The components of an snc family on a smooth scheme, at a point and at a generic point #
An snc family E on a scheme X smooth over a field k (regular stalks): Kollár's snc coordinates
at a point of E^i (exists_stalkIdeal_component_eq_span_of_isSnc), the stalk of E^i at a point
of E^i is prime, the reduced ideal of the component through x has the member's stalk
(stalkIdeal_vanishingIdeal_closure_eq_of_specializes), the components of a member are pairwise
disjoint (genericPoint_eq_of_specializes), the inner product of a monomial's components at a point
of E^i (finprod_stalk_eq_of_specializes); at the generic point η of a component:
𝔪_η = (E^i)_η (maximalIdeal_eq_stalkIdeal_component), no other member passes through η
(notMem_support_of_ne), the component's subscheme is integral
(isIntegral_componentIdeal_subscheme), the membership half of [Kol07, Definition–Lemma 110]
(stalkIdeal_le_pow_of_specializes) and, in characteristic zero, the orders
ord ((E^i)_η^a · J) = a + ord J and ord ((E^i)_η^a) = a. These are the facts about the
components of the boundary that the splitting of an ideal into its monomial and non-monomial parts
(Hironaka.MarkedOrderReduction, Step 3 of the proof of [Kol07, Theorem 107]) uses; the forms
without a structure morphism are in Hironaka.Resolution.Algebraic.Snc.DictionaryRegular.
An snc family E on a scheme X smooth over k (regular stalks); the statements on triples in
Hironaka.MarkedOrderReduction are the instances of these at T.X.left ↘ Spec k, T.E, T.I.
B. The components of a member at a point #
Kollár's snc coordinates at a point of E^i: E^i_x = (z_c) for a regular system of parameters
z of 𝒪_{X,x} [Kol07, Definition 24 (2)].
The stalk of E^i at a point of E^i is prime (one member of a regular system of parameters of
the regular local ring 𝒪_{X,x}).
The reduced ideal of an irreducible component of E^i through x has, at x, the stalk of
E^i itself — it is a minimal prime over the prime (E^i)_x = (z_c)
(stalkIdeal_vanishingIdeal_mem_minimalPrimes), hence that prime.
The irreducible components of a member are pairwise disjoint ([Kol07, Definition 24 (1)], E^i
smooth): two components through x coincide, their reduced ideals having the same (prime) stalk at
x (eq_of_stalkIdeal_vanishingIdeal_closure_eq); the corollary of
Hironaka.Sequence.eq_of_specializes_of_isRegular.
At a point x of E^i the inner product of the monomial over the components of E^i is the
power of (E^i)_x at the exponent of the one component through x.
F. The component at its generic point #
At the generic point η of a component of E^i, the maximal ideal of 𝒪_{X,η} is the stalk of
E^i — the component is a divisor and 𝒪_{X,η} its local ring.
No member other than E^i passes through the generic point of a component of E^i
[Kol07, Definition 24] — its parameter would generate the maximal ideal (z_c) and be divisible by
z_c, against the independence of the parameters.
The subscheme of a component's reduced ideal is integral — irreducible (the support is the
closure of the generic point) and with regular stalks (at a point w of the component the stalk is
𝒪_{X,w} / (z_c), a regular local ring).
The membership half of [Kol07, Definition–Lemma 110], with the order along a subvariety of
[Kol07, Definition 47]: at a point x of a component D of E^i with generic point η,
I_x ⊆ (E^i)_x^{ord_η I} — the order along the prime (E^i)_x = (z_c) is ord_η I
(ordAlong_stalkIdeal_eq_ord_stalkIdeal on the integral component subscheme) and membership follows
for the prime element z_c.
G. Orders at the generic point #
At the generic point η of a component of E^i, ord ((E^i)_η^a · J) = a + ord J
([Kol07, Definition 59, (3)]; the multiplicativity of the order on a regular local ring containing
ℚ).
ord ((E^i)_η^a) = a at the generic point of a component of E^i.