Kollár's Example 82: going down is not onto, and a general H need not give an equivalence #
[Kol07, Example 82] illustrates the two defects of going down along a hypersurface of maximal
contact. First part: for I = (xy − zⁿ) and H = (x = 0), (H, I|_H) ≅ (𝔸², (zⁿ)) has a
one-dimensional locus of order n, while I has an isolated point of order 2, so a blow-up of
order ≥ n of (H, I|_H, 2) centred at another point of that line has no counterpart of order
2 on (X, I); for the general H_g = (x − y = 0), (H_g, I|_{H_g}) ≅ (𝔸², (x² − zⁿ)) has the
same isolated point of order 2. Second part: for I = (x³ + xy⁵ + z⁴) and the general
H = (x + u₁xy³ + u₂y⁴ + u₃z² = 0), after two chart blow-ups the transform of I has order 2
while its restriction to the transform of H still has order 3.
All statements are computations in polynomial rings, in the vocabulary of
Hironaka/Resolution/Algebraic/Balanced/:
- Orders at a
ℚ-pointpare read as𝔪_p-adic orders ([Kol07, Definition 47] in the polynomial ring). Membershipsg ∈ 𝔪_p^kare exhibited by witnesses; non-membershipsg ∉ 𝔪_p^kfollow uniformly from two tools, a partial derivative lowers the power of an ideal by one (pderiv_mem_pow_of_mem_pow_succ, the Leibniz rule) and evaluation atpkills𝔪_p(eval_eq_zero_of_mem_span; the simp lemmapderiv_ofNatofHironaka/Scheme/IdealSheaf/Order/AffineSpace.leanhandles the numerals), so a(k−1)-fold partial derivative ofgthat does not vanish atpshowsg ∉ 𝔪_p^k. - The derivative ideals are
Ideal.derivative, computed from the partial derivatives of the generators (derivative_span_pderiv);D(xy − zⁿ) = (x, y, z^{n−1})isderivative_example52ofHironakaExamples/Balanced/Example52.lean(the ideal of [Kol07, Example 52.2]). - The chart blow-ups are the substitution
σ : x ↦ x₁y₁, y ↦ y₁, z ↦ z₁y₁, and Kollár's rows are polynomial identitiesσ g = y₁ᵐ · g'; the elimination ofx₂by the equation ofH₂is stated with the unit1 + u₁y₂³cleared, as the identity(1 + u₁y₂³)³ · f₂ = g + h₂ · qwith explicitg,q.
Two generic tools for orders at a point #
The Leibniz rule for a partial derivative: g ∈ N^{a+1} implies ∂ᵢ g ∈ N^a.
Evaluation at a point kills every polynomial of an ideal whose generators vanish there.
Example 82, first part, in ℚ[x, y, z] #
x ∈ 𝔪₀ = (x, y, z). HironakaExamples/MaximalContact/Example14.lean, which names the
variables x₁, x₂, y, uses this statement and the next two for its own generators.
ord₀ (xy − zⁿ) = 2.
x, x − y ∈ MC(I) = D(I): H = (x = 0) and H_g = (x − y = 0) are of maximal contact.
The locus of order 2, V(D(I)), has the origin as its only ℚ-point.
(zⁿ) has order ≥ n at every ℚ-point of the line z = 0: the one-dimensional locus of
order n on H.
I has order < 2 at (0, 1, 0) ∈ H, a point of the line of order n on H.
I|_{H_g} = (x² − zⁿ) for H_g = (x − y = 0).
D(x² − zⁿ) in ℚ[x, z]: the partial derivatives of the generator.
D(x² − zⁿ) = (x, z^{n−1}) in ℚ[x, z].
The locus of order 2 of I|_{H_g} has the origin as its only ℚ-point.
Example 82, second part, over a nontrivial commutative ℚ-algebra #
x ∈ 𝔪₀ = (x, y, z) in R[x, y, z].
y ∈ 𝔪₀ in R[x, y, z].
z ∈ 𝔪₀ in R[x, y, z].
A nontrivial ℚ-algebra has characteristic zero.
ord₀ f = 3 and ord₀ h = 1, for f = x³ + xy⁵ + z⁴ and h = x + u₁xy³ + u₂y⁴ + u₃z².
h ∈ D²(f) = MC(I). Over the ℚ-algebra R the partial derivatives are ℚ-derivations, and
6x = ∂ₓ∂ₓ f, 5y⁴ = ∂_y∂ₓ f, 20xy³ = ∂_y∂_y f, 12z² = ∂_z∂_z f lie in D²(f); dividing by
the nonzero rationals gives the four monomials of h.
The first blow-up (Kollár's second row): σ f = y₁³·f₁ and σ h = y₁·h₁.
The second blow-up (Kollár's third row): σ f₁ = y₂³·f₂ and σ h₁ = y₂·h₂.
ord₀ f₂ = 2: after two blow-ups the transform of I has order 2, as Kollár says.
The elimination identity: (1 + u₁y₂³)³ · f₂ = g + h₂ · q with g free of x₂, so f₂
restricted to h₂ = 0 is g up to the unit (1 + u₁y₂³)³.
Kollár's I₂|_{H₂} ⊂ (y₂³, y₂²z₂⁴): g ∈ (y₂³, y₂²z₂⁴).
Kollár's "its restriction to the birational transform H₂ of H still has order 3": g has
order ≥ 3, and exactly 3 when u₂ is a unit (∂_y³ g at the origin is −6u₂).