
here is stalkwise and coordinate-free, like that of the algebraic form in Hironaka/: at a point
p' of the strict transform S', both marked transforms are colon ideals (IsDivExceptional),
the total transforms and the exceptional ideal sheaves correspond under the restriction of germs
ρ = 𝒪_{M', p'} → 𝒪_{S', p'} (pull-back functoriality; exceptionalIdealSheaf_restrictMap,
idealSheaf_preimageVal_eq_pullback), and the colon by u^m commutes with ρ
(Ideal.map_colon_singleton_pow_eq): the total transform is divisible by u^m and ρ(u) ≠ 0
because the exceptional coordinate u = u_{σ i} does not vanish identically on S' — at a point
of S' the blow-up chart is a good chart (i ∉ range τ),
where S' = {u_{σ (τ j)} = 0} and u_{σ i} lies outside the ideal of S'
(coord_notMem_span_coord_sub_const). Off the centre both sides are plain pull-backs. This is
Kollár's argument: it does not matter whether one first sets x_1 = 0 and computes the transform
or first computes the transform and then sets y_1 = 0, and the bad chart contains no point of
B_Z H [Kol07, Lemma 62, proof].
Also: the order can only go up under restriction (the opening remark of [Kol07, §9];
ord_le_ord_pullback at the closed embedding), and the centres of a succession with centres in the
strict transforms of H are recovered from those of its restriction to H (the injection of the
going-down property of maximal contact [Kol07, 51.1], read on FiniteSuccession).
The order can only go up under restriction to a closed submanifold (ord_le_ord_pullback at
the closed embedding S ↪ M; the opening remark of [Kol07, §9]).
The two closed embeddings and the two blow-downs commute: ι_S ∘ π|_{S'} = π ∘ ι_{S'}
(restrictMap_apply, rfl).
The total transform of the restriction is the restriction of the total transform (pull-back
functoriality along ι_S ∘ π|_{S'} = π ∘ ι_{S'}).
The exceptional ideal sheaf of π|_{S'} is the restriction to S' of the exceptional ideal
sheaf I_F of π (the exceptional divisor of the restricted blowing-up is the trace of F; in
Kollár's chart the birational transform of H is (y_1 = 0), [Kol07, Lemma 62, proof]).
The mark m ≤ ord_Y J is legitimate for the restriction (J|_S, m) along Y ⊆ S
(restriction is a ring map, so J_a ⊆ I_{Y,a}^m restricts).
The exceptional coordinate u = u_{σ i} of a good blow-up chart does not vanish identically on
the strict transform S' near its points: S' = {u_{σ (τ j)} = 0} there and u_{σ i} lies outside
the ideal spanned by the u_{σ (τ j)}, the kernel of the restriction of germs.
Kollár's Lemma 62 for one blowing-up [Kol07, Lemma 62]: the marked transform of (J|_S, m)
under the restricted blow-down π|_{S'} : S' → S is the restriction to S' of the marked
transform π_*^{-1}(J, m).
Every centre Z_i of a succession with centres in the strict transforms of H is the image
of the centre Z_i ∩ H_i of the restricted succession under the embedding H_i ↪ M_i: the
injection of the going-down property of maximal contact [Kol07, 51.1], on FiniteSuccession.