The exceptional divisor at its generic points #
Four lemmas on one blow-up, used in the proof of the statement CP3 of the embedded
desingularization (Hironaka.Resolution.Algebraic.Wlo05.EmbeddedChainRelative;
Hironaka.Resolution.Algebraic.Wlo05.EmbeddedCP3Weak):
for the blow-up π : B_Z X → X of a smooth X along a smooth centre Z of any shape with
exceptional divisor F = π⁻¹Z, at a generic point ε of F:
π εis a generic point ofZ(blowUpπ_mem_genericPoints_of_mem_genericPoints_exceptionalDivisor);ord_ε (π^* J) = ord_{π ε} J(ord_comap_blowUpπ_of_mem_genericPoints_exceptionalDivisor; [Kol07, Definition 47]: the order alongZis the multiplicity of the pullback along the exceptional divisor);- the weak transform of an ideal of constant order
dalongZhas order0atε(ord_weakTransform_eq_zero_of_mem_genericPoints_exceptionalDivisor; the weak transform of [Kol07, 58], formula (58.1)); - the weak transform of an ideal vanishing generically on
Dhas order0at the generic points of the strict transform ofD(ord_weakTransform_eq_zero_of_mem_genericPoints_strictTransform).
The local computation (exists_chart_of_mem_genericPoints_exceptionalDivisor). F is a
smooth divisor of the smooth B_Z X, so F_ε = 𝔪_ε at a generic point ε of F. In the chart at
z = π ε (exists_stalk_equiv_localization_chart_of_smooth): 𝒪_{B,ε} ≅ R'_𝔮 with
R' = 𝒪_{X,z}[y_i/y_ρ], Z_z = P = (y_0, …, y_ρ) a partial centre of the regular system of
parameters y, 𝔮 a prime of R' over 𝔪_z. The ideal F_ε = Z_z 𝒪_{B,ε} corresponds to
P R'_𝔮 = (y_ρ) R'_𝔮 (map_chartCenter), and (y_ρ) is a prime of R' since
R'/(y_ρ) ≅ (𝒪_{X,z}/P)[Y] (exists_quotient_span_algebraMap_x_ringEquiv_chartCenter). As
F_ε = 𝔪_ε, (y_ρ) R'_𝔮 is the maximal ideal, whence 𝔮 = (y_ρ) and
𝔪_z ≤ 𝔮 ∩ 𝒪_{X,z} = P = Z_z: the centre is the whole maximal ideal at z, ρ is the last
parameter (𝔪_z needs dim generators), and z is a generic point of Z
(mem_genericPoints_of_stalkIdeal_eq_maximalIdeal). Then the order identity is the computation of
the order along the exceptional divisor in the chart of the last coordinate
(ordAlong_span_algebraMap_x_map) transported through the isomorphism, and the vanishing of the
weak transform is the order computation in the regular local ring 𝒪_{B,ε} with 𝔪_ε = (a):
π^* J has order d and lies in (a)^d, so its colon by (a)^d is the unit ideal
(colon_span_singleton_pow_eq_top_of_ord_le). The chart is that of [Kol07, Definition 60] and
[Hau14, Definition 6.2] (the strict transform); the order along a centre is [Kol07, Lemma 61].
The generic-point criterion Z_z = 𝔪_z #
A point z of V(Z) with Z_z = 𝔪_z is a generic point of V(Z) (the converse of
stalkIdeal_eq_maximalIdeal_of_mem_genericPoints, on any scheme): a generization ξ ∈ V(Z) of z
is, on an affine U ∋ z, a prime q ⊆ p of Γ(U) containing Z(U); both q and p become
𝔪_z = Z_z in 𝒪_{X,z} = Γ(U)_p, so q = p and ξ = z.
The chart identification, algebraically #
Transport of an extended ideal through a commutative square e ∘ φ = α ∘ ψ:
e(φ(I) B) ⊆ α(ψ(I) R').
A prime 𝔮 of R' whose localization is identified with a local ring B in such a way that
the maximal ideal of B lands in P R'_𝔮 for a prime P ⊆ 𝔮 is P: for a ∈ 𝔮, a/1 is in the
maximal ideal of R'_𝔮 ≅ B, hence in P R'_𝔮, whose trace on R' is P (P prime, P ⊆ 𝔮).
