Kollár's Example 106 for the order-reduction functor: the first blow-up #
[Kol07, Example 106] continues in the chart x₁ = x, y₁ = y/x, z₁ = z/x, w₁ = w/x of the blow-up of
the origin, where the transform is I₁ = (x₁ − y₁², x₁(x₁ + z₁² − w₁³)) with exceptional divisor
E₁ = (x₁ = 0), and states that "the order has dropped to 1", continuing with (I₁, 1, E₁). That
holds on the x-chart Kollár displays but not on the whole blow-up: on the chart x = x₁z₁,
y = y₁z₁, z = z₁, w = w₁z₁ the transform is I₁ = (x₁³z₁ − y₁², z₁(x₁⁴z₁ + x₁ − w₁³)), and
at the origin q = [0:0:1:0] of that chart, a point of the exceptional divisor E₁, the marked
transform
(I₁, 2) still has order exactly 2. The order-2 run of BO_{4,2}(𝔸⁴_ℚ, I, ∅) therefore does
not stop after the first blow-up. This module computes the order-2 locus of I₁ at every scheme
point of the blow-up and derives the second centre by the cosupport forcing of
HironakaExamples/Sequence/CosuppForcing.lean: the tail of the sequence after the blow-up of the
origin is an order-2 sequence for the weak transform I₁ = π_*^{-1}(I, 2), so its first centre is
supported at the order-2 locus of I₁, the single point q.
- The blow-up
blowUp 𝔸⁴ (ofIdealTop 𝔪₀)is identified with the affine model of the blow-up of the origin (HironakaExamples/BlowUp/Glue/SpecIso.lean), whose four chartsU_a,a ∈ {x, y, z, w}, are copies ofSpec ℚ[x, y, z, w]with the substitutionsσ_aofHironakaExamples/MaximalContact/Example106Charts.lean(chartSubst; the charts of Hau14, Definition 4.12). - On the chart of
a,I₁is the ideal sheaf of the colon(σ_a I : a²), the controlled transform of [Kol07, Definition 60] on an affine chart; the identitiesσ_a g = a² g'make it Kollár'sI₁on thex-chart and give they-,z- andw-chart ideals (chartIdeal_zerotochartIdeal_three). - By the derivative criterion at primes ([Kol07, Lemma 74 (3)];
le_ord_specIdealSheaf_iff), the order-2locus is empty on thex-,y- andw-charts (D(I₁) = (1)) and is the origin of thez-chart (a prime containingD(I₁)containsy₁,x₁³, thenz₁andw₁³), whereord_q I₁ = 2exactly (D²(I₁) ∋ ∂²_{y₁}(x₁³z₁ − y₁²) = −2). qis the intersection ofE₁with the strict transform of thez-axis(x = y = w = 0), a closed point (mem_modelST_and_EA_iff,isClosed_singleton_qA), and everything is transported to the blow-up of𝔸⁴_ℚalong the identification (qB,two_le_ord_weakTransform_iff,ord_weakTransform_qB).secondCenter_of_axioms: for any order-2sequence with the three properties of the functor's values that begins with the blow-up of the origin, the continuation is a blow-up of a centre supported atq, at which(I₁, 2)has order exactly2.
The same facts at the ℚ-points of the charts are in Example106Charts.lean; the order is a
notion at scheme points, so they are proved here at primes, reusing the chart identities. The
later stages are in HironakaExamples/OrderReduction/Example106Length.lean.
A general helper #
In a domain, the colon of (a g₁, a g₂) by a ≠ 0 is (g₁, g₂): the controlled transform
of [Kol07, Definition 60] on an affine chart.
The model blow-up of the origin of 𝔸⁴_ℚ #
The partial derivative ∂/∂x_i as a ℚ-derivation of ℚ[x, y, z, w].
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The centre (x_0, …, x_3) of the model blow-up is 𝔪₀ = (x, y, z, w).
The chart substitution of x: (x, y, z, w) ↦ (x, yx, zx, wx).
The chart substitution of y: (x, y, z, w) ↦ (xy, y, zy, wy).
The chart substitution of z: (x, y, z, w) ↦ (xz, yz, z, wz).
The chart substitution of w: (x, y, z, w) ↦ (xw, yw, zw, w).
On the chart of x_j the total transform of V(P) is the ideal sheaf of σ_j P.
On the chart of x_j the exceptional ideal is (x_j).
The weak transform of I on the model blow-up, as the controlled transform of control 2
([Kol07, Definition 60]; weakTransform_eq_markedTransform_of_smooth identifies it with the weak
transform of the blow-up sequences).
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The controlled transform on the chart of x_j is the ideal sheaf of the colon (σ_j I : x_j²)
([Kol07, Definition 60] on the chart).
The chart ideals of Kollár's I₁ #
The image of I under a chart substitution σ, with the generators rewritten.
On the x-chart, I₁ = (x₁ − y₁², x₁(x₁ + z₁² − w₁³)), as printed in [Kol07, Example 106].
On the y-chart, I₁ = (x₁³y₁ − 1, y₁(x₁⁴y₁ + x₁z₁² − w₁³)).
On the z-chart, I₁ = (x₁³z₁ − y₁², z₁(x₁⁴z₁ + x₁ − w₁³)), the chart not displayed in
[Kol07, Example 106].
