Z has simple normal crossings with E: the consequences #
[Kol07, Definition 24 (4)]: a closed subscheme Z ⊆ X has simple normal crossings with the divisor
E = ∑ E^i if at every x ∈ Z there are local coordinates z_1, …, z_n ∈ 𝔪_x (a regular system of
parameters of 𝒪_{X,x}) with every component through x of the form E^i = (z_{c(i)} = 0), c
injective, and Z = (z_j = 0 : j ∈ s) for a set s of indices. The consequences Kollár draws — "in
particular, Z is smooth", some E^i may contain Z, and if none does then "E|_Z is again a
simple normal crossing divisor on Z" — are all local at a point w of V(Z) over x = ι w,
where ι : V(Z) ⟶ X is the closed immersion and the stalk map φ = ι^♯_w : 𝒪_{X,x} → 𝒪_{Z,w} is
surjective with kernel I_{Z,x} = (z_j : j ∈ s) (ker_stalkMap_subschemeι). Everything is then the
ring-level statement of Hironaka.Scheme.Snc.ParameterSubset for this φ:
- "
Zis smooth":𝒪_{Z,w}is regular, of dimensionn − |s|(isRegularLocalRing_stalk_subscheme,natCast_card_compl_eq_ringKrullDim_stalk_subscheme); - containment:
E^i = (z_j = 0)containsZnearx((E^i)_x ⊆ I_{Z,x}) iffj ∈ s(stalkIdeal_le_stalkIdeal_iff_mem); - transversality: for
j ∉ sthe imageφ z_jgenerates the stalk ofE^i ∩ Z = (E^i).comap ιatw, lies in𝔪_w ∖ 𝔪_w², and𝒪_{Z,w}/(φ z_j) = 𝒪_{X,x}/(z_l : l ∈ s ∪ {j})is regular (stalkIdeal_comap_subschemeι_eq_span,stalkMap_notMem_maximalIdeal_sq,isRegularLocalRing_quotient_stalkIdeal_comap); a closed subscheme all of whose stalk quotients are regular is regular (isRegular_subscheme_of_isRegularLocalRing_quotient); - restriction: when no component through
xcontainsZ, the images of thez_l,l ∉ s, are a regular system of parameters of𝒪_{Z,w}and the snc datacrestrict to the family(E^i).comap ι(span_range_stalkMap_comp_eq_maximalIdeal,exists_snc_data_comap).
In the sections at a point and on the subscheme, as in Hironaka.Scheme.Snc.Family, the snc data
are explicit hypotheses. The last section states the consequences for the predicates
DivisorFamily.IsSncAt and HasSncWith on a scheme smooth over k: hasSncWith_iff,
IsSncAt.stalkIdeal_le_iff_mem, HasSncWith.isRegularLocalRing_quotient_stalkIdeal,
HasSncWith.isRegular, HasSncWith.smooth, IsSncAt.exists_generator_stalkIdeal_comap_of_notMem,
HasSncWith.isSmoothDivisor_comap, IsSncAt.exists_isSncAt_comap, HasSncWith.isSnc_comap.
Every point of V(D) lies over a point of the support of D.
A closed subscheme V(D) is regular as soon as every stalk quotient 𝒪_{Y,w}/D_w, w ∈ V(D),
is regular — 𝒪_{V(D),v} ≅ 𝒪_{Y,ι v}/D_{ι v} (stalkQuotientEquiv).
At a point x of X #
A component E^i = (z_j = 0) through x contains Z near x iff j is one of the
coordinates of Z.
𝒪_{X,x}/I_{Z,x} is regular (Kollár's "in particular, Z is smooth").
At a point w of V(Z) over x = ι w #
The stalk at w of the restriction D ∩ Z = D.comap ι is generated by the image of a generator
of D_x.
When no component through x = ι w contains Z, the snc data c of E at x restrict to snc
data of the family E^i ∩ Z = (E^i).comap ι at w, with coordinates the images of the z_l,
l ∉ s, enumerated by sᶜ.orderEmbOfFin (Kollár's "E|_Z is again a simple normal crossing
divisor on Z").
𝒪_{Z,w} = 𝒪_{X,x}/(z_j : j ∈ s) is a regular local ring (Kollár's "Z is smooth").
dim 𝒪_{Z,w} = n − |s|.
dim 𝒪_{Z,w} = |sᶜ|, the number of remaining coordinates.
The images of the coordinates z_l, l ∉ s, generate 𝔪_w.
For j ∉ s, the image of z_j in 𝒪_{Z,w} lies outside 𝔪_w² — E^i ∩ Z = (z_j = 0) is a
smooth divisor of Z at w.
𝒪_{Z,w}/(E^i ∩ Z)_w = 𝒪_{X,x}/(z_l : l ∈ s ∪ {j}) is regular.
