The dictionary, part III: only normal crossings is simple normal crossings of the components #
The core of the translation between Hironaka's and Kollár's normal crossing conditions:
[Hir64, Definition 2] asks, at a point x, for a regular system of parameters generating "the
ideal of each irreducible component of E containing x", that is the minimal primes of the
stalk E_x, while [Kol07, Definition 24] speaks of the members of a family through x. The bridge
is the correspondence, in the local ring 𝒪_{X,x}, between the minimal primes of E_x and the
irreducible components of V(E) through x:
- on an affine open
U ∋ x, primes of𝒪_{X,x}are the primes ofΓ(U)inside the prime ofx(the stalk is the localization, Mathlib'sIsAffineOpen.isLocalization_stalk), points ofUare the primes ofΓ(U)(IsAffineOpen.fromSpec,primeIdealOf), specialization is inclusion of primes (PrimeSpectrum.le_iff_specializes), and a point lies inV(E)iff its prime containsE(U); - the reduced irreducible component through a generic point
ηofV(E), the vanishing ideal sheaf ofclosure {η}, has onUthe prime ofη(vanishingIdeal_closure_singleton_ideal) and atxits image, a prime of𝒪_{X,x}containingE_x; it is minimal because a prime between them is the prime of a point ofV(E)generizingη, henceηitself (stalkIdeal_vanishingIdeal_mem_minimalPrimes); conversely a minimal prime ofE_xis the prime of a point ofV(E) ∩ Ugenerizingx, and that point is a generic point ofV(E)by minimality (exists_mem_genericPoints_of_mem_minimalPrimes); distinct generic points give distinct primes (eq_of_stalkIdeal_vanishingIdeal_closure_eq). This is the same affine-local argument asexists_genericPoint_of_isPrime_stalkIdeal.
Hence Kollár's local condition for the component family at x (IsSncAt) is Hironaka's
(componentFamily_isSncAt_iff): the injective coordinate assignment to the members through x is
the assignment of a coordinate to each minimal prime, and the members of the component family are
regular subschemes because their stalks are generated by a coordinate. The two dictionaries follow
(isSncBoundaryWith_iff_hasSncWith_componentFamily,
isSncBoundary_iff_isSnc_componentFamily), and with them the first part of clause (iv)
of [Hir64, Main Theorem II] along a sequence of order d
(IsOrderSeq.isSncBoundary_boundarySeq: Kollár's E_i are snc at every stage, and
Hironaka's boundaries are their reduced unions).
The prime of Γ(U) at a point of the affine open U (Mathlib's
IsAffineOpen.primeIdealOf).
Equations
Instances For
The point of the prime of a point of U is the point.
Specialization between points of U is inclusion of their primes (Mathlib's
PrimeSpectrum.le_iff_specializes transported along the open embedding fromSpec).
A point of U lies in V(E) iff E(U) lies in its prime.
The vanishing ideal sheaf of the closure of a point η ∈ U has, on U, the prime of η: the
preimage of closure {η} in Spec Γ(U) is the zero locus of that prime, whose vanishing ideal is
the prime (radical).
The stalk at x ∈ U of the reduced component through η ∈ U is the image in 𝒪_{X,x} of the
prime of η.
Minimal primes of E_x and the irreducible components of V(E) through x #
For a generic point η of V(E) specializing to x, the stalk at x of the reduced
component closure {η} is a minimal prime of E_x (the ideal of an "irreducible component of E
containing x" in [Hir64, Definition 2]): it is the image of the prime of η, prime and
containing E_x, and a prime between E_x and it is the prime of a point of V(E) generizing
η, which is η by genericity.
Every minimal prime of E_x is the stalk at x of the reduced component through a generic
point of V(E) specializing to x: the prime is the image of a prime of Γ(U) inside the prime
of x, whose point lies in V(E), generizes x, and is generic in V(E) by minimality.
Two points specializing to x whose reduced components have the same stalk at x coincide
(their primes in Γ(U) are recovered from the stalk ideals).
The dictionary for the family of irreducible components #
Kollár's local condition [Kol07, Definition 24 (1)–(3)] at x for the family of irreducible
components, an injective assignment of coordinates to the members through x with
(E^i)_x = (z_{c(i)}), is Hironaka's condition on the components [Hir64, Definition 2], every
minimal prime of E_x being some (z_i): the members through x are exactly the minimal primes
of E_x (stalkIdeal_vanishingIdeal_mem_minimalPrimes,
exists_mem_genericPoints_of_mem_minimalPrimes), distinct members giving distinct primes.
E has only normal crossings with D [Hir64, Definition 2] iff D has simple normal
crossings with the family of irreducible components of E [Kol07, Definition 24 (1)–(4)]:
pointwise componentFamily_isSncAt_iff with the same clause on the ideal of D.
On X smooth over k, E has only normal crossings Hir64, Definition 2 with D = X iff
the family of its irreducible components is a simple normal crossing divisor
[Kol07, Definition 24 (1)–(3)]: its members are regular subschemes (at each of their points the
stalk is a coordinate (z_i), so the quotient is a regular local ring) with Kollár's coordinates
at every point.
The first part of clause (iv) of [Hir64, Main Theorem II], which Hironaka remarks follows
from (iii): along a smooth blow-up sequence of order d starting with (X, J, E.componentFamily)
for E with only normal crossings, every boundary E_i has only normal crossings. Kollár's E_i
are simple normal crossing divisors at every stage ([Kol07, Definition 25];
IsOrderSeq.isSnc_totalTransformSeq) and Hironaka's boundaries are their reduced unions.