On the w-chart, I₁ = (x₁³w₁ − y₁², w₁(x₁⁴w₁ + x₁z₁² − 1)).
The order-2 locus on each chart, at every prime #
On the x-chart the order has dropped to 1 at every point, as [Kol07, Example 106] says:
D(I₁) ∋ ∂_{x₁}(x₁ − y₁²) = 1, so no prime contains D(I₁).
The y-chart has no point of order 2: x₁³y₁ − 1 ∈ I₁ and ∂_{y₁}(x₁³y₁ − 1) = x₁³ ∈ D(I₁),
so 1 ∈ D(I₁).
The w-chart has no point of order 2: a prime containing D(I₁) contains
∂_{w₁}(x₁³w₁ − y₁²) = x₁³, hence x₁, and ∂_{w₁}(w₁(x₁⁴w₁ + x₁z₁² − 1)) = 2x₁⁴w₁ + x₁z₁² − 1,
hence 1.
On the z-chart a prime contains D(I₁) iff it is the origin q = [0:0:1:0]: from
∂_{y₁}(x₁³z₁ − y₁²) = −2y₁, ∂_{z₁}(x₁³z₁ − y₁²) = x₁³,
∂_{x₁}(z₁(x₁⁴z₁ + x₁ − w₁³)) = 4x₁³z₁² + z₁ and ∂_{z₁}(z₁(x₁⁴z₁ + x₁ − w₁³)) = 2x₁⁴z₁ + x₁ − w₁³
a prime containing D(I₁) contains y₁, x₁, z₁, w₁; conversely I₁ ⊆ 𝔪_q²
(ord_zero_example106_z) gives D(I₁) ⊆ 𝔪_q.
ord_q I₁ is not more than 2: 1 ∈ D²(I₁) on the z-chart, from
∂²_{y₁}(x₁³z₁ − y₁²) = −2.
The order-2 locus of I₁ on the model blow-up #
D^{2−1} = D at the ring level, for the derivative criterion.
Off the z-chart no scheme point of the blow-up has ord I₁ ≥ 2.
On the z-chart the order-2 locus of I₁ is the origin q = [0:0:1:0], at every scheme
point.
The order of I₁ at q is not ≥ 3.
Every point of the model blow-up lies on one of the four charts.
Kollár's point q = [0:0:1:0] ∈ E₁, the origin of the z-chart, on the model blow-up.
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On the model blow-up the order-2 locus of I₁ is the single point q, at every scheme
point.
The strict transform of the z-axis meets E₁ at q #
The z-axis (x = y = w = 0) ⊆ 𝔸⁴.
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The strict transform of the z-axis (x = y = w = 0) on the model blow-up.
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The strict transform of the z-axis on the chart of x_j.
The z-axis is prime: ℚ[x, y, z, w]/(x, y, w) ≅ ℚ[z].
σ_z on the generators of the z-axis: σ_z x = xz.
On the z-chart the total transform of the z-axis is contained in (x₁, y₁, w₁).
On the z-chart, x₁z₁, y₁z₁ and w₁z₁ lie in the total transform of the z-axis.
Off the z-chart the total transform of the z-axis contains the exceptional generator.
Kollár's q = [0:0:1:0] is the intersection of E₁ with the strict transform of the z-axis,
at every scheme point of the model blow-up: a point lies on both iff it is q.
Transport to the blow-up of record #
The identification Φ is a bijection on points (inverse then hom).
The identification Φ is a bijection on points (hom then inverse).
The exceptional divisor of the blow-up transports to the model's.
The weak transform of I, which is its marked transform π_*^{-1}(I, 2) since ord_0 I = 2
([Kol07, 58] against [Kol07, Definition 60]; weakTransform_eq_markedTransform_of_smooth),
transports to modelW.
The strict transform of the z-axis transports to the model's.
Kollár's point q on the blow-up of 𝔸⁴_ℚ.
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The order of I₁ at a point of the blow-up is the order of modelW at its image under Φ.
On the blow-up of 𝔸⁴_ℚ the order-2 locus of I₁ is the single point q, at every scheme
point; it lies off the x-chart displayed in [Kol07, Example 106].
ord_q I₁ = 2 on the blow-up of 𝔸⁴_ℚ.
Membership in a support on the blow-up, read on the model through Φ.
On the blow-up of 𝔸⁴_ℚ, q is the intersection of E₁ with the strict transform of the
z-axis, at every scheme point.
The second centre #
The second centre ([Kol07, Example 106], continued beyond the displayed chart): for a
smooth blow-up sequence of order 2 for (𝔸⁴_ℚ, I, E) that begins with the blow-up of the origin,
has no empty centre and ends with maximal order < 2, the continuation after the first blow-up is
again a blow-up, of a centre supported at the single point q = [0:0:1:0] ∈ E₁, the intersection
of E₁ with the strict transform of the z-axis, at which the marked transform
I₁ = π_*^{-1}(I, 2) has order exactly 2. The centre J of the first blow-up is passed with
hJ : J = 𝔪₀ (as centerIdeal ℚ 4 4) so that the statement applies to ofIdealTop 𝔪₀.