Kollár's Example 12: the three point blow-ups on the z-chart #
[Kol07, Example 12], m = 3: after the blow-up of the origin, on the z-chart,
f₁ = x₁³ + z₁(y₁² − z₁⁴)² + z₁¹⁸; two more point blow-ups (the origin of the z-chart each time)
give f₂ = x₂³ + z₂²(y₂² − z₂²)² + z₂¹⁵ and f₃ = x₃³ + z₃³((y₃² − 1)² + z₃⁹), Kollár's three
displays, the third regrouped (printed x₃³ + z₃³(y₃² − 1)² + z₃¹²). Computed exactly: I₁ = (f₁)
and I₂ = (f₂) have order exactly 3 at the origin,
their only ℚ-point of order ≥ 3; the locus of order 3 of I₃ = (f₃) is the line
x₃ = z₃ = 0; at every stage Hᵢ = (xᵢ) and Hᵢ' = (xᵢ + …) lie in MC(Iᵢ) = D²(Iᵢ)
(∂²fᵢ/∂xᵢ² = 6xᵢ, and ∂²fᵢ/∂yᵢ² is 4 times the second generator); the restriction
I₃|_{H₃} = (z₃³·g) with g = (y₃² − 1)² + z₃⁹ Kollár's equation of the birational transform of
S ∩ H, the factor z₃³ being the exceptional part that the marked transform carries.
The three stage polynomials.
Equations
- Hironaka.Examples.Example12Stages.f1 = MvPolynomial.X 0 ^ 3 + MvPolynomial.X 2 * (MvPolynomial.X 1 ^ 2 - MvPolynomial.X 2 ^ 4) ^ 2 + MvPolynomial.X 2 ^ 18
Instances For
Equations
- Hironaka.Examples.Example12Stages.f2 = MvPolynomial.X 0 ^ 3 + MvPolynomial.X 2 ^ 2 * (MvPolynomial.X 1 ^ 2 - MvPolynomial.X 2 ^ 2) ^ 2 + MvPolynomial.X 2 ^ 15
Instances For
Equations
- Hironaka.Examples.Example12Stages.f3 = MvPolynomial.X 0 ^ 3 + MvPolynomial.X 2 ^ 3 * ((MvPolynomial.X 1 ^ 2 - 1) ^ 2 + MvPolynomial.X 2 ^ 9)
Instances For
x ∈ D²(⟨g⟩) whenever ∂²g/∂x² = 6x.
q ∈ D²(⟨g⟩) whenever ∂²g/∂y² = 4q.
Stage 1 #
The first blow-up on the z-chart: σ f = z₁³ f₁ with f₁ = x₁³ + z₁(y₁² − z₁⁴)² + z₁¹⁸,
Kollár's display, and the transforms of the equations of H and H'.
I₁ = (f₁) has order exactly 3 at the origin.
Stage 1: the origin is the only ℚ-point of order ≥ 3, from ∂²f₁/∂x² = 6x,
∂²f₁/∂y² = 4z(3y² − z⁴), ∂f₁/∂z|_{z=0} = y⁴, ∂²f₁/∂y∂z = 4y(y² − 5z⁴).
Stage 1: x₁ ∈ MC(I₁), so H₁ = (x₁) is of maximal contact.
Stage 1: x₁ + 3y₁²z₁ − z₁⁵ ∈ MC(I₁), so H₁' is of maximal contact
(∂²f₁/∂y² = 4(3y²z − z⁵)).
Stage 2 #
The second blow-up: σ f₁ = z₂³ f₂ with f₂ = x₂³ + z₂²(y₂² − z₂²)² + z₂¹⁵, Kollár's display,
and the transforms of the equations of H₁ and H₁'.
I₂ = (f₂) has order exactly 3 at the origin.
Stage 2: the origin is the only ℚ-point of order ≥ 3, from ∂²f₂/∂y² = 4z²(3y² − z²),
∂²f₂/∂z²|_{z=0} = 2y⁴, ∂²f₂/∂y∂z = 8yz(y² − 2z²).
Stage 2: x₂ ∈ MC(I₂), so H₂ = (x₂) is of maximal contact.
Stage 2: x₂ + 3y₂²z₂² − z₂⁴ ∈ MC(I₂), so H₂' is of maximal contact
(∂²f₂/∂y² = 4(3y²z² − z⁴)).
Stage 3 #
The third blow-up: σ f₂ = z₃³ f₃ with f₃ = x₃³ + z₃³((y₃² − 1)² + z₃⁹), Kollár's
x₃³ + z₃³(y₃² − 1)² + z₃¹², and the transforms of the equations of H₂ and H₂'.
Stage 3: the locus of order 3 contains the line x₃ = z₃ = 0, f₃ ∈ 𝔪_{(0,b,0)}³.
Stage 3: the locus of order 3 lies in the line x₃ = z₃ = 0, from ∂²f₃/∂x² = 6x,
∂²f₃/∂y² = 4z³(3y² − 1), ∂f₃/∂y = 4z³y(y² − 1); a point with z ≠ 0 would need 3y² = 1 and
y(y² − 1) = 0.
Stage 3: x₃ ∈ MC(I₃), so H₃ = (x₃) is of maximal contact.
Stage 3: x₃ + z₃³(3y₃² − 1) ∈ MC(I₃), so H₃' is of maximal contact
(∂²f₃/∂y² = 4z³(3y² − 1)).
The restriction at stage 3, I₃|_{H₃} = (z₃³·g) with g = (y₃² − 1)² + z₃⁹ Kollár's equation
of the birational transform of S ∩ H; the marked transform carries the exceptional factor
z₃³.