The maximal-contact theorem: a smooth hypersurface in MC(I) has maximal contact #
For an ideal sheaf I with m = max-ord I, a smooth hypersurface H with
𝒪(−H) ⊆ MC(I) = D^{m−1}(I) and I|_H ≠ 0 is a hypersurface of maximal contact
[Kol07, Theorem 80 (1)], in the sense of [Kol07, Definition 78]: the centres of every smooth
blow-up sequence of order m lie in the birational transforms of H. This module proves that
clause on the whole manifold (IsOfOrder.centersIn_of_idealSheaf_le_iteratedDeriv), the static
form of maximal contact that order reduction uses. Kollár's proof: being a hypersurface of
maximal contact is a local question; for a smooth blow-up sequence of order m starting with
(X, I), his Corollary 77 at j = m − 1 gives a smooth blow-up sequence of order
≥ 1 starting with (X, J_0 := MC(I), 1); with H_i := (Π_i)_*^{-1} H ⊂ X_i the birational
transform of H, 𝒪_{X_0}(−H_0) ⊂ J_0 and the smoothness of H_0 give 𝒪_{X_i}(−H_i) ⊂ J_i for
every i; since ord_{Z_i} I_i ≥ m, one has ord_{Z_i} J_i ≥ ord_{Z_i} MC(I_i) ≥ 1 and hence
ord_{Z_i} H_i ≥ 1, i.e. Z_i ⊂ H_i for every i.
The proof here is an induction on the stage keeping the invariant "H_i is a smooth hypersurface
and I_{H_i} ⊆ J_i": the order clause of Corollary 77 (isOfOrderGe_iteratedDeriv at
j = m − 1) gives ord_{Z_i} I_{H_i} ≥ 1, i.e. Z_i ⊆ H_i
(center_subset_strictTransformSeq_of_le: at a point of Z_i off H_i the stalk of I_{H_i} is
the unit ideal, which would force the stalk of the centre to be the unit ideal); then H_{i+1} is
the strict transform of a smooth
hypersurface through the centre, a smooth hypersurface with
I_{H_{i+1}} = π_*^{-1}(I_{H_i}, 1) ⊆ π_*^{-1}(J_i, 1) = J_{i+1} (the ideal sheaf of the strict
transform of a smooth hypersurface through the centre is the marked transform of its ideal sheaf
with the mark 1, the analytic form of Giraud's lemma [Wlo05, Lemma 2.7.4], and the marked
transform is monotone).
This theorem, with the local construction of maximal contact [Kol07, 51.2], supplies the hypersurfaces on which order reduction is carried out by induction on the dimension.
One stage of the proof of [Kol07, Theorem 80]: along a sequence of order ≥ 1 for (J, 1), a
smooth hypersurface H_i with I_{H_i} ⊆ J_i contains the centre Z_i
(ord_{Z_i} I_{H_i} ≥ 1; at a point of Z_i off H_i the stalk of I_{H_i} is the unit
ideal).
A smooth blow-up sequence of order m ≥ 1 for I is a smooth blow-up sequence of order ≥ 1
starting with (X, J_0 := MC(I), 1): Kollár's Corollary 77 at j = m − 1, the first step of the
proof of [Kol07, Theorem 80].
The invariant of the induction in the proof of [Kol07, Theorem 80]: along a smooth blow-up
sequence of order m ≥ 1 starting with (M, I, E₀), for a smooth hypersurface H with
I_H ⊆ D^{m−1}(I), every strict transform H_i is a smooth hypersurface whose ideal sheaf lies
in the marked transform J_i of (D^{m−1}(I), 1).
Every strict transform H_i of a smooth hypersurface H with I_H ⊆ D^{m−1}(I) along a
smooth blow-up sequence of order m ≥ 1 is a smooth hypersurface (the smoothness of H_0
carried along the sequence in the proof of [Kol07, Theorem 80]; cf. [Wlo05, Lemma 2.7.4 (2)]).
The ideal sheaf of the strict transform H_i lies in the marked transform J_i of
(MC(I), 1) (Kollár's "𝒪_{X_i}(−H_i) ⊂ J_i for every i" in the proof of
[Kol07, Theorem 80]).
Every centre lies in the strict transform of H at its stage (Kollár's "thus Z_i ⊂ H_i for
every i" in the proof of [Kol07, Theorem 80]).
The pullback of ideal sheaves is monotone (stalkwise, Ideal.map is monotone): the same-model
case of IdealSheaf.pullback_le_pullback.