Normal crossings #
The two readings of normal crossings in which the algebraic main theorems are stated, both at the
stalk, with a regular system of parameters (IsLocalRing.IsRegularSystemOfParameters) for
"local coordinates z_1, …, z_n ∈ 𝔪_x".
- Kollár's divisors with ordered index set [Kol07, Definition 31],
DivisorFamily X: a finite linearly ordered family of closed subschemes, each given by its ideal sheaf.IsSncAtis the simple normal crossing condition of [Kol07, Definition 24] at one point (an injective assignment of local coordinates to the components through the point),IsSncquantifies it over all points and asks the components to be regular, andHasSncWithis clause (4) of that definition. The support, the singular locusSing E(the points on at least two components), the inverse image familycomapand the empty family complete the vocabulary;totalTransformis the total transform of [Kol07, Definition 25] under one blow-up (the strict transforms of the components followed by the exceptional divisor, which receives the largest index, as in [Kol07, Definition 65]).IsIdealOfSncDivisor I:Iis the ideal sheaf𝒪(-∑ a_j F_j)of an effective divisor supported on a simple normal crossing family. - Hironaka's Definition 2 [Hir64, Ch. 0, §5, Definition 2],
IsSncBoundaryWith E DandIsSncBoundary E(his "Ehas only normal crossings withD" and "Ehas only normal crossings"): at every pointxofDsome regular system of parameters of𝒪_{X,x}contains a generator of the ideal of each irreducible component ofEthroughx(the minimal primes ofE_x) and generators of the ideal ofD. WithD := ⊥(the zero ideal, i.e.D = X) this is "Ehas only normal crossings". On a scheme Hironaka's components are the global irreducible components ofE, so the condition is simple normal crossings of the reduced subschemeE: it agrees withDivisorFamily.IsSncandDivisorFamily.HasSncWithfor the family of irreducible components ofE[Kol07, Definition 24] (Hironaka.Sequence.isSncBoundary_iff_isSnc_componentFamily,Hironaka.Sequence.isSncBoundaryWith_iff_hasSncWith_componentFamily). The analytic counterparts areAnalyticManifold.IdealSheaf.IsSncBoundaryandAnalyticSpace.ClosedSubspace.IsSncBoundary.
Divisors with ordered index set (Kollár Definitions 24 and 31) #
Kollár's divisor E = ∑ E^i with ordered index set [Kol07, Definition 31]: a finite linearly
ordered family of closed subschemes E^i ⊆ X, each given by its ideal sheaf.
- ι : Type u
The (finite, linearly ordered) index set.
- linearOrder : LinearOrder self.ι
- component : self.ι → X.IdealSheafData
The ideal sheaf of the component
E^i.
Instances For
The empty divisor E = ∅.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The set of points lying on at least one component.
Instances For
The singular locus Sing E of the divisor ∑ E^i: the points lying on at least two components
(where a simple normal crossing divisor is singular). The principalization of Theorem 35 is an
isomorphism over the complement of cosupp I ∪ Sing E
(AlgebraicGeometry.exists_functorial_principalization).
Equations
Instances For
The inverse image family h⁻¹(E) of [Kol07, 34.1].
Equations
Instances For
The simple normal crossing condition of [Kol07, Definition 24] at one point: the coordinates
z : Fin n → 𝒪_{X,x} form a regular system of parameters, and there is an injective assignment c
of a coordinate index to each component through x (and only to those) with E^i = (z_{c(i)} = 0)
near x, that is (E^i)_x = (z_{c(i)}).
Equations
- One or more equations did not get rendered due to their size.
Instances For
Kollár's simple normal crossing divisor [Kol07, Definition 24 (1)–(3)]: each E^i is smooth,
and at each point x there are local coordinates z_1, …, z_n ∈ 𝔪_x (a regular system of
parameters of 𝒪_{X,x}) such that every component through x is (z_{c(i)} = 0) near x, with
c(i) ≠ c(i') for distinct components through x (IsSncAt).
