Kollár's Example 12: S = (x³ + (y² − z⁶)² + z²¹ = 0), m = 3 #
[Kol07, Example 12]: I = (x³ + (y² − z⁶)² + z²¹) ⊂ ℚ[x, y, z], m = 3, H = (x), and a second
hypersurface of maximal contact H' = (x + 3y² − z⁶). Computed exactly:
MC(I) = D²(I) = (x, y³, y²z⁴, yz⁵, 3y² − z⁶);Ihas order exactly3at the origin, its onlyℚ-point of order≥ 3;x,x + 3y² − z⁶ ∈ MC(I);I|_H = ((y² − z⁶)² + z²¹), Kollár's trace "of multiplicity 4", andI|_{H'} = ((z⁶ − 3y²)³ + (y² − z⁶)² + z²¹)have the origin as their only point of order3.
The three point blow-ups on the z-chart are in
HironakaExamples/MaximalContact/Example12Stages.lean. Orders at ℚ-points read the value and the
partials up to order two (vanish_of_mem_m3_cube); the equations are solved by cases on which
factor vanishes.
g ∈ 𝔪₀⁴ in 𝔸³ kills every third partial derivative of g at 0.
g ∈ 𝔪₀⁵ in 𝔸² kills every fourth partial derivative of g at 0.
A partial derivative of a member of an ideal lies in its derivative ideal (Fin 3 form).
MC(I) = D²(I) and the cosupport #
Kollár's f = x³ + (y² − z⁶)² + z²¹.
Equations
- Hironaka.Examples.Example12Charts.f12 = MvPolynomial.X 0 ^ 3 + (MvPolynomial.X 1 ^ 2 - MvPolynomial.X 2 ^ 6) ^ 2 + MvPolynomial.X 2 ^ 21
Instances For
The ideal M = (x, y³, y²z⁴, yz⁵, 3y² − z⁶), to be shown equal to MC(I) = D²(I).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The generators of D²(I) lie in M: f and its first and second partials.
MC(I) = D²(I) = (x, y³, y²z⁴, yz⁵, 3y² − z⁶). ⊆: f, its partials and second partials lie
in M (z⁶ ≡ 3y² modulo 3y² − z⁶ absorbs the pure powers of z); ⊇: 6x = ∂²f/∂x²,
4(3y² − z⁶) = ∂²f/∂y², −24yz⁵ = ∂²f/∂y∂z, then 4y³ = ∂f/∂y + 4z·yz⁵ and
336·y²z⁴ = ∂²f/∂z² + (132z⁴ + 420z¹³)(3y² − z⁶) − 1260yz⁸·yz⁵.
I has order exactly 3 at the origin (∂³f/∂x³ = 6).
cosupp(I, 3) = {0}: at a ℚ-point of order ≥ 3, ∂²f/∂x² = 6x gives x = 0,
∂²f/∂y² = 12y² − 4z⁶ and ∂f/∂y = 4y(y² − z⁶) give y = 0, then z = 0.
x ∈ MC(I): H = (x) is a hypersurface of maximal contact.
x + 3y² − z⁶ ∈ MC(I): H' = (x + 3y² − z⁶) is a hypersurface of maximal contact.
The restrictions #
I|_H = ((y² − z⁶)² + z²¹).
Kollár's trace "has multiplicity 4": order exactly 4 at 0 (∂⁴/∂y⁴ = 24).
cosupp((I|_H, 3)) = {0}: the trace lies in 𝔪₀³, and the origin is its only point of order
3 (12y² − 4z⁶ and 4y(y² − z⁶)).
I|_{H'} = ((z⁶ − 3y²)³ + (y² − z⁶)² + z²¹) (x ↦ z⁶ − 3y²).
cosupp((I|_{H'}, 3)) = {0}: with u = z⁶ − 3y², v = y² − z⁶, the partials of
g = u³ + v² + z²¹ are ∂_y g = y(4v − 18u²), ∂²_y g = 4v − 18u² + 8y² + 216uy²,
∂_z g = z⁵(18u² − 12v + 21z¹⁵), ∂_y∂_z g = −24yz⁵(9u + 1); the cases z = 0, z ≠ 0 ∧ y = 0
and 9u = −1 each force the origin or a contradiction over ℚ.