The predicates IsSncAt and HasSncWith on a scheme smooth over k #
[Kol07, Definition 24 (4)] asks, at each point x of the closed subscheme Z, for local
coordinates z_1, …, z_n as in (1)–(3) — a regular system of parameters of 𝒪_{X,x} with each
component E^i through x equal to (z_{c(i)} = 0) — such that moreover Z = (z_j = 0 : j ∈ s)
near x. The predicate HasSncWith says exactly this (hasSncWith_iff is Iff.rfl). On X
smooth over k every stalk is a regular local ring (AlgebraicGeometry.isRegularLocalRing_stalk),
so Kollár's consequences are the ring-level statements of the two sections above:
Zis smooth.𝒪_{Z,x} = 𝒪_{X,x}/(z_j : j ∈ s)is the quotient of a regular local ring by part of a regular system of parameters, hence regular (after moving thez_j,j ∈ s, to the front:Hironaka.Scheme.Snc.ParameterSubset). A pointwofV(Z)lies overx = ι w ∈ Z, and𝒪_{Z,w}is that quotient through the surjective stalk map of the closed immersionι(kernelI_{Z,x}); soV(Z)is regular and, over the perfect fieldk, smooth. The regular quotient𝒪_{X,x}/I_{Z,x}atxis stated separately.- Containment.
E^i = (z_j = 0)containsZnearx, that is(z_j) ⊆ (z_l : l ∈ s), iffj ∈ s: the classes ofz_1, …, z_nare a basis of𝔪_x/𝔪_x², soz_jis a combination of thez_l,l ∈ s, only when it is one of them. - Transversal components. If
j ∉ s, the restrictionE^i ∩ Z = (E^i).comap ιhas atwthe stalk(φ z_j)(stalkIdeal_comap),φ z_j ∈ 𝔪_w ∖ 𝔪_w²(again the basis of𝔪_x/𝔪_x²:z_j ∉ 𝔪_x² + (z_l : l ∈ s)), and𝒪_{Z,w}/(φ z_j) = 𝒪_{X,x}/(z_l : l ∈ s ∪ {j})is regular. WhenE^icontainsZnear no point ofZ ∩ E^i, this makesE^i ∩ Za smooth divisor ofV(Z): its closed subscheme is regular because all its stalk quotients are. - Restriction. If no component through
xcontainsZ, the images of thez_l,l ∉ s, are a regular system of parameters of𝒪_{Z,w}(they generate𝔪_w = φ(𝔪_x), anddim 𝒪_{Z,w} = n − |s|is their number), and the assignmentcrestricts: each componentE^i ∩ Zthroughwis(φ z_{c(i)} = 0)withc(i) ∉ s, injectively. HenceE|_Zis again a simple normal crossing divisor onV(Z).
The namespace is reopened so that the section variables are bound in the order {k} [Field k] {X}.
[Kol07, Definition 24 (4)] unfolded: Z has simple normal crossings with E iff at every
x ∈ Z there are local coordinates satisfying (1)–(3) and an index set s with
I_{Z,x} = (z_j : j ∈ s).
Under (4), a component E^i = (z_j = 0) through x contains Z near x iff j is one of the
coordinates of Z — z_j ∈ (z_l : l ∈ s) iff j ∈ s, the classes of the coordinates being a basis
of 𝔪_x/𝔪_x².
At x ∈ Z, the stalk 𝒪_{Z,x} = 𝒪_{X,x}/I_{Z,x} is a regular local ring — the quotient of a
regular local ring by part of a regular system of parameters.
V(Z) is regular.
V(Z) is smooth over the perfect field k (Kollár's "in particular, Z is smooth").
The transversal case at a point w of V(Z) over x = ι w: for a component E^i = (z_j = 0)
through x with j ∉ s, the restriction E^i ∩ Z (the inverse image (E^i).comap ι) has at w a
stalk generated by the image of z_j, an element of 𝔪_w ∖ 𝔪_w², with regular quotient —
E^i ∩ Z = (z_j = 0) ⊆ Z is a smooth divisor of Z at w.
Globally: if the component E^i contains Z near no point of Z ∩ E^i, then E^i ∩ Z is a
smooth divisor of V(Z).
At a point w of V(Z) over x = ι w: if no component through x contains Z near x, the
restricted family E|_Z = E.comap ι satisfies Kollár's (1)–(3) at w with coordinates the images
of the z_l, l ∉ s — enumerated by an injective σ : Fin m → Fin n with values outside s.
Kollár's "if E does not contain Z, then E|_Z is again a simple normal crossing divisor on
Z" [Kol07, Definition 24]: if Z has simple normal crossings with E and no component of E
contains Z near any point of Z, the inverse image family E.comap ι on V(Z) is a simple
normal crossing divisor.