Equations
Instances For
[Kol07, Definition 24 (4)]: the closed subscheme Z has simple normal crossings with E if at
each point x of Z one can choose the coordinates z_1, …, z_n as in IsSnc such that in
addition Z = (z_{j_1} = ⋯ = z_{j_s} = 0) near x.
Equations
- E.HasSncWith Z = ∀ x ∈ Z.support, ∃ (n : ℕ) (z : Fin n → ↑(X.presheaf.stalk x)), E.IsSncAt x z ∧ ∃ (s : Finset (Fin n)), Z.stalkIdeal x = Ideal.span (z '' ↑s)
Instances For
The total transform of [Kol07, Definition 25] under one blow-up with centre D: the birational
transforms of the components of E, followed by the exceptional divisor f⁻¹(D), which receives
the largest index (new divisors come after old ones, [Kol07, Definition 65]).
Equations
- One or more equations did not get rendered due to their size.
Instances For
I is the ideal sheaf of an effective divisor ∑ a_j F_j supported on a simple normal crossing
family F [Kol07, Theorem 35 (2) and its proof in 72].
Equations
Instances For
Normal crossings (Hironaka's Definition 2) #
E has only normal crossings with D [Hir64, Ch. 0, §5, Definition 2]
(equivalently, D has only normal crossings with E): at every point x of D there is a
regular system of parameters (z_1, …, z_n) of 𝒪_{X,x} such that the ideal in 𝒪_{X,x} of
each irreducible component of E containing x (a minimal prime of E_x) is generated by one
of the z_i, and the ideal of D in 𝒪_{X,x} is generated by some of the z_i.
On a scheme Hironaka's components are the global irreducible components of E (their local
ideals at x are the minimal primes of E_x), each of which is asked to be cut out by a
parameter at each of its points of D; so at the points of D the condition is that the reduced
subscheme E has simple normal crossings there, and has them with D. On a Noetherian scheme it
agrees with DivisorFamily.HasSncWith of the family of irreducible components of E
[Kol07, Definition 24 (4)] (Hironaka.Sequence.isSncBoundaryWith_iff_hasSncWith_componentFamily),
which is also a condition at the points of D only.
Nothing is asserted at the points off D: for the unit ideal D = ⊤ (the empty subscheme) the
condition holds for every E. The manifold predicate of the same name
(AnalyticManifold.IdealSheaf.IsSncBoundaryWith) says more: that its boundary is the ideal of a
simple normal crossings hypersurface family at every point of the manifold, and in addition has
simple normal crossings with D at the points of D. On a scheme the condition at every point is
IsSncBoundary below, asserted separately where it is wanted.
Relation to the source.
- Translation.
IsSncBoundaryWith E Dis Hironaka's "$E$ has only normal crossings with $D$", andIsSncBoundary E, which isIsSncBoundaryWith E ⊥, his "$E$ has only normal crossings". - Restatement. The predicate demands a regular system of parameters at every point of
D(at every point ofXforIsSncBoundary), so the regularity of $\mathcal{O}_{X,x}$ there is part of it. On a regular scheme a reducedEwhose components are cut out by parameters has an invertible ideal sheaf, so Hironaka's "everywhere of codimension one" follows from reducedness and normal crossings at the points ofD(everywhere forIsSncBoundary) and is not a separate hypothesis.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Hironaka's "E has only normal crossings" [Hir64, Ch. 0, §5, Definition 2], his Definition 2
with D = X ("we simply say that E has only normal crossings at x") at every point of X ("with
no reference to a point, if it is the case at every point of X"): IsSncBoundaryWith E ⊥, the
zero ideal sheaf ⊥ being the ideal of X itself.
On a scheme Hironaka's components are the global irreducible components of E, whose local
ideals are asked to be parameters, so the condition is simple normal crossings of the reduced
subscheme E. On a Noetherian scheme smooth over a field it agrees with DivisorFamily.IsSnc of
the family of irreducible components of E [Kol07, Definition 24 (1)–(3)]
(Hironaka.Sequence.isSncBoundary_iff_isSnc_componentFamily).