Simple normal crossings restricted to a smooth hypersurface #
The proof of [Kol07, Corollary 85]: for Z ⊆ H smooth with H + E snc at every point, Z has
simple normal crossings with E in X iff it has them with H + E in X iff it has them with
E|_H in H — Kollár: "this poses the same restriction on order reduction", since E|_H is again
snc. Going down to a hypersurface of maximal contact
(Hironaka.Resolution.Algebraic.MaximalContact.GoingDown) needs the passage to H: when the centre
Z_i ⊆ H_i of a blow-up has simple normal crossings with E_i + H_i on X_i (from Z_i snc with
E_i and H_i + E_i snc, hasSncWith_append_of_le), the same centre, viewed in H_i, has simple
normal crossings with E_i|_{H_i}. This is clause (3′) of [Kol07, Definition 66] for the restricted
sequence.
The point lemma. Let g : Y ⟶ X be a closed immersion with image H = V(ker g), and let Z
have simple normal crossings with E + H at x = g(y) ∈ Z: coordinates z_1, …, z_n (a regular
system of parameters of 𝒪_{X,x}), E^i = (z_{c(i)}) for the components through x, H = (z_h),
and Z = (z_s : s ∈ S). Since Z ⊆ H, z_h ∈ (z_s : s ∈ S), so h ∈ S (mem_span_image_iff of
Hironaka.Scheme.Snc.ParameterSubset). The stalk map φ : 𝒪_{X,x} → 𝒪_{Y,y} is surjective with
kernel (z_h) (ker_stalkMap_of_isClosedImmersion of
Hironaka.Scheme.BlowUpSequence.RestrictDivisors), so the images of the z_j, j ≠ h, form a
regular system of parameters of 𝒪_{Y,y} (span_range_comp_compl_eq_maximalIdeal,
natCast_sub_card_eq_ringKrullDim_of_ker), the components of E|_H through y are the (φ z_{c(i)}) with c(i) ≠ h (injectivity of c), and Z|_H = (φ z_s : s ∈ S ∖ {h}) — the generator
φ z_h = 0 drops out. The bookkeeping of the surviving indices is complIndex of
Hironaka.Scheme.Snc.ParameterSubset.
The smooth divisor as a family. For the case E = ∅ of [Kol07, Definition 78] the restricted
sequence's clause (3′) is the normal crossing of the centres with the exceptional divisors on H,
and the restricted sequence needs "∅ + H is snc": a smooth divisor H alone is a simple normal
crossing family, because a generator of H_x in 𝔪_x ∖ 𝔪_x² extends to a regular system of
parameters (exists_span_eq_maximalIdeal_of_notMem_sq of Hironaka.Algebra.Local.RegularSystem),
and at a point off H any regular system of parameters serves
(isSnc_append_empty_of_isSmoothDivisor).
Sources: [Kol07, Corollary 85] (the proof); [Kol07, Definition 24; Definition 66; Definition 78].
The restriction direction of the proof of [Kol07, Corollary 85]: for a closed immersion
g : Y ⟶ X with image H = V(ker g) and a closed subscheme Z ⊆ H having simple normal crossings
with E + H, the subscheme Z of Y has simple normal crossings with E|_H = E.comap g.
A smooth divisor H alone is a simple normal crossing family — Kollár's H + E with E = ∅:
at a point of H the generator of H_x in 𝔪_x ∖ 𝔪_x² extends to a regular system of parameters,
and off H any regular system of parameters serves; the only component is H.