The prime of the chart identified: for a regular system of parameters y of R, the chart
ring R' = R[y_i/y_ρ] of the partial centre P = (y_0, …, y_ρ), a prime 𝔮 of R' over 𝔪, and
a local ring B ≅ R'_𝔮 compatibly with φ : R → B — if P B is the maximal ideal of B, then
𝔮 = (y_ρ): P B corresponds to P R'_𝔮 = (y_ρ) R'_𝔮 (map_chartCenter), and (y_ρ) is a prime
of R' below 𝔮 (isPrime_span_algebraMap_x_of_chartCenter), so le_of_map_maximalIdeal_le
applies.
If the partial centre (y_0, …, y_ρ) of a regular system of parameters y is the whole
maximal ideal, ρ is the last parameter: 𝔪 needs dim R = n' generators (spanFinrank), and
(y_0, …, y_ρ) has ρ + 1.
The chart at a generic point of the exceptional divisor #
At a generic point ε of the exceptional divisor F = π⁻¹Z of the blow-up of the smooth X
along the smooth centre Z: F_ε = 𝔪_ε — F is a smooth divisor of B_Z X
(isSmoothDivisor_exceptionalDivisor_of_smooth), and the stalk of a smooth divisor at a generic
point of its support is the maximal ideal.
The local computation at a generic point ε of the exceptional divisor, z = π ε (the chart
exists_stalk_equiv_localization_chart_of_smooth at z, with its prime 𝔮 identified):
Z_z = 𝔪_z; adapted parameters y at z with Z_z = (y_0, …, y_ρ); 𝒪_{B,ε} ≅ R'_𝔮 for
R' = 𝒪_{X,z}[y_i/y_ρ], compatibly with π^♯; ρ is the last parameter; and 𝔮 = (y_ρ). The
identification: F_ε = 𝔪_ε is (y_0, …, y_ρ) 𝒪_{B,ε} (stalkIdeal_comap), so
prime_eq_span_of_map_chartCenter_eq_maximalIdeal gives 𝔮 = (y_ρ); then
𝔪_z ≤ 𝔮 ∩ 𝒪_{X,z} = (y_0, …, y_ρ) = Z_z (comap_span_algebraMap_x_eq_chartCenter), and 𝔪_z
needs dim 𝒪_{X,z} generators (val_add_one_eq_of_chartCenter_eq_maximalIdeal).
The image under π of a generic point of the exceptional divisor is a generic point of the
centre — Z_{π ε} = 𝔪_{π ε} by the local computation
(exists_chart_of_mem_genericPoints_exceptionalDivisor), and a point of V(Z) where the stalk of
Z is the maximal ideal is a generic point (mem_genericPoints_of_stalkIdeal_eq_maximalIdeal).
The order along the exceptional divisor ([Kol07, Definition 47]: the order of I along Z
equals the multiplicity of π^* I along the exceptional divisor; here for a smooth centre of any
shape): at a generic point ε of the exceptional divisor, ord_ε (π^* J) = ord_{π ε} J — in the
chart at z = π ε, the centre is 𝔪_z and ε is the prime (y_ρ) of the last coordinate
(exists_chart_of_mem_genericPoints_exceptionalDivisor), so this is
ordAlong_span_algebraMap_x_map transported through 𝒪_{B,ε} ≅ R'_{(y_ρ)}.
The weak transform ([Kol07, 58], formula (58.1), read against [Kol07, Definition 60]; for a
smooth centre of any shape): if J has order exactly d at every generic point of the centre, its
weak transform has order 0 at every generic point ε of the exceptional divisor — the weak
transform is the marked transform (π^* J : F^d) (weakTransform_eq_markedTransform_of_smooth);
at ε, F_ε = 𝔪_ε = (a) and π^* J has order d by the two previous lemmas, so
(π^* J)_ε ⊆ (a)^d and the colon ((π^* J)_ε : a^d) is the unit ideal
(colon_span_singleton_pow_eq_top_of_ord_le, the order being multiplicative on the regular local
ring 𝒪_{B,ε} containing ℚ).
(W3): the strict transform of a divisor on which the ideal vanishes generically #
If J has order 0 at every generic point of V(D), its weak transform has order 0 at
every generic point ε of the strict transform of D — no smoothness or order hypothesis is
needed. The support of the strict transform is the union of the strict transforms of the closures
of the generic points ξ of V(D) (coe_support_strictTransform_biUnion); ε lies in one of
them, ξ ∉ V(Z) (else that strict transform is empty), the generic point ξ' of the strict
transform of closure {ξ} maps to ξ
(exists_isGenericPoint_strictTransformSeq_and_isIso_of_le_firstCenterIndex for the one-step
sequence) and specializes to ε, so ξ' = ε, π ε = ξ ∉ V(Z), ε ∉ F, and π is a local
isomorphism at ε (ord_weakTransform_of_